Antalya Cebir Günleri IX

www.math.metu.edu.tr/~antalya/index2007.html 22–27 Mayıs 2007

Brian Hartley’in anısına

CONTENTS  AAD IX

Welcome to Antalya Algebra Days. This year’s meeting, the ninth meeting in the series, is dedicated to the great mathematician and good friend Brian Hartley. Each of us has a reason to attend this year’s conference. Some of us were Hartley’s students; some, his friends; some know him from his papers. When I first sent out invitations to attend and contribute to this conference, everybody accepted within two days. This is a sign of the of the friendship Hartley has kept in our hearts after thirteen years without him. I thank everybody who has come, especially those who have travelled long distances. I understand and appreciate your support for this conference. I also would like to thank heartily my colleagues and friends Ali Nesin, Alev Topuzoğlu and Sinan Sertöz for allowing me to invite more speakers in group theory this year. I would like to thank the indispensible couple of organizers Ayşe Berkman and David Pierce: without their constant support my job would have been difficult. We are indebted to Otto Kegel, Alexandre Zalesskii and Victor Mazurov for their scientific guidence and help in the organization of this conference. They were with us whenever we needed help. Finally we are deeply grateful to all the scientific committee members for their support. Our special thanks go to Tamer Koç and Şükran Demir of Tivrona Tours, who have met all of our requests smilingly. I hope you enjoy the conference.

Mahmut Kuzucuoğlu

Contents

1 Hour talks 4

1. Alexandre Borovik ...........4

2. Ken Brown .................. 4

3. Roger Bryant ................ 5

4. Rostislav Grigorchuk .........5

5. J. I. Hall .....................6

6. Otto Kegel ...................6

7. Mahmut Kuzucuoğlu .........7

8. Alexander A. Lashkhi ........8

9. Felix Leinen ................. 9

10. James D. Lewis ............11 11. A. Yu. Olshanskii ..........11 12. Donald S. Passman ........ 12 13. Piotr Pragacz ..............13 14. Peter Roquette ............ 14 15. Peter Rowley ..............15 16. Aner Shalev ............... 16 17. Pavel Shumyatsky ......... 16 18. Howard Smith .............16 19. Patrick Smith ............. 17 20. Henning Stichtenoth .......17 21. Ralph Stöhr ............... 18 22. Simon Thomas ............ 19 23. Efim Zelmanov ............ 20

May 22–27, 2007  CONTENTS

2 Half-hour talks 20

1. Bernhard Amberg .......... 20

2. A. O. Asar ..................21

3. Houshang Behravesh ........23

4. Ya. I. Diasamidze ...........25

5. Gülin Ercan ................ 26

6. V. M. Levchuk ..............26

7. Natalia Makarenko ..........27

8. A. I. Papistas ...............29

9. Stephen J Pride ............ 29

10. G. S. Suleymanova .........30

11. Ergün Yalçın .............. 31

12. Şükrü Yalçınkaya .......... 32

13. A. E. Zalesski ..............33

3 Contributed talks 34

1. Rafail Alizade .............. 34

2. Murat Altunbulak .......... 35

3. Ahmet Arıkan .............. 36

4. Şerban A. Basarab ..........36

5. Veronica Crispin Quiñonez ..42

6. Ayça Çeşmelioğlu ...........43

7. Adrien Deloro .............. 44

8. A. S. Dzhumadil’daev .......46

9. Aslı Güçlükan .............. 47

10. Marianne Johnson .........47

11. Semra Öztürk Kaptanoğlu .48 12. Bilal Khan ................ 49 13. Elisabetta Pastori ......... 50 14. Dilek Pusat-Yılmaz ........ 50 15. Francesco Russo ........... 51 16. Mahmood Sohrabi .........54 17. Vladimir Tolstykh ......... 54 18. Anna Torstensson ......... 55 19. Bahattin Yıldız ............ 56

4 Posters 57

1. Haci Civciv .................57 2. Bounabi Daoud .............58 3. Yılmaz Mehmet Demirci .... 60 4. Betul Gezer .................61 5. Ghodrat Ghaffarzadeh ...... 62 6. İlker İnam .................. 62 7. Fatih Koyuncu ..............63 8. Carmela Sica ............... 64 9. Ahmet Tekcan ..............66 10. S. Eylem Toksoy ...........67 11. Maria Tota ................ 68 12. Nadir Trabelsi .............69 13. E. Türkmen ............... 69 14. Ramazan Türkmen ........ 71

5 Participants 72

Committees, support 75

Timetable 76

1 HOUR TALKS  AAD IX

1 Hour talks

Black box groups and groups of finite Morley rank

Alexandre Borovik

My talk is a brief survey of strange and still mysterious connections between the theory of groups of finite Morley rank [3] and a chapter of computational group theory known as black box recognition of finite groups [1, 2, 6].

I will also briefly mention a new development: theory of asymptotic classes and measurable theories [4, 5] which suggest a new bridge between the theory of finite simple groups and model theory of groups.

References

[1] L. Babai, W. M. Kantor, P. P. Palfy, A. Seress, Black-box recognition of finite simple groups of Lie type by statistics of element orders, J. Group Theory 5, 2002, 383Ű401.

[2] A. V. Borovik, Centralisers of involutions in black box groups, Contemp. Math., 298, Amer. Math. Soc., Providence, RI, 2002, 7–20.

[3] A. V. Borovik, J. Burdges, and G. Cherlin, Involutions in groups of finite Morley rank of degenerate type, to appear in Selecta Math.

[4] R. Elwes and D. Macpherson, A survey of asymptotic classes and measurable structures, preprint, available at http://www.amsta.leeds.ac.uk/pure/staff/macpherson/hdm36.pdf.

[5] H.D. Macpherson and C. Steinhorn, One-dimensional asymptotic classes of finite structures, to appear in Trans. Amer. Math. Soc.

[6] Ş. Yalçınkaya, Black box recognition of finite groups and related group theoretic constructions, in preparation.

University of Manchester

mailto:borovik@manchester.ac.uk

http://www.maths.manchester.ac.uk/~avb/

Noetherian Hopf algberas

Ken Brown

Following the discovery of quantum groups in the 1980s and the exploration of many Hopf algebras related to quantum groups in the years following, a very large menagerie of noetherian Hopf algebras had become available for study by the mid-1990s. I will review some of what is known about noetherian Hopf algebras, and explain some open questions about them.

University of Glasgow, United Kingdom

May 22–27, 2007  1 HOUR TALKS

mailto:kab@maths.gla.ac.uk http://www.maths.gla.ac.uk/~kab/

Necklaces and free Lie algebras

Roger Bryant

I shall first talk about the well-known formula (Witt’s formula) that gives the number of primitive necklaces of length n and the dimension of the nth homogeneous component of a free Lie algebra. I shall then describe some recent results concerned with the module structure of a free Lie algebra under the action of a group. Let G be a group, K a field and V a finite-dimensional KG-module. Let L(V) be the free Lie algebra over K that has V as a subspace and every basis of V as a free generating set. The action of each element of G on V extends to a Lie algebra automorphism of L(V). Thus L(V) becomes a KG-module, and each homogeneous component L^{n}(V) is a KG-submodule called the nth Lie power of V. I shall describe joint work with Manfred Schocker that gives a general decomposition theorem for Lie powers in terms of modules denoted by B_{n}. I shall also report on recent work with Marianne Johnson that clarifies the structure of these modules B_{n}.

University of Manchester

mailto:roger.bryant@manchester.ac.uk

On Brian Hartley problem on maximal subgroups in branch groups.

Rostislav Grigorchuk

I will speak about the problem posted by Brian Hartley on existence of maximal subgroups in certain branch self-similar groups.

The solution of the problem was found by Ekaterina Pervova for the main case, that interested Bryan Hartley (including the first example of a group of intermediate growth, and Gupta-Sidki p-groups). The topic was farther developed in 2002 in joint work of J.S.Wilson and speaker. Namely the result of Pervova was generalized to all groups which are abstractly commensurable with the 3-generated 2-group G of intermediate growth. Unexpected consequence of this (and the developed technique) is the subgroup separability of the group G (quite rare property in group theory) and the fact that any infinite finitely generated subgroup of G is abstractly commensurable to G. The overview of all this results, some new observation and a list of open questions will be presented in the talk.

Texas A&M University

mailto:grigorch@math.tamu.edu http://www.math.tamu.edu/~grigorch

Locally finite simple Moufang loops

J. I. Hall

A Moufang loop is a binary system close to a group but satifying only a weak form of the associative law. Doro and Glauberman [1, 2] observed that there is a direct connection between simple Moufang loops and simple groups with triality. Using this correspondence, Liebeck [5] proved that nonassociative finite simple Moufang loops arise from split octonian algebras over finite fields. The techniques used in the study of locally finite simple groups (say, [3, 4]) can then be used to prove that Liebeck’s result remains true with “locally finite” in place of “finite.”

References

[1] S. Doro, Simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 83 (1978), 377–392.

[2] G. Glauberman, On loops of odd order, I, J. Algebra 1 (1964), 374–396; II, J. Algebra 8 (1968), 393–414.

[3] J.I. Hall, Locally finite simple groups of finitary linear transformations, in: Finite and locally finite groups (Istanbul, 1994), eds.: B. Hartley, G.M. Seitz, A.V. Borovik, and R.M. Bryant, 147–188, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 471, Kluwer Acad. Publ., Dordrecht, 1995.

[4] J.I. Hall, Periodic simple groups of finitary linear transformations. Ann. of Math. (2) 163 (2006), 445–498.

[5] M.W. Liebeck, The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 102 (1987), 33–47.

Michigan State University, East Lansing, MI 48824 USA

mailto:jhall@math.msu.edu

http://www.math.msu.edu/~jhall/

Locally linear simple groups

Otto Kegel

The group algebra F G and the group G are two quite different structures (although closely related). Questions like:

Is there an element h not equal to 1 of an extension H of G such that the element 1 − h belongs to the twosided Ideal (F H)I where I is a given ideal of F G? mix the two structures and ask for a suitable larger group H (so that the ideals of F G are induced by the ideals of H. (Brian Hartley was involved in the first paper treating this type of question, Passman: The algebraic structure of group rings (1977), p. 387).

Freiburg, Germany

mailto:Otto.H.Kegel@t-online.de

Barely Transitive Groups

Mahmut Kuzucuoğlu

A group G has a barely transitive representation if G acts on an infinite set Ω transitively and faithfully and every orbit of every proper subgroup is finite. A group is called a barely transitive group if it has a barely transitive representation. One can show easily that a group G is barely transitive if and only if G has a subgroup H such that |G : H| is infinite, ⋂_{x∈G} H^{x} = {1} and for every proper subgroup K of G, |K : K ∩ H| < ∞: here H denotes the stabilizer of a point.

The concept of a barely transitive permutation group was introduced by Hartley in [1], in the context of research on Heineken–Mohamed groups. There, it was pointed out that, an infinite group is barely transitive in its regular permutation representation if all proper subgroups of that group are finite, and so the study of barely transitive groups can be thought of as a natural generalization of the research arising from the Schmidt problem. Besides the trivial examples of groups of type C_{p^{∞}}, there are examples of non-abelian imperfect locally finite barely transitive groups given by Hartley. On the other hand infinite quasi-finite groups in their right regular representations are barely transitive. In particular the p-groups constructed by Olshanskii are the examples of simple, periodic, finitely generated barely transitive groups.

The structure of imperfect barely transitive groups is fairly well understood. In fact they are locally finite. The situation with perfect barely transitive groups is quite different even in the class of locally finite groups. Does there exist a locally finite barely transitive group that coincide with its derived subgroup? This question, posed in [1] see also [3, Question 15.20] is still not answered. Basic properties of locally finite barely transitive groups are studied in [2] We will give a survey on barely transitive groups.

References

[1] B. Hartley, The normalizer condition and barely-transitive permutation groups,ą Algebra Logika, 13, No. 5, 539-602 (1974).

[2] M. Kuzucuoğlu, Barely transitive permutation groups, Arch. Math. 55, 521-532 (1990) .

[3] Unsolved Problems in Group Theory The Kourovka Notebook, 15th edition, Novosibirsk 2002;

Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey

mailto:matmah@metu.edu.tr

http://www.math.metu.edu.tr/people/mahmut_kuzucuoglu.shtml

Geometry of classical groups over rings

Alexander A. Lashkhi

Classical Groups can be defined when the scalars only form a ring (commutative in most cases). The methods used in their study when the scalars form a field can be slightly extended to local rings; but for more general rings, they don’t apply any more, and new ideas were needed. They were brilliantly provided by O. T. O’Meara, J. Tits and F. Veldkamp. More recently, unexpected connection of Classical Groups with K-theory has been discovered (A. Hahn, D. James, U. Brehm, A. Bak and others) [1]–[5].

One of the main problems in this field of study is to translate the specific maps from the geometrical point of view (perspectivities, collineations, harmonic maps) into algebraic language (representation by the [semi]-linear isomorphisms) —the Fundamental Theorems of Geometric Algebra. For different geometries (affine, projective, symplectic, orthogonal, unitary) and different rings we will consider this and other similar problems.

A. Defining the perceptivity (in the classical way) it’s essential that the geometry is the lattice with complements. This property is false for rings. Consequently, it’s necessary to change the definition of perspectivities. Note that none single ring versions of the theorem about perspective maps is not known. With this goal should be important the definition: submodule A ⊆ X is D-complement to the submodule B ⊆ X if A ∩ B = 0 and PG(K, A ⊕ B) ≅ PG(K, X). Taking into consideration this one for the left Ore domains desirable results could be obtained.

B. For the projective line harmonic maps and existence of K. von Staudt’s theorem was considered by various of authors. Positive answers were given for some specific rings. However for the (commutative) principal ideal domains the theorem of K. von Staudt is invalid. Defining the projective space as the set of all free one-dimensional submodules for some general noncommutative rings in condition for non-bijective harmonic maps is given and the complete analog of von Staudt’s theorem is obtained.

C. For free modules over the rings with invariant basis property (IBN-ring) the collineations and lattice isomorphisms of affine spaces and affine geometries was studied and thus the fundamental Theorem of Affine Geometry was proved.

References

[1] F. Buekenhout. (eds.) Handbook of Incidence geometry. Buildings and Foundations. Elsevier, Amsteram–Lausanne–New York–Oxford–Shanno–Tokyo, 1995.

[2] U. Brehm. Representation of Sum-preserving mappings between submodule lattices. Manuscript. Hawaii, 1984, see also J. Math Sci. plenum Publ. CO. New York. 2006 (in printing).

[3] J. Dieudonné. La Geometrie des Groups Classiques. Springer, Berlin - Heidelberg–New York, 1963.

[4] A. J. Hahn, O. Timothy O’Meara. The Classical Groups and K-Theory. Springer-Verlag, Berlin–Heidelberg–New York–London–Paris–Tokyo, 1989.

[5] J. Tits. Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Math. v. 386, Springer, Berlin–Heidelberg–New York, 1974.

Batumi State University

mailto:lashkhi@spider.gtu.edu.ge

Asymptotic representation theory of finite groups: Evolution of a representation theory for locally finite groups?

Felix Leinen

In the early 1990s, A. E. Zalesskii discovered a neat and natural way to generalize notions from the representation theory of finite groups to locally finite groups – see his survey [15] in the notes of the Istanbul Conference.

For a locally finite group G, the role of representations is taken by so-called inductive systems, that is, by families of sets Φ_{F} of irreducible representations of the finite subgroups F of G with the compatability condition that Φ_{F_{1}} be precisely the set of irreducible constituents of the restrictions to F_{1} of the representations in Φ_{F_{2}} whenever F_{1} ⩽ F_{2}. Clearly, inductive systems reflect the asymptotic behaviour of the representations of the finite subgroups of G. And A. E. Zalesskii proved that the ideals in the complex group algebra of G are in 1-1-correspondence with the inductive systems of complex representations of the finite subgroups of G.

Further, for our locally finite group G, the role of characters is taken by normalized positive definite class functions ℂG ⟶ ℂ, which are also studied in the theory of C^{⋆}-algebras. Due to a theorem by A. M. Vershik and S. V. Kerov [14], convexly indecomposable such functions (the “irreducible characters” of G) arise as direct limits of irreducible normalized characters of the finite subgroups of G. Every normalized positive definite class function gives rise to an ideal in ℂG; the converse fails.

In principle, the above connections open the possibility to generalize classical representation theory to locally finite groups. But in practice, it is very hard to understand the immersion of the finite subgroups in a given locally finite group G and their effect on the branching rules for irreducible characters. At least, this information is available when the group G is built up from a well-understood family of finite subgroups in a systematic uniform way. This happens for example with many simple locally finite groups.

First results about positive definite class functions for simple locally finite groups were obtained by E. Thoma [13] and H.-L. Skudlarek [12] more than 30 years ago. Around that time, K. Bonvallet, B. Hartley, D. S. Passman, and M. K. Smith [1] observed that the complex group algebra of P. Hall’s universal locally finite group ULF admits only the three unavoidable ideals.

In the early 1990s, A. E. Zalesskii [16] and B. Hartley [5, 4, 3] pursued the topic further. This was the starting point for a series of papers [11, 10, 9, 8, 7, 6] by O. Puglisi and F. Leinen with the aim to settle the situation completely for simple locally finite groups. Our work relies on S. Delcroix’s distinction of simple locally finite groups into various classes [2]. At present, open questions remain for so-called groups of 1-type (these groups are built up from “tame” Kegel covers with alternating quotients), but also for finitary isometry groups over finite fields.

References

[1] K. Bonvallet − B. Hartley − D. S. Passman − M. K. Smith, Group rings with simple augmentation ideals, Proc. Amer. Math. Soc. 56 (1976), 79–82.

[2] S. Delcroix, Non-finitary locally finite simple groups, Ph. D. thesis, University of Gent, Belgium 2000.

[3] B. Hartley − A. E. Zalesskii, Confined subgroups of simple locally finite groups and ideals of their group rings, J. London Math. Soc. (2) 55 (1997) 210–230.

[4] B. Hartley − A. E. Zalesskii, The ideal lattice of the complex group ring of finitary special and general linear groups over finite fields, Math. Proc. Camb. Phil. Soc. 116 (1994) 7–25.

[5] B. Hartley − A. E. Zalesskii, On simple periodic linear groups: Dense subgroups, permutation representations and induced modules, Israel J. Math. 82 (1993) 299–327.

[6] F. Leinen − O. Puglisi, Positive definite functions of finitary isometry groups over fields of odd characteristic, J. Pure Appl. Algebra 208 (2007), 1003–1021.

[7] F. Leinen − O. Puglisi, Some results concerning simple locally finite groups of 1-type, J. Algebra 287 (2005), 32–51.

[8] F. Leinen − O. Puglisi, Positive definite functions of diagonal limits of finite alternating groups, J. London Math. Soc. 70 (2004), 678–690.

[9] F. Leinen − O. Puglisi, Diagonal limits of finite alternating groups: confined subgroups, ideals, and positive definite functions, Illinois J. Math. 47 (2003), 345–360.

[10] F. Leinen − O. Puglisi, Ideals in group algebras of simple locally finite groups of 1-type, Pacific J. Math. 207 (2002), 433–445.

[11] F. Leinen − O. Puglisi, Confined subgroups in periodic simple finitary linear groups, Israel J. Math. 128 (2002), 285–324.

[12] H.-L. Skudlarek, Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann. 223 (1976) 213–231.

[13] E. Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40–61.

[14] A. M. Vershik − S. V. Kerov, Locally semisimple algebras. Combinatorial theory and the K_{0}-functor, Sovrem. Probl. Mat. Nov. Dost. 26 (1985), 3–56 (Russian).

[15] A. E. Zalesskii, Group rings of simple locally finite groups, pp. 219–246 in: B. Hartley − G. M. Seitz − A. V. Borovik − R. M. Bryant (eds.), Finite and locally finite groups, NATO ASI Series C 471, Kluwer Academic Publishers, Dordrecht - Boston - London 1995.

[16] A. E. Zalesskii, Group rings of inductive limits of alternating groups, Leningrad Math. J. 2 (1991), 1287–1303.

Institute of Mathematics, University of Mainz, D-55099 Mainz, Germany.

mailto:Leinen@uni-mainz.de

http://www.mathematik.uni-mainz.de/~leinen

Variants of the Cycle Class Map

James D. Lewis

The celebrated Hodge conjecture in its classical form, is a statement about the image of a particular cycle class map from an algebraic cycle group on a nonsingular complex projective variety to singular cohomology. We give a precise statement of this conjecture from an algebraic point of view. We then discuss a variant of this conjecture, called the Hodge-Deligne conjecture. Next, and motivated in part by constructions in algebraic K-theory, we consider twisted variants of this cycle class map construction, and explain the advantages of this construction over the original cycle class map.

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

mailto:lewisjd@ualberta.ca

http://www.math.ualberta.ca/Lewis_JD.html

Quotients of large Lie Groups and Lie Algebras

A. Yu. Olshanskii

We present recent results obtained jointly with D. I. Osin and Yu. A. Bahturin.

Recall that a group G is large if some finite index subgroup of G admits a surjective homomorphism onto a non-cyclic free group. Given a subset S of a group G, we denote by ⟨S⟨^{G} the normal closure of S in G. In the recent paper [1], Lackenby proved the following theorem.

Let G be a large group, H is a normal subgroup of G of finite index admitting a surjective homomorphism onto a non-cyclic free group, g_{1}, …, g_{k} a collection of elements of H. Then the quotient group G/⟨g_{1}^{n}, …, g_{k}^{n}⟩^{G} is large for all but finitely many n ∈ ℕ.

We provide a shorter proof of this theorem based just on the Schreier formula and Baumslag–Pride theorem of 1978 on the largeness of a group given by n ⩾ 2 generators and n − 2 relations.

Thus our approach is very elementary. However it leads to new constructions. We obtain new examples of finitely generated infinite torsion groups. In particular, our examples enjoys in addition the following properties: the orders of elements are square free; every nilpotent section (i.e., a factor-group of a subgroup) is abelian; every section is residually finite.

Also an analog of the theorem and other results are obtained for restricted Lie algebras over fields of prime characteristic.

References

[1] M. Lackenby. Adding high powered relations to large groups. Prep., 2005; arXiv: mathGR/0512356.

[2] A. Yu. Olshanskii and D. V. Osin. Large groups and their periodic quotients. To appear in Proc. of the AMS. 7 pp.

[3] Yu. A. Bahturin and A. Yu. Olshanskii. Large restricted Lie algebras. To appear in Journal of Algebra, 19 pp.

Vanderbilt University and Moscow State University

mailto:alexander.olshanskiy@vanderbilt.edu

Invariant ideals in abelian group algebras

Donald S. Passman

If H is a nonidentity group, then the group algebra K[H] always has at least three distinct ideals, namely 0, the augmentation ideal ωK[H], and K[H] itself. Thus it is natural to ask if groups exist for which the augmentation ideal is the unique nontrivial ideal. In such cases, we say that ωK[H] is simple. Certainly H must be a simple group for this to occur and, since the finite situation is easy enough to describe, we might as well assume that H is infinite simple. The first such examples, namely algebraically closed groups and universal groups, were offered in 1976. From this, it appeared that such groups would be quite exotic and rare. But A. E. Zalesskiĭ has shown that, for locally finite groups, this phenomenon is really the norm. Indeed, for all locally finite infinite simple groups, the characteristic 0 group algebras K[H] tend to have very few ideals.

While some work still remains to be done on the simple group case, it nevertheless makes sense to move on to the second stage of this program. As suggested by B. Hartley and A. E. Zalesskiĭ, we should next consider certain abelian-by-(quasi-simple) groups. Specifically, these are the locally finite groups H having a minimal normal abelian subgroup V, with H/V infinite simple (or perhaps just close to being simple). Note that G = H/V acts as automorphisms on V, and hence on the group algebra K[V]. Furthermore, if I is any nonzero ideal of K[H], then it is easy to see that I ∩ K[V] is a nonzero G-stable ideal of K[V]. Thus, for the most part, this second stage is concerned with classifying the G-stable ideals of K[V]. Even in concrete cases, this turns out to be a surprisingly difficult task. Fortunately, there has been some progress on this problem, and the goal of this talk is to survey these results. For the most part, the methods used here are quite different from the usual group ring techniques.

University of Wisconsin–Madison

mailto:passman@math.wisc.edu

http://www.math.wisc.edu/~passman

Thom polynomials and their positivity

Piotr Pragacz

The global behavior of singularities is governed by their Thom polynomials [16], [1], [2], [4], [5]. Knowing the Thom polynomial of a singularity, one can compute the cohomology class represented by the points of a map, having this singularity. For a more detailed account, we refer to [8] in the booklet of Antalya Algebra Days 2006.

Computation of the Thom polynomial of a singularity is a difficult task, even for “easiest” singularities. We are especially interested in the computation of the Thom polynomials containing as a parameter the difference of dimensions between the target and source of a map [15].

We widely use the Schur function techniques [6].

The talk has two goals:

(i) New, explicit computations of Thom polynomials for the singularities I_{2,2} and A_{i} via their Schur function expansions [9], [10], [11], [12], [7]. These computations use the “method of restriction equations” of Rimanyi et al. for computing Thom polynomials [15], [2].

(ii) Investigation of the structure of the Thom polynomials expanded in the Schur function basis. Combining the approach to Thom polynomials via classifying spaces of singularities [5] with the Fulton-Lazarsfeld theory of cone classes and positive polynomials for ample vector bundles [3], we show – with Andrzej Weber – that the coefficients of the Schur function expansions of the Thom polynomials of nontrivial stable singularities are nonnegative with positive sum [13]. Then we present a generalization of this result to invariant cones in representations [14].

Research supported by a KBN grant, by TÜBİTAK (during the stay at METU in Ankara), and by the Humboldt Stiftung (during the stay at the MPIM in Bonn).

References

[1] V. Arnold, V. Vasilev, V. Goryunov, O. Lyashko: Singularities. Local and global theory, Enc. Math. Sci. vol. 6 (Dynamical Systems VI), Springer, 1993.

[2] L. Feher, R. Rimanyi, Calculation of Thom polynomials and other cohomological obstructions for group actions, in: “Real and complex singularities (San Carlos 2002)” (T. Gaffney and M. Ruas eds.), Contemporary Math. 354, (2004), 69–93.

[3] W. Fulton, R. Lazarsfeld, Positive polynomials for ample vector bundles, Ann. of Math. 118 (1983), 35–60.

[4] M.E. Kazarian, Characteristic classes of singularity theory, in: “The Arnold-Gelfand mathematical seminars: Geometry and singularity theory” (1997), 325–340.

[5] M.E. Kazarian, Classifying spaces of singularities and Thom polynomials, in: “New developments in singularity theory”, NATO Sci. Ser. II Math. Phys. Chem., 21, Kluwer Acad. Publ., Dordrecht (2001), 117–134.

[6] A. Lascoux, Symmetric functions and combinatorial operators on polynomials, CBMS/AMS Lectures Notes 99, Providence (2003).

[7] O. Ozturk, On Thom polynomials for A_{4}(−) via Schur functions, Preprint, IM-PAN Warszawa 2006 (670); to appear in Serdica Math. J. 1 (2007).

[8] O. Ozturk, Thom polynomials via Schur functions, in the booklet of “Antalya Algebra Days 2006”, 39–40.

[9] P. Pragacz, Thom polynomials and Schur functions I, math.AG/0509234.

[10] P. Pragacz, Thom polynomials and Schur functions: the singularities I_{2,2}(−), Preprint MPIM Bonn 2006 (83), to appear in Ann. Inst. Fourier 57 (2007).

[11] P. Pragacz, Thom polynomials and Schur functions: towards the singularities A_{i}, Preprint MPIM Bonn 2006 (139).

[12] P. Pragacz, Thom polynomials and Schur functions: the singularities A_{3}(−), in preparation.

[13] P. Pragacz, A. Weber, Positivity of Schur function expansions of Thom polynomials, Preprint, math.AG/0605308 and MPIM Bonn 2006 (60).

[14] P. Pragacz, A. Weber, Positivity of Thom polynomials revisited, in preparation.

[15] R. Rimanyi, Thom polynomials, symmetries and incidences of singularities, Inv. Math. 143 (2001), 499–521.

[16] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier 6 (1955–56), 43–87.

Institute of Mathematics of Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

mailto:P.Pragacz@impan.gov.pl

From FLT to Finite Groups. The remarkable career of Otto Grün.

Peter Roquette

Students who start to learn group theory will soon be confronted with the theorems of Grün (or some generalization). Immediately after publication in the mid 1930s [1] these theorems found their way into group theory textbooks [2], and they turned out to be of fundamental importance in connection with the classical Sylow theorems. But little is known about the mathematician whose name is connected with those theorems.

In my talk I shall report on the remarkable mathematical career of Otto Grün. The talk will be based largely on the letters which were exchanged between Otto Grün and Helmut Hasse. I found 50 such letters in the archives of the University of Göttingen, from 1932 to 1972. They document the fascinating story of a mathematician, quite rare in our time, who was completely self-educated, without having attended university, and nevertheless succeeded, starting at age 44, to give important contributions to mathematics, in particular to group theory. Grün’s first papers are concerned with class field theory around Fermat’s Last Theorem (FLT). It was Hasse who pointed out to him the group theoretical problems in this connection, which then turned Grün’s main interest to group theory.

We shall mention several documents about Grün which were found after publication of our paper [3].

References

[1] Grün, O. Beiträge zur Gruppentheorie I. J. Reine Angew. Math. 174 (1935) 1-14 [2] Zassenhaus, H. Lehrbuch der Gruppentheorie. Teubner Leipzig (1937)

[3] Roquette, P. From FLT to finite groups. Jber. DMV 107 (2005) 117-154

Heidelberg University

mailto:roquette@uni-hd.de

http://www.roquette.uni-hd.de

Coxeter Length

Peter Rowley

The notion of the length of an element in a Coxeter group frequently plays an important role in understanding various aspects of the Coxeter group. Indeed, there are many results concerning Coxeter groups which are proved by induction on the length of elements. In this talk we introduce a length function on subsets of a Coxeter group (which generalizes the length function on elements) and investigate some of its properties.

University of Manchester, UK

mailto:peter.j.rowley@manchester.ac.uk

Commutator maps, measure preservation and character methods

Aner Shalev

Given a group G we study the commutator map f: G × G → G where f(x, y) = x^{−1}y^{−1}xy. When G is finite and has very few representations in some sense, we show that the map f is almost measure preserving. In particular almost all elements of such groups are commutators.

The main applications of this technique are for finite simple groups. We prove results towards a longstanding conjecture of Ore (which is still open), and confirm a conjecture of Guralnick and Pak on the Product Replacement Algorithm. This is joint work with Shelly Garion.

I will also mention recent results with Liebeck and O’Brien on Ore’s conjecture, and with Larsen on more general word maps and Waring type problems.

Our methods combine representation theory with probabilistic arguments. In particular the so called Witten zeta function encoding character degrees plays a key role in the proofs.

Hebrew University

mailto:shalev@math.huji.ac.il

Centralizers in Locally Finite Groups

Pavel Shumyatsky

It is well-known that centralizers play fundamental role in the theory of locally finite groups. The main theme of our talk will be the following question.

Let G be a locally finite group admitting an automorphism ϕ of finite order such that the centralizer C_{G}(ϕ) satisfies certain finiteness conditions. What impact does this have on the structure of the group G?

Sometimes the impact is quite strong and we will give a survey of results illustrating this phenomena. In particular, we concentrate on results where G is shown to have a large nilpotent or soluble subgroup.

Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil

mailto:pavel@mat.unb.br

Countably recognizable classes of groups

Howard Smith

A subgroup-closed class C of groups is countably recognizable if, whenever a group G has all of its countable subgroups in C, then G too belongs to C. (For a class C of subgroups that is not subgroup-closed there is an appropriately amended version of the definition of countable recognizability.) We will trace briefly the development of a theory of countable recognizability, highlighting several of the main results along the way, before presenting some new results on classes related to (local) solubility/nilpotency and finiteness of MalâĂŹcev rank.

Department of Mathematics, Bucknell University, Pennsylvania 17837, USA

mailto:howsmith@bucknell.edu

Modules with Ascending Chain Condition on

Submodules with a Bounded Number of Generators

Patrick Smith

Given a positive integer n, a module satisfies n-acc provided every ascending chain condition of n-generated submodules terminates. This notion seems to have arisen first in the theory of Abelian groups and the Baumslag brothers generalized it to more general groups and to other algebraic systems. We shall consider it in the context of modules. A typical theorem is that if R is a commutative Noetherian one-dimensional domain then every direct sum of cyclic R-modules satisfies n-acc for every positive integer n. On the other hand if R is a (not necessarily commutative) right and left Noetherian ring then every direct product of copies of the right R-module R satisfies n-acc for every positive integer n.

University of Glasgow

mailto:pfs@maths.gla.ac.uk

Asymptotically good codes with transitive automorphism groups

Henning Stichtenoth

One of the main open problems in algebraic coding theory is the question whether the class of cyclic codes is asymptotically good. This means: Is there a sequence of cyclic codes (C_{i})_{i⩾0} over the finite field 𝔽_{q} whose parameters [n_{i}, k_{i}, d_{i}] (length, dimension, minimum distance) satisfy

n_{i} → ∞, lim inf_{i→∞}(k_{i}/n_{i}) > 0 and lim inf_{i→∞}(d_{i}/n_{i}) > 0.

Cyclic codes C ⊆ 𝔽_{q}^{n} have the property that their automorphism group is a transitive subgroup of the symmetric group S_{n}. Hence it is natural to consider the class of transitive codes. Our main result is:

Theorem 1 The class of codes over 𝔽_{q} (where q is a square) with transitive automorphism group is asymptotically good.

The proof of the theorem uses Goppa’s algebraic-geometric (AG) codes over function fields. The main ingredient is a new asymptotically good tower of function fields F_{0} ⊆ F_{1} ⊆ F_{2} ⊆ … over 𝔽_{q} with the property that all extensions 𝔽_{n}/𝔽_{0} are galois. With a careful choice of the divisors that define the corresponding AG codes one then obtains a transitive action of the Galois group on the codes.

Sabancı University

mailto:henning@sabanciuniv.edu

http://www.stat-math.uni-essen.de/~stichtenoth/henning/

On torsion in free central extensions of groups

Ralph Stöhr

Let G be a group given by a free presentation G = F/R and consider the quotient

F/[γ_{c}R, F]  (∗)

where γ_{c}R denotes the c-th term of the lower central series of R and c ⩾ 2. This quotient is a free central extension of F/γ_{c}R, which is in its turn an extension of G = F/R with free nilpotent kernel R/γ_{c}R. While the quotient F/γ_{c}R is always torsion-free, elements of finite order may occur in γ_{c}R/[γ_{c}R, F], the centre of (∗). In the case where c = 2 and R = F′, that is for the free centre-by-metabelian group F/[F′′, F], this was discovered by C.K. Gupta who proved in 1973 that the free centre-by metabelian group of rank n ⩾ 4 contains an elementary abelian 2-group of rank \binom{n}{4} in its centre. This was a major surprise at the time as it was the first example of a relatively free group that is given by a single multilinear commutator identity and is not torsion-free. In a pioneering paper of 1977 Yu.V. Kuz’min identified Gupta’s torsion subgroup with the fourth homology group of the free abelian group F/F′ reduced modulo 2, thus linking the problem to homology of groups. In the eighties there was a series of papers by Yu.V. Kuz’min focusing on the case c = 2, and there was a series of papers by the speaker, including a joint paper [1] with the late Brian Hartley, on the general case c ⩾ 2. One of the main achievements of all these papers was a precise identification of the torsion subgroup t(γ_{c}R/[γ_{c}R, F]) of (∗) in the case when c is a prime, say c = p, in terms of homology. Namely, if G has no elements of order p, then

t(γ_{p}R/[γ_{p}R, F]) ≅ H_{4}(G, ℤ_{p})

where ℤ_{p} = ℤ/pℤ regarded as a trivial G-module. In the early nineties there was a single addition to this body of results for the case c = 4: for G without elements of order 2 there is an isomorphism t(γ_{4}R/[γ_{4}R, F]) ≅ H_{6}(G, ℤ_{2}). This was the last word on the subject for a long time. The methods used in the eighties and nineties appeared to be exhausted. Very recently, new results by Bryant, Erdmann and Schocker on modular Lie representations made it possible to make further progress on the torsion subgroup of (∗). In joint work with Marianne Johnson we have been able to identify this torsion subgroup for c = 6 (for G without elements of order 2 and 3). The result was a big surprise to us. In my talk I will review the history of the problem, and then I will focus on these recent developments.

References

[1] Hartley, B. ; Stöhr, R. Homology of higher relation modules and torsion in free central extensions of groups. Proc. London Math. Soc. (3) 62 (1991), no. 2, 325–352.

University of Manchester

mailto:ralph.stohr@manchester.ac.uk

http://www.maths.manchester.ac.uk/~rs/

Simple Locally Finite Groups Of Diagonal Type

Simon Thomas

It is generally accepted that it is impossible to classify all countable simple locally finite groups, at least in the conventional sense of parametrizing the isomorphism types in some reasonable way. Instead, according to Hartley [5], we should look for well-behaved classes that can be classified, and try to increase the scope of these as much as possible. For example, the simple locally finite linear groups were explicitly classified in the early 1980s by several authors [1, 2, 4, 6]; and a little later, the simple locally finite groups of finitary linear transformations were explicitly classified by Hall [3]. In this talk, I will consider the question of whether a satisfactory classification exists for the class of simple locally finite groups of diagonal type; i.e. those simple locally finite groups which can be expressed diagonal limits of products of finite alternating groups.

References

[1] V. V. Belyaev, Locally finite Chevalley groups, in “Studies in Group Theory”, Urals Scientific Centre of the Academy of Sciences of USSR, Sverlovsk, 1984, pp. 39–50 (in Russian).

[2] A. V. Borovik, Periodic linear groups of odd characteristic, Dokl. Akad. Nauk. SSSR 266 (1982), 1289–1291.

[3] J.I. Hall, Periodic simple groups of finitary linear transformations, Ann. Math. 163 (2006), 445–498.

[4] B. Hartley and G. Shute, Monomorphisms and direct limits of finite groups of Lie type, Quart. J. Math (2) 35 (1984), 49–71.

2 HALF-HOUR TALKS  AAD IX

[5] B. Hartley, Simple locally finite groups, in “Finite and locally finite groups” (edited by B. Hartley, G. M. Seitz, A. V. Borovik and R. M. Bryant), Klumer Academic Publ., Dordrecht, 1995, pp. 1–44.

[6] S. Thomas, The classification of the simple periodic linear groups, Arch. Math. 41 (1983), 103–116.

Rutgers University mailto:sthomas@math.rutgers.edu http://www.math.rutgers.edu/~sthomas/

Lie algebras of infinite differential operators

Efim Zelmanov

I will discuss some new examples of nil algebras of finite Gelfand–Kirillov dimension and their possible connections with

(i) old problems, (ii) Grigorchuk groups, (iii) Physics. University of California, San Diego mailto:ezelmano@math.ucsd.edu

2 Half-hour talks

Triply factorized groups and nearrings

Bernhard Amberg

In the theory of factorized groups G = AB with subgroups A and B one often has to deal with groups of the following form G = AB = AM = BM, where the normal subgroup M of G satisfies the intersection property A∩M = A∩M = 1. Such triply factorized groups—even with nonabelian M—may be constructed by using (local) nearrings. A nearring R is an algebraic structure with two operations “ + ” and “ · ” such that R is a not necessary abelian group under the operation “ + ” and R is a semigroup under the operation “ · ” so that a one-sided distributive law holds. A nearring R with a unit element 1 is called local if the set of all invertible elements forms a subgroup of the additive group of R. If L is the subgroup of all non-invertible elements of the local nearring R, then the set 1 + L is a subgroup of the multiplicative group of R acting on L, such that the semidirect product of (1 + L) with L is a triply factorized group G, where M is isomorphic to L and A and B are both isomorphic to 1 + L.

May 22–27, 2007  2 HALF-HOUR TALKS

Conversely, it can be shown that for every triply factorized group G with A∩B = 1 there exists a sub-nearring of the nearring of all mappings from M into itself, such that the given triply factorized group can be obtained via this construction.

Mathematisches Institut der Universitaet Mainz, D-55099, Mainz, Germany

mailto:amberg@mathematik.uni-mainz.de

Subgroups of finitary permutation groups

A. O. Asar

This work continues the study of the subgroups of totally imprimitive groups of finitary permutations which was started in [1]. Some characterizations of non F C-subgroups and subnormal subgroups of stabilizers of these groups are obtained, and sufficient conditions are given for these groups to have proper subgroups having an infinite orbit.

Let G be a totally imprimitive subgroup of FSym(Ω), where Ω is infinite. In this work we continue the study of the following problem which was started in [1]: Can G contain a proper non F C-subgroup such that every orbit of every proper subgroup of it is finite or does every subgroup of G has an infinite orbit? It is well known that if every orbit of every proper subgroup of G is finite, then G is a minimal non F C-group (see [12, Theorem 1] or [3, Lemma 8.3D]). In [1], some characterizations of G were obtained based on the properties of point stabilizers. Also it was shown that if G satisfies the normalizer condition, then it is a p-group and G′ is a minimal non F C-group. (The converse of this result is also true.) Furthermore sufficient conditions were given for G to contain a proper subgroup having an infinite orbit.

In the present work the following properties of subgroups of G are obtained. The intersection of two distinct conjugates of a subgroup of G is always an F C-group and in particular, G is a minimal non F C-group if and only if any two proper subgroups of G generate a proper subgroup. A subnormal subgroup of a stabilizer which is either solvable or of finite exponent is contained in a normal subgroup of finite exponent. (This together with [1, Lemma 2.5] gives a complete characterization of the subnormal subgroups of a stabilizer). Now suppose that G is a p-group. Then the kernel of a permutation representation with respect to a non-trivial block is always non-trivial. Finally sufficient conditions are given for G, when it is a p-group, to contain a proper subgroup having an infinite orbit. Also it is shown that these conditions are satisfied by Wiegold’s example given in [12]. In fact it is easy to see that this example and its derived subgroup contain many proper subgroups having infinite orbits.

In view of the important reduction theorems given in [2] and [6] a complete determination of the types of the subgroups of G will settle the following well known problem: Does there exist a perfect locally finite minimal non F C-(p-group)?

For the definition and the important properties of totally imprimitive groups the reader is referred to [3, 8, 9] and [10]. Before stating the main results of this work we need to make a definition. Let G ⩽ Sym(Ω). For any non-trivial block Δ for G let

S_{G}(Δ) = {x ∈ G : supp(x) ⊆ Δ}.

It is easy to see that S_{G}(Δ) is a normal subgroup of G_{{Δ}} (see [1, Lemma 2.3(a)]). If S_{G}(Δ) is not contained in G_{{Π}} for any block Π ⊂ Δ, then Δ will be called a dominant block for G. Note that if S_{G}(Δ) is transitive on Δ then Δ is dominant. Let

Δ_{1} ⊂ Δ_{2} ⊂ ⋯ ⊂ Δ_{n} ⊂ …

be an ascending chain of non-trivial blocks for G whose union is Ω. If each Δ_{i} is dominant, then this chain is called a dominant chain for G. We also let Ker(Δ) denote the kernel of the natural permutation representation of G with respect to Δ (i.e, into FSym(Σ), where Σ = {x(Δ) : x ∈ G}).

The main results of this work are stated below.

Theorem 1 Let G be a totally imprimitive subgroup of FSym(Ω), where Ω is infinite. Then the following hold:

(i) a Let W be a subgroup of G and let g ∈ G \ N_{G}(W). Then W ∩ W^{g} is an F C-group. Thus if W is not an F C-group and if W ⩽ Y ⩽ G, then N_{G}(W) ⩽ N_{G}(Y);

(ii) G is a minimal non F C-group if and only if any two proper subgroups of G generate a proper subgroup of G.

Theorem 2 Let G be a totally imprimitive subgroup of FSym(Ω), where Ω is infinite. Let Δ be proper block for G and let S be a subnormal subgroup of G_{{Δ}}. Then ⟨S^{x} : x ∈ G⟩ has finite exponent if one of the following holds:

(i) S is solvable;

(ii) S has finite exponent.

Theorem 2 (i) generalizes [1, Theorem 1.1]. If G is a minimal non F C-group, then subnormality is not needed when exp(S) is finite by [1, Remark 1.10]. But subnormality is needed when S is only solvable since a point stabilizer always contains an abelian subgroup of infinite exponent. Note also that if G is a minimal non F C-group then it is a p-group, and so ⟨S^{x} : x ∈ G⟩ is nilpotent of finite exponent by [8, Theorem 2.4(ii)] or [3, Lemma 8.3D(ii)].

Theorem 3 Let G be a totally imprimitive p-subgroup of FSym(Ω), where Ω is infinite. Let Δ be a finite block for G and let M be the kernel of the representation of G with respect to Δ. Then Core_{G}(N_{G}(G_{{Δ}}) ≠ M. In particular if Δ is non-trivial then Core_{G}(G_{{Δ}}) ≠ 1.

(For more on the kernels the reader is referred to [7].)

Theorem 4 Let G be a totally imprimitive p-subgroup of FSym(Ω), where Ω is infinite. Suppose that G has a non-trivial block Δ such that any non-trivial block for G containing Δ is dominant. Then G contains a proper subgroup that has an infinite orbit.

Corollary 1 Let G be a totally imprimitive p-subgroup of FSym(Ω), where Ω is infinite. Suppose that G has an ascending chain of non-trivial blocks

Δ = Δ_{o} ⊂ Δ_{1} ⊂ ⋯ ⊂ Δ_{n} ⊂ …

whose union is equal to Ω such that for each n ⩾ 0 the following holds:

• G_{{Δ_{n+1}}} = N_{G}(G_{{Δ_{n}}}) and

• there exists a cycle y_{n+1} ∈ G_{{Δ_{n+1}}} such that G_{{Δ_{n+1}}} = ⟨y_{n+1}⟩G_{{Δ_{n}}}.

Then G contains a proper subgroup that has an infinite orbit.

It can be shown that the hypotheses of Theorem 4 Corollary 1 are satisfied by the example given in [12].

The notation and the definitions are standard and may be found in [3] and [11] (see also [1]).

References

[1] A.O. Asar, ‘On finitary permutation groups’, Turk. J. Mat. 30 (2006) 101-116.

[2] V. V. Belyaev, ’On the question of existence of minimal non F C-groups’, Siberian Math. J. 39 (1998) 1093-1095.

[3] J. D. Dixon and B. Moretimer, Permutation Groups (Springer-Verlag, New York, 1996)

[4] B. Hartley and M. Kuzucuoglu, ‘Non-simplicity of locally finite barely transitive groups’, Proc. Edinburg Math. Soc.(1997) 40, 483-490.

[5] M Kuzucuoglu, ‘Barely transitive permutation groups’, Arch. Math. 55(1990) 521-532

[6] F. Leinen, ‘A reduction theorem for perfect locally finite minimal non F C-groups’, Glasgow Math. J. 41 (1999) 81-83.

[7] Leinen, F., Puglisi, O.: Finitary representations and images of transitive finitary permutation groups. J. Algebra 222, 524-549 (1999).

[8] P. M.Neumann, ‘The lawlessness of groups of finitary permutations’, Arch. Math. 26 (1975) 561-566.

[9] P. M.Neumann, ‘The structure of finitary permutation groups’, Arch. Math. 27 (1976) 3-17.

[10] C.J.E Pinnock, ‘Irreducible and transitive locally-nilpotent by abelian groups’, Arch. Math. 74 (2000) 168-172.

[11] D. J. S. Robinson, A Course in the Theory of Groups, (Springer-Verlag, New York, Heidelberg, Berlin, 1980)

[12] J. Wiegold, ‘Groups of finitary permutations’, Arch. Math. 25 (1974) 466-469(1974).

Gazi University, Gazi Egitim Faculty, 06500 Teknikokullar, Ankara, Turkey

mailto:aliasar@gazi.edu.tr

Quasi-permutation representations of finite groups

Houshang Behravesh

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently , a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [2].

By a quasi-permutation matrix we mean a square matrix over the complex field ℂ with non-negative integral trace. Thus every permutation matrix over ℂ is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field ℚ, and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. See [1].

By a rational valued character we mean a character χ corresponding to a complex representation of G such that χ(g) ∈ ℚ for all g ∈ G. As the values of the character of a complex representation are algebraic numbers, a rational valued character is in fact integer valued. A quasi-permutation representation of G is then simply a complex representation of G whose character values are rational and non-negative. The module of such a representation will be called a quasi-permutation module. We will call a homomorphism from G to GL(n, ℚ) a rational representation of G and its corresponding character will be called a rational character of G. It is easy to see that

c(G) ⩽ q(G) ⩽ p(G)

where G is a finite group.

We will give algorithms for calculating p(G), q(G) and c(G) where G is a finite group and also when G has a unique minimal normal subgroup. Also we will give the old and new results on the quasi-permutation representation of finite groups. Finally we will state a question from Brian Hartley.

References

[1] J. M. Burns, B. Goldsmith, B. Hartley, R. Sandling. On quasi-permutation representations of finite groups. Glasgow Math.J. 36 (1994), 301-308.

[2] W. J. Wong. Linear groups analogous to permutation groups. J.Austral. Math.Soc. Ser.A 3 (1963), 180-184.

Department of Mathematics, University of Urmia, Urmia, Iran

mailto:hbehravesh@yahoo.com

Complete Semigroups of Binary Relations

Ya. I. Diasamidze

This is joint work with Sh. I. Makharadze.

Let X be an arbitrary nonempty set. D be complete X−semilattice of unions, i.e. it is some nonempty set of subsets from X which is closed with respect to the set-theoretic union of elements from D, and f be arbitrary mapping of the set X in the set D. To each mapping f we assign a binary relation α_{f} on the set X which satisfies conditions

α_{f} = ⋃_{x∈X}({x} × f(x)).

We denoted by B_{X}(D) the set of all such α_{f} (f : X ⟶ D). It is well-known that B_{X}(D) is subsemigroup of the semigroup B_{X} which is called a complete semigroup of binary relations, defined by an X−semilattice of unions D.

The report is dedicated to semigroups B_{X}(D). The issue on substantiation of researched semigroups and some of their properties is studied. Some early well-known classes of semigroups that proved to be complete semigroups of binary relations are pointed out. Divisibility conditions of binary relations are found out. Describtion of idempotent and regular elements; right units; exterior and irreducible elements; maximal subgroup of semigroups B_{X}(D) is given. Some of the properties of finite X−semilattice unions are studied.

Further, semigroups B_{X}(D) that are defined by semilattices, generated by sets of nonchainwise paires; X−chaines; elementary semilattices; nodal and generalized-elementary semilattices are studied.

References

[1] Diasamidze Ya., “Complete semigroups of binary relations", J. of Math. Sciences, Plenum Publ. Cor., New York, 117, No 4, 2003, 4271-4319.

[2] Diasamidze Ya., Makharadze Sh., “Complete semigroups of binary relations, defined by elementary and nodal X−semilattices of unions", Journal of Mathematical Sciences, Plenum Publ. Cor., New York , V. 111, No 1, 2002, 3171-3226.

[3] Diasamidze Ya., Sirabidze T., “Complete semigroups of binary relations, defined by 3 elemented chains", J. of Math. Sciences, Plenum Publ. Cor., New York, 117, No 4, 2003, 4320-4350.

[4] Diasamidze Ya., Makharadze Sh., Fartenadze G., Givradze O., “On finite X−semilattices of unions", Bull. Georg. Acad. Sci., 169, No 2. 2004, 263-266.

[5] Diasamidze Ya., Makharadze Sh., Fartenadze G., Givradze O., “Description of Σ(X, m) Class (m−finite number)", Bull. Georg. Acad. Sci., 169, No 3. 2004, 463-465.

Batumi State University

mailto:diasamidze_ya@mail.ru

On fixed point free action

Gülin Ercan

Let A be a finite group acting on a (solvable) group G fixed point freely. A long standing conjecture is that then the Fitting height of G is bounded by the longest chain of subgroups of A. Even though this conjecture is known to hold for large classes of groups A when (|G|, |A|) = 1, it is still not much known without this condition. In this presentation the above conjecture without the coprimeness condition will be discussed.

This is joint work with İsmail Ş. Güloğlu.

Middle East Technical University

mailto:ercan@metu.edu.tr

QF-schemes and quadratic forms over local rings

V. M. Levchuk

The results of [1, 5] receive a development. The solution of the construction’s task of a normal view for quadratic forms over a local principal ideal ring R = 2R with a QF -scheme of order 2 is shown. Recall, that group G = R^{∗}/R^{∗2} together with its particular map of subsets from G refers to QF -scheme or the quadratic form scheme of ring R. There are QF -schemes with abstract group G of exponent 2 are considered. We give a combinatorial representation for number of projective congruence quadrics classes of the projective space over R with a nilpotent maximal ideal. For the projective planes the quadric’s enumerations up to projective equivalence is given; we consider also the projective planes over rings with a non-principal maximal ideal. Some problems for QF -schemes are opened.

This research is supported by Russian fund of fundamental researches, grant No. 06-01-00824.

References

[1] Milnor J., Huismoller D. Symmetric bilinear forms. 1985.

[2] Kula M. Fields and quadratic form schemes. – Ann. Math. Sil. 1985, 1 (13), 7-22.

[3] Marshall M. The elementary type conjecture in quadratic form theory. – Contemp. Math. 344 (2004), 275-293.

[4] Becher K.J., Hoffmann D.W. Symbol lengths in Milnor K-theory. – Homology, Homotopy and Applications. 6 (2004), 17-31.

[5] Levchuk V.Ň., Starikova Ő.Ă. Quadratic forms of projective spaces over rings. – Matem. sbornik. 2006, no. 6, 97-110.

Krasnoyarsk, Siberian Federal University

mailto:levchuk@lan.krasu.ru

Large characteristic subgroups and their applications to almost regular automorphisms

Natalia Makarenko

I present our recent joint results with E. I. Khukhro, which enable us to solve certain important problems concerning groups with almost regular automorphisms.

In the theory of groups with almost regular automorphisms the final aim is often to prove the existence of a nilpotent subgroup with bounds for the nilpotency class and index. But in many situations we can construct such a nilpotent subgroup, say K, of bounded class c and bounded index n only in some normal subgroup, say H, and not in the group G itself. To use induction we need to consider the quotient G/K and therefore it is important to have K to be characteristic in H. Sometimes we can consider the automorphic closure ∏_{α∈Aut H} K^{α}, which is a characteristic nilpotent subgroup of class ⩽ cn. However in an induction on the length of a certain subnormal series it is desirable not to increase the nilpotency class of the subgroup at each step. We show that if a group has a nilpotent subgroup of class c and index n, then there is a characteristic nilpotent subgroup of the same class c whose index is bounded in terms of n and c. Moreover, we obtain the analogous result for a subgroup satisfying an arbitrary multilinear commutator identity. We also prove similar theorems for a nilpotent normal subgroup (or an ideal in a Lie algebra) with a bound for the rank (dimension) of the quotient group (algebra).

We now give precise formulations. We say for short that a certain quantity is (a, b, …)-bounded if it is bounded above by some function depending only on a, b, … Let x_{1}, x_{2}, … be group variables.

Definition 1 Multilinear commutators of weight 1 in the variables x_{i} are the variables x_{i} themselves. By induction, multilinear commutators of weight w > 1 in the variables x_{i} are commutators of the form κ = [κ_{1}, κ_{2}], where κ_{1} and κ_{2} are multilinear commutators with disjoint sets of variables of weights w_{1} and w_{2} for w = w_{1} + w_{2}.

A group G satisfies a multilinear commutator identity κ = 1 if and only if the commutator (verbal) subgroup κ(G) obtained by replacing all the variables in the commutator κ by the group G is trivial.

Theorem 1 Let κ be a multilinear commutator of weight w. If a group G contains a subgroup H of finite index |G : H| = n satisfying the identity κ(H) = 1, then G also contains a characteristic subgroup C satisfying the identity κ(C) = 1 whose index is finite and (n, w)-bounded.

Such a result was so far known in folklore only for abelian subgroups, that is, for κ = [x_{1}, x_{2}].

We say that an ideal J of a Lie algebra L is automorphically-invariant if J is invariant under all automorphisms of L. A Lie algebra L is nilpotent of class

⩽ c if γ_{c+1}(L) := [\underbrace{L, …, L}_{c+1} ] = 0.

Theorem 2 If a Lie algebra L over any field has a nilpotent ideal of nilpotency class c and of finite codimension r, then L has also an automorphically-invariant ideal of finite (r, c)-bounded codimension that is nilpotent of class ⩽ c.

This result, as well as ideas in its proof, are used to prove a similar result for groups, where the role of dimension is played by rank.

Recall that a group has rank r if every finitely generated subgroup of it can be generated by r elements.

Theorem 3 Suppose that a group G has a nilpotent normal subgroup H of nilpotency class c such that the quotient group G/H has finite rank r. If either

(i) H is periodic, or

(ii) H is torsion-free,

then G has also a characteristic nilpotent subgroup C of nilpotency class ⩽ c with quotient group G/C of finite (r, c)-bounded rank.

We apply the above theorems for obtaining some results on almost regular automorphisms, which had previously stalled precisely because of the need for converting subnormal series into normal. Theorem 1 proved to be useful in the study of almost regular automorphisms of order 4.

Theorem 4 There exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism of order 4 with exactly m fixed points, then G has characteristic subgroups N ⩽ H ⩽ G such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c.

This result gives an affirmative answer to Shumyatsky’s question 11.126 in the “Kourovka Notebook”.

Theorem 3 is applied to automorphisms of prime order that are almost regular in the sense of small rank of the fixed-point subgroup. In [1, Theorem 2] it was proved that if a finite nilpotent group G of derived length d admits an automorphism of prime order p with centralizer of rank r, then G has a subnormal series of (p, r, d)-bounded length whose smallest term is nilpotent of p-bounded class, while the other factors have (p, r, d)-bounded rank. Now this result can be strengthened. Using also the result in [2], where Hall–Higman type theorems were combined with powerful p-groups to obtain almost nilpotency of a finite group with such an automorphism, we now have the following corollary.

Corollary 1 If a finite soluble group G of derived length d admits an automorphism of prime order p with centralizer of rank r, then G has characteristic subgroups R ⩽ C ⩽ G such that the quotient group C/R is nilpotent of p-bounded class, the subgroup R has (p, r)-bounded rank, and the quotient G/C has (p, r, d)-bounded rank; if G is nilpotent, then one can put R = 1.

References

[1] E. I. Khukhro, Finite soluble and nilpotent groups with a restriction on the rank of the centralizer of an automorphism of prime order, Sibirsk. Mat. Zh., 41 (2000), 451–469; Engl. transl., Siberian Math. J., 41 (2000), 373–388.

[2] E. I. Khukhro and V. D. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank, J. Algebra, 301 (2006), 474–492.

Institute of Mathematics, Novosibirsk

mailto:makarenk@math.nsc.ru

A construction of non-tame automorphisms of a free group of rank 3 in the variety A_{p}A

A. I. Papistas

Let F_{3} be a free group of rank 3. For a non-negative integer m, A_{m} denotes the variety of all abelian groups of exponent of dividing m, interpreted in such a way that A_{0} = A is the variety of all abelian groups. Write V_{m} for the variety of all extensions of groups in A_{m} by groups in A. For a prime integer p, let M_{3} = F_{3}(V_{p}). The natural mapping from F_{3} onto M_{3} induces a homomorphism α from Aut(F_{3}) into Aut(M_{3}). An automorphism of M_{3} which belongs to the image of α is called tame. We give a way of constructing non-tame automorphisms of M_{3}.

Aristotle University of Thessaloniki, Faculty of Science, Department of Mathematics, GR 541 21, Thessaloniki, Greece

mailto:apapist@math.auth.gr

The geometry of Zappa-Szep products of groups

Stephen J Pride

A group G is a semidirect product of its two subgroups H and K if:

(i) H is normal in G;

(ii) H ∩ K = 1;

(iii) HK = G.

In such a situation we have a homomorphism

ψ : K ⟶ Aut(H), ψ_{k}(h) = khk^{−1} (h ∈ H, k ∈ K),

and then the multiplication is given by

(hk)(h′k′) = hψ_{k}(h′)kk′.

Conversely, given a pair of groups H, K and a homomorphism ψ : K ⟶ Aut(H), we can construct a group Ĝ consisting of the ordered pairs (h, k) with multiplication

(h, k)(h′, k′) = (hψ_{k}(h′), kk′).

Then Ĥ = {(h, 1) : h ∈ H}, K̂ = {(1, k) : k ∈ K} are isomorphic copies of H, K, and Ĝ is the semidirect product of Ĥ, K̂.

A Zappa-Szep product occurs when condition (1) above is not required to hold. To construct such a product from given groups H, K, two functions K × H ⟶ K, K × H ⟶ H are needed with certain conditions. Most of the work on this construction has been algebraic (see [1, 2] and the references cited there). However, using techniques of geometric group theory one can get a different perspective on the construction. I will describe this in my talk, and use such techniques to obtain new results concerning these groups.

References

[1] Matthew G. Brin, On the Zappa-Szep product, Comm. in Alg. 33 (2005), 393-424.

[2] M. Kunze, Zappa products, Acta Math. Hung. 41 (1983), 225-239.

[3] Stephen J Pride, The geometry of Zappa-Szep products, in preparation.

University of Glasgow

mailto:sjp@maths.gla.ac.uk

Abelian normal subgroups of unipotent subgroups of Chevalley groups

G. S. Suleymanova

Throughout this paper G(K) denotes a Chevalley group over an arbitrary field K, associated with the root system G. In this work the maximal abelian normal subgroups of the unipotent subgroup UG(K) are described. The terminology from [2, 1, 4] will be used. The description was partially announced in [5]. For the classical types G = A_{n}, B_{n}, C_{n}, D_{n}, ^{2}A_{m}, ^{2}D_{n+1} the results were obtained with V.M. Levchuk.

For example, the following theorem is proved:

Theorem 1 Suppose that M is a maximal abelian normal subgroup of UG(K).

(i) If G = ^{2}B_{2}, then M = Z_{1}⟨α⟩ with an arbitrary α ∉ Z_{1};

(ii) If G = G_{2}, or 3K = 0, or G = ^{2}G_{2}, then M = Z_{2} = U_{2};

(iii) If G = G_{2} and 2K = 0 and |K| > 2, then either M = {x_{a+b}(t)x_{2a+b}(td) | t ∈ K}Z_{2} with an arbitrary d ∈ K and Z_{2} = U_{4}, or M = U_{3};

(iv) If G = G_{2} and |K| = 2, then M = ⟨x_{a}(1)x_{2a+b}(1)⟩Z_{2}, where Z_{2} = ⟨x_{a+b}(1)x_{2a+b}(1)⟩U_{4};

(v) If G = G_{2}, or 6K = K, or G = ^{3}D_{4}, then M = Z_{3} = U_{3};

(vi) If G = ^{2}F_{4}, then M = {R_{3,−2}(t)R_{42}(td) | t ∈ K}Z_{4} with an arbitrary d ∈ K, or M = ⟨R_{43}(a)⟩R_{42}(K)Z_{4}, where a ∈ K^{∗}.

The full description of the set A_{N}(U) of large abelian normal subgroups of U = UG(K) over the finite field K = GF (q) [3, Problem (1.6)]) may be obtained from these results. For example, the next list of subgroups from A_{N}(U) and their order a(U) may be obtained from Theorem 1:

G  A_{N}(U)  a(U) K
2B_{2}(q) Z_{1}⟨α⟩ (α ∉ Z_{1})  2q
2G_{2}(q) U_{2}  q^{2}
3D_{4}(q^{3}) U_{3}  q^{5}
G_{2}(q) U_{3}  q^{3}  6K = K
U_{2}  q^{4}  3K = 0
U_{3}, {x_{a+b}(t)x_{2a+b}(td) | t ∈ K}U_{4} (|K| > 2) q^{3}  2K = 0
⟨x_{a}(1)x_{2a+b}(1), x_{a+b}(1)x_{2a+b}(1)⟩U_{4}
or M = U_{3} or M = X_{a+b}U_{4} q^{4}  |K| = 2
2F_{4}(q) ⟨R_{43}(a)⟩R_{42}(K)U_{5}  2q^{5}

This research is supported by Russian fund of Fundamental researches, grant No. 06-01-00824a.

References

[1] N. Bourbaki, Groupes et algebres de Lie (IV–VI), Hermann, Paris, 1968.

[2] R. Carter, Simple groups of Lie type, New York, Wiley and Sons, 1972.

[3] A. S. Kondratyev, Subgroups of finite Chevalley groups, Uspekhi Mat. Nauk, 41:1(247) (1986), 57-96.

[4] V.M. Levchuk, Automorphisms of unipotent subgroups of Chevalley groups, Algebra and Logic, 29:2 (1990), 141-161 (Part. I); 29:3, 315-338 (Part. II).

[5] V.M. Levchuk, Generalized unipotent subgroups of classical groups, Fund. and Appl. Math., 2:2 (1996), 625-627.

Siberian Federal University

mailto:suleymanova@list.ru

Vanishing Products of One Dimensional Classes in Group Cohomology

Ergün Yalçın

Let G be a 2-group. We are interested in the algebra structure of the cohomology of G in mod 2 coefficients, denoted usually by H^{∗}(G, ℤ/2). Let σ_{G} denote the the product of all (nontrivial) one dimensional classes in H^{∗}(G, ℤ/2). A classical theorem by Serre [1] states that (σ_{G})^{2} = 0 provided that G is not an elementary abelian 2-group. We recently improved this result in the following way:

Theorem 1 ([2]) If G is a nonabelian 2-group, then σ_{G} = 0

In the proof, we use a cohomological characterization of abelian p-groups. Namely, we show that a p-group G is nonabelian if and only if the inflation map

inf_{G/Φ(G)}^{G} : H^{3}(G/Φ(G), ℤ) → H^{3}(G, ℤ)

has a nontrivial kernel (here Φ(G) denotes the Frattini subgroup of G). The key lemma which leads to this characterization is purely group theoretical.

References

[1] J. P. Serre, Une relation dans la cohomologie des p-groupes, C.R. Acad. Sci. Paris 304 (1987), 587–590.

[2] E. Yalçın, A note on Serre’s theorem in group cohomology, preprint 2007, available through author’s web page.

Bilkent University

mailto:yalcine@fen.bilkent.edu.tr

http://www.fen.bilkent.edu.tr/~yalcine/

From proofs about finite groups to probabilistic algorithms for black box groups

Şükrü Yalçınkaya

The main purpose of this talk is to develop an analogy between the methods for recognizing a black box group and the classification of the finite simple groups. Following this talk, Alexandre Borovik will talk about the relations between black box groups and groups of finite Morley rank.

A black box group G is a finite group whose elements are encoded as 0-1 strings of uniform length and the group operations are performed by an oracle (‘black box’). Given strings representing g, h ∈ G, the black box can compute the strings representing g · h, g^{−1} and decide whether g = h. In problems naturally arising in computational group theory, some additional information is usually available, for example, the group can be given as a group of matrices over a finite field; another natural assumption is that we have an order oracle which determines the orders of elements.

In this context, a natural task is to find a probabilistic algorithm which determines the isomorphism type of a group within given (arbitrarily small) probability of error. Under various mild assumptions, there are positive answers to this question in class of finite simple groups; for example, one can determine the isomorphism type of a finite simple group of Lie type with an order oracle [2].

In this talk, we propose a uniform approach for recognizing simple groups of Lie type which follows the computational version of the classification of the finite simple groups. Similar to the inductive argument on centralizers of involutions which plays a crucial role in the classification project, our approach is based on a recursive construction of the centralizers of involutions in black box groups [4, 5].

We present an algorithm which constructs a long root SL_{2}(q)-subgroup in a finite simple group of Lie type of odd characteristic. Following this construction, we take the Aschbacher’s “Classical Involution Theorem” [1] as a model in the final recognition algorithm and we construct all root SL_{2}(q)-subgroups corresponding to the nodes in the extended Dynkin diagram, that is, we construct the extended Curtis - Phan - Tits [6, 7, 8] presentation of the finite simple groups of Lie type of odd characteristic. In particular, we construct all subsystem subgroups which can be read from the extended Dynkin diagram. With this approach it is also possible to determine whether the p-core (or “unipotent radical”) O_{p}(G) of a black box group G is trivial or not where G/O_{p}(G) is a finite simple group of Lie type of odd characteristic p answering a well-known question of Babai and Shalev [3].

References

[1] M. Aschbacher, A characterization of Chevalley groups over fields of odd order I, II, Ann. of Math. 106 (1977), 353–468.

[2] L. Babai, W. M. Kantor, P. P. Palfy, A. Seress, Black-box recognition of finite simple groups of Lie type by statistics of element orders, J. Group Theory 5, 2002, 383Ű401.

[3] L. Babai and A. Shalev, Recognizing simplicity of black-box groups and the frequency of p-singular elements in affine groups, in Groups and Computation III (W. Kantor and A. Seress, eds.), de Gruyter, Berlin (2001), 39–62.

[4] A. V. Borovik, Centralisers of involutions in black-box groups, Contemp. Math. 298 (2002), 7–20.

[5] J. N. Bray, An improved method of finding the centralizer of an involution, Arch. Math. (Basel) 74 (2000), 241–245.

[6] C. Curtis, Central extensions of groups of Lie type, J. Reine Angew. Math. 220 (1965), 174–185.

[7] K. W. Phan, On groups generated by three-dimensional special unitary groups I, II, J. Austral. Math. Soc. Ser. A 23 (1977).

[8] J. Tits, Groupes semi-simples isotropes, Colloque sur la théorie des groupes algébriques, Bruxelles (1962), 137–146.

Middle East Technical University

mailto:sukru@math.metu.edu.tr

Hartley’s theorem on representations of the general linear groups: a new approach

A. E. Zalesski

In 1986 Brian Hartley [1] obtained the following interesting result:

Theorem 1 Let K be a field, E the standard GL_{n}(K)-module, and V be an irreducible finite-dimensional GL_{n}(K)-module over K with dim V > 1. Let H be a finite subgroup of GL_{n}(K). Suppose that H is not of exponent 2 and the restriction of E to H contains a free KH-module. Then V contains a free KH-submodule.

3 CONTRIBUTED TALKS  AAD IX

His proof is based on deep properties of the duality between irreducible representations of the general linear group GL_{n}(K) and the symmetric group S_{n}. We suggest another proof based on general principles of representation theory of algebraic groups, more precisely, on properties of the weight systems of algebraic group representations. Therefore, it seems to be more conceptual and in a sense more transparent. We also expect that our method will have further applications.

References

[1] B. Hartley, Relative higher relations modules, and a property of irreducibble K-representations of GL_{n}(K), J. Algebra 104(1986), 113–125.

University of East Anglia, Norwich, UK

mailto:h054@uea.ac.uk

3 Contributed talks

Extensions of weakly supplemented modules

Rafail Alizade

Throughout, R is a ring with identity and modules are unital left R-modules. If N + K = M and N ∩ K ≪ M, then K is called a weak supplement of N (see, [2]). M is a weakly supplemented module if every submodule of M has a weak supplement. A submodule N of M is called coclosed in M if N/K ≪ M/K for some K ⊆ M then K = N.

Theorem 1 Let 0 → L → M → N → 0 be a short exact sequence. If L and N are weakly supplemented and L has a weak supplement in M then M is weakly supplemented.

If L is coclosed in M then the converse holds, that is if M is weakly supplemented then L and N are weakly supplemented.

The hypothesis that L has a weak supplement can not be omitted. To give an example we firstly prove the following lemmas.

Lemma 1 Let R be a Dedekind domain and K be the field of quotients of R. Then _{R}K is weakly supplemented.

Lemma 2 Let R be a Dedekind domain and {𝔭_{i}}_{i∈I} be an infinite collection of distinct maximal ideals of R. Let M = ∏_{i∈I}(R/𝔭_{i}) be the direct product of the

simple R-modules R/𝔭_{i} and T = T (M) be the torsion submodule of M. Then M/T is divisible, therefore M/T ≅ K^{(J)} for some index set J and Rad M = 0.

May 22–27, 2007  3 CONTRIBUTED TALKS

Example 1 Let R and M be as in Lemma 2 and T = ⊕_{i∈I}(R/𝔭_{i}) be the torsion submodule of M. Note that T is semisimple, so it is weakly supplemented. Let N be a submodule of M such that N/T ≅ K. Then N/T is weakly supplemented by Lemma 1. Note that Rad N = 0 by Lemma 2 and N is not semisimple because N/T ≅ K is not semisimple. Hence by Corollary 2.3 in [1], N is not weakly supplemented.

Moreover we have proved that a Dedekind domain R is semilocal iff every direct product of simple R-modules is weakly supplemented.

This is joint work with Engin Büyükaşık, İzmir Institute of Technology, enginbuyukasik@iyte.edu.tr.

References

[1] C. Lomp, On semilocal modules and rings, Communications in Algebra. 27:4 (1999), 1921-1935.

[2] H. Zöschinger, Invarianten wesentlicher Überdeckungen, Math. Annalen. 237 (1978), 193-202.

İzmir Institute of Technology

mailto:rafailalizade@iyte.edu.tr

http://www.iyte.edu.tr/~rafailalizade

One particle N-representability

Murat Altunbulak

The talk will be a presentation of the joint-paper with Prof. Alexander Klyachko. The problem in question is about relation between spectra of a mixed state ρ of N fermion system and its one point reduced matrix ρ^{(1)}. We briefly review the history of the problem and give two different solutions, one in terms of linear inequalities on spectra of ρ and ρ^{(1)} and another in terms of irreducible components of certain plethysms of the unitary group. We combine these solutions to produce all constraints on one point reduced matrix of a pure state for fermion systems up to rank 8. Some of them were conjectured by Borland and Dennis [1] more than 30 years ago.

References

[1] Borland, R. E. and Dennis, K.: The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. J. Phys. B: Atom Molec. Phys. 5, 7–15 (1972)

Bilkent University

mailto:murata@fen.bilkent.edu.tr

Characterizations of Minimal Non-Solvable Fitting p-Groups

Ahmet Arıkan

Let G be a minimal non-solvable Fitting p-group. We mainly prove that G has 2^{ℵ_{0}} subgroups and G has a homomorphic image \overline{G} with a proper subgroup \overline{X} such that C\overline{G}(\overline{X}) = 1 which gives a generalization of Asar’s result in [1]. We also define the solvabilizer of a solvable group and consider minimal non-solvability problem in another context.

References

[1] Asar A. O., Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Černikov. J. London Math. Soc. 61, 412-422 (2000).

[2] Ol’šhanskiǐ, A. Yu., Geometry of defining relations in groups . (Kluwer Academic Publishers, Dordrecht-Boston-London 1991).

Gazi Üniversitesi, Gazi Eǧitim Fakültesi, Matematik Eğitimi Anabilim Dalı, 06500 Teknikokullar, Ankara, Turkey

mailto:arikan@gazi.edu.tr

From abstract co-Galois theory to field theoretic co-Galois theory

Şerban A. Basarab

An abstract (topological and group theoretic) approach of the field theoretic co-Galois theory is initiated in the recent papers [1] and [3]. Given a profinite group Γ and a quasi-cyclic discrete group A on which Γ acts continuously, several classes of closed subgroups of Γ and related classes of subgroups of the cocycle group Z^{1}(Γ, A), called radical, hereditarily radical, Kneser, hereditarily Kneser and co-Galois, are defined, investigated and characterized.

A more general frame for the abstract co-Galois theory is the object of a recent project of the author [4], [5]. In this more general setting one considers a continuous action of a profinite group Γ on a (not necessarily commutative) profinite group G together with a continuous 1-cocycle η : Γ ⟶ G with the property that the profinite group G is topologically generated by the image η(Γ). The main object of study is the natural Galois connection between the topological lattice of all closed subgroups of Γ and the topological lattice of all Γ-invariant closed normal subgroups of G. The theory developed in [1], [3], which we may call cyclotomic abstract co-Galois theory is the first stage of this more general theory, where G := Hom(Z^{1}(Γ, A), A) is the Pontryagin dual of the discrete Abelian torsion group Z^{1}(Γ, A) with the action induced by the action of Γ on the quasi-- cyclic group A, while the cocycle η : Γ ⟶ G is defined by η(γ)(α) = α(γ) for α ∈ Z^{1}(Γ, A), γ ∈ Γ.

The main results of the field theoretic co-Galois theory are easily obtained from their cyclotomic abstract co-Galois theoretic versions. Moreover, new results in the field theoretic co-Galois theory are derived from specific results of the cyclotomic co-Galois theory [5]. On the other hand, the more general frame for the abstract co-Galois theory provides a suitable extension of the frame of the classical field theoretic co-Galois theory.

We illustrate the introductory remarks above with an extension of the abstract Kneser criterion [1, Theorem 1.20], an application of a result concerning the so called quasi-co-Galois subgroups to the description of the minimal non-co-- Galois hereditarily Kneser algebraic separable field extensions and an example of strongly co-Galois algebraic separable field extensions related to the model theory of Henselian fields and p-adically closed fields.

Surjective cocycles and minimal non-Kneser structures

The next result is an extension of [1, Theorem 1.20].

Proposition 1 [4, Proposition 6.23] (Abstract Kneser Criterion) Let G be a profinite Γ-operator group and η ∈ Z^{1}(Γ, G) be a generating cocycle, i.e. the profinite group G is topologically generated by η(Γ). The following assertions are equivalent for any Γ-invariant closed normal subgroup a of G.

(1) The induced cocycle η_{a} ∈ Z^{1}(Γ, G/a) is surjective.

(2) a ⊈ m for all Γ-invariant open normal subgroups m of G wich are maximal with the property that the induced cocycle η_{m} ∈ Z^{1}(Γ, G/m) is not surjective.

The following class of finite algebraic structures arises naturally from the abstract Kneser criterion above.

Definition 1 A triple (Γ, G, η) consisting of a finite group Γ, a finite Γ-operator group G and a generating cocycle η ∈ Z^{1}(Γ, G), with Δ := Ker(η), is called a minimal non-Kneser structure if the following conditions are satisfied :

(1) η(Γ) ≠ G,

(2) for every Γ-invariant normal subgroup a ≠ {1}, (Γ : η^{−1}(a)) = (G : a), and

(3) the cocycle η is normalised, i.e. ⋂_{γ∈Γ} γΔγ^{−1} = Fix_{Γ}(G) ∩ Δ = {1}.

Problem 1 Classify up to isomorphism the minimal non-Kneser structures.

A proper subclass of minimal non-Kneser structures is provided by the next obvious lemma.

Lemma 1 [4, Lemma 7.3] Let η ∈ Z^{1}(Γ, G) be a normalised generating cocycle such that η(Γ) ≠ G. Then (Γ, G, η) is a minimal non-Kneser structure whenever the Γ-operator group G is simple, i.e. {1} and G ≠ {1} are its only Γ-invariant normal subgroups.

The next results provide some classes of minimal non-Kneser structures including the very simple ones which occur in [1, Lemma 1.18 and Theorem 1.20].

Lemma 2 [4, Lemma 7.7] Let (Γ, G, η) be a minimal non-Kneser structure. Assume that Γ and G are Abelian. The following assertions hold.

(1) G is an Abelian p-group for some prime number p, Γ acts faithfully on G and the generating cocycle η is injective. Let p^{n}, n ⩾ 1, be the exponent of G.

(2) There exists a unique minimal non-zero Γ-submodule of G.

(3) The image R of the canonical ring morphism ℤ/p^{n}ℤ [Γ] ⟶ End(G) is a finite commutative local ring of characteristic p^{n}, and Γ, identified with a subgroup of R^{∗}, generates R as ℤ/p^{n}ℤ-module, in particular as ring.

Proposition 2 [4, Proposition 7.8] Let η ∈ Z^{1}(Γ, G), where Γ and G are Abelian and finite, G is a p-group for some prime number p, while Γ is not a p-group. Then, the following assertions are equivalent.

(1) (Γ, G, η) is a minimal non-Kneser structure.

(2) The local ring R ⊆ End(G) is a finite field k ≅ 𝔽_{q}, q = p^{f}, f ⩾ 1 for p ≠ 2, f ⩾ 2 for p = 2, G is a one-dimensional k-vector space identified with k^{+}, the group Γ, identified with a subgroup of the multiplicative group k^{∗}, is cyclic of order 1 ≠ r | (q−1) such that f is the order of p mod r ∈ (ℤ/rℤ)^{∗}, and η ∈ Z^{1}(Γ, k^{+}) = B^{1}(Γ, k^{+}) ≅ (ℤ/pℤ)^{f} is, up to multiplication by elements in k^{∗}, the coboundary u ∈ Γ ↦ u − 1 ∈ k^{+}.

Proposition 3 [4, Proposition 7.9] Let η ∈ Z^{1}(Γ, G), where Γ and G are finite Abelian p-groups for some prime number p. Assume that the finite local ring R ⊆ End(G) defined in Lemma 3 is principal. Then, the following assertions are equivalent.

(1) (Γ, G, η) is a minimal non-Kneser structure.

(2) One of the following conditions is satisfied:

(i) n = 1, i.e. the characteristic of R is p, and R ≅ 𝔽_{p}[x]/(x^{m}) with m ⩾ 2 and (m, p) = 1.

(ii) p = 2, n = 2, i.e. the characteristic of R is 4, and R ≅ ℤ/4ℤ[x]/(f(x), 2x^{e}), where f ∈ ℤ/4ℤ[x] is an Eisenstein polynomial of degree e ⩾ 1, in particular, the nilpotency index m = 2e is even and R is free of rank e as ℤ/4ℤ-module.

(iii) p = 2, n = 2, i.e. the characteristic of R is 4, and R ≅ ℤ/4ℤ[x]/(f(x), 2x^{t}), where f ∈ ℤ/4ℤ[x] is an Eisenstein polynomial of degree e ⩾ 2, 0 < t < e, and the nilpotency index m = e + t is odd.

In all three cases above, the R-module G can be identified with R^{+}, R/𝔪 ≅ 𝔽_{p}, Γ = 1 + 𝔪, and the injective cocycle η ∈ Z^{1}(Γ, R^{+}) is unique up to multiplication with elements from R^{∗} and summation with homomorphisms from Γ to H^{0}(Γ, R^{+}) ≅ ℤ/pℤ.

The next result is an immediate consequence of Propositions 4 and 5.

Corollary 1 [4, Corollary 7.11] Let (Γ, G, η) be a minimal non-Kneser structure. Assume that G is Abelian of exponent e and the action of Γ is induced by a character χ : Γ ⟶ (ℤ/eℤ)^{∗}, i.e. γg = χ(γ)g for all γ ∈ Γ, g ∈ G. Then G ≅ ℤ/eℤ, where e is either an odd prime number p or 4. In the first case, (Γ, G, η) ≅ (U, 𝔽_{p}^{+}, u ↦ u−1), where U ≅ Γ ≅ ℤ/rℤ, 2 ⩽ r | (p−1), is the unique subgroup of order r = |Γ| of the multiplicative group 𝔽_{p}^{∗} of the prime field 𝔽_{p}, acting by multiplication. In the latter case, (Γ, G, η) ≅ (ℤ/2ℤ, ℤ/4ℤ, 1 mod 2 ↦ 1 mod 4) with the unique nontrivial action of ℤ/2ℤ on ℤ/4ℤ.

As a consequence of Corollary 6, we find again [1, Lemma 1.18, Theorem 1.20].

From quasi-co-Galois subgroups to quasi-co-Galois field extensions

Let Γ be a profinite group acting continuously on a discrete quasicyclic group A ⊆ ℚ/ℤ. Put ℨ := Z^{1}(Γ, A)^{∨} = Hom(Z^{1}(Γ, A), A). The Abelian profinite group ℨ becomes a compact Γ-module with the action defined by (γψ)(g) = γ(ψ(g)) for γ ∈ Γ, ψ ∈ ℨ, g ∈ Z^{1}(Γ, A), while the continuous map η : Γ ⟶ ℨ, γ ↦ η(γ) : g ↦ g(γ) is a generating cocycle. We denote by 𝕃(G) the lattice of all subgroups of a subgroup G ⊆ Z^{1}(Γ, A) and by 𝕃(Γ | Δ) the lattice of all closed subgroups of Γ lying over a closed subgroup Δ ⊆ Γ. The lattices 𝕃(Z^{1}(Γ, A)) and 𝕃(Γ) := 𝕃(Γ | { 1 }) are related through the canonical reversing maps G ↦ G^{⊥} := ⋂_{g∈G} Ker(g), Δ ↦ Δ^{⊥} := { g ∈ Z^{1}(Γ, A) | Δ ⊆ Ker(g) := g^{⊥} }.

First we recall some basic notions of the cyclotomic abstract co-Galois theory.

Definition 2 A subgroup G ⊆ Z^{1}(Γ, A) is Kneser if the induced cocycle η_{G} ∈ Z^{1}(Γ, ℨ/G^{∨}) is surjective, so the induced map Γ/G^{⊥} ⟶ ℨ/G^{∨} is a homeomorphism of profinite spaces.

G is co-Galois if G is Kneser and the lattices 𝕃(G) and 𝕃(Γ | G^{⊥}) are canonically anti-isomorphic.

A closed subgroup Δ ⊂ Γ is Kneser if there exists a Kneser subgroup G ⊆ Z^{1}(Γ, A) such that Δ = G^{⊥} = Ker(η_{G}).

Δ is hereditarily Kneser if each Λ ∈ 𝕃(Γ | Δ) is Kneser.

Δ is co-Galois if there exists a co-Galois subgroup G ⊆ Z^{1}(Γ, A) such that Δ = G^{⊥}. [Such a co-Galois subgroup G is unique].

Δ is quasi-co-Galois if Δ is maximal with the property that it is hereditarily Kneser but not co-Galois.

The quasi-co-Galois subgroups of Γ are exactly the open Kneser subgroups Δ which are not co-Galois, but each Λ ∈ 𝕃(Γ | Δ) \ { Δ } is co-Galois. A more precise description of the quasi-co-Galois subgroups is given by the next statement.

Proposition 4 [5, Lemma 3.1] The quasi-co-Galois subgroups of Γ are exactly those open subgroups Δ ⊆ Γ which satisfy one of the following four conditions.

(1) Δ ◁ Γ, ˆ1/4 ∈ A^{Δ} \ A^{Γ}, and Γ/Δ ≅ (ϵ_{4}^{⊥}/Δ) ⋊ (Γ/ϵ_{4}^{⊥}) ≅ 𝔻_{8}.

(2) Δ ◁ Γ, ˆ1/8 ∈ A^{Δ}, ˆ1/4 ∉ A^{Γ}, and the factor group Γ/Δ ≅ Q (the group of quaternions) has the presentation

Γ/Δ ≅ ⟨ σ, τ | σ^{4} = 1, σ^{2} = τ^{2}, στσ^{−1} = τ^{−1} ⟩ =

⟨ σ, τ | στ = τ^{−1}σ, τσ = σ^{−1}τ ⟩,

with the action on (1/8)ℤ/ℤ ⊆ A^{Δ} given by σ ˆ1/8 = ˆ3/8 and τ ˆ1/8 = ˆ1/8.

(3) ˆ1/8 ∈ A\A^{Δ}, ˆ1/4 ∈ A^{Δ} \A^{Γ}, Δ∩ϵ_{8}^{⊥} ◁ Γ, and the factor group Γ/(Δ∩ϵ_{8}^{⊥}) has the presentation

Γ/(Δ ∩ ϵ_{8}^{⊥}) ≅ (ϵ_{4}^{⊥}/(Δ ∩ ϵ_{8}^{⊥})) ⋊ (Γ/ϵ_{4}^{⊥}) ≅
⟨ σ, τ, δ | σ^{2} = τ^{4} = δ^{2} = (στ)^{2} = [σ, τδ] = [τ, δ] = 1 ⟩,

with the action on (1/8)ℤ/ℤ given by σ ˆ1/8 = −ˆ1/8, τ ˆ1/8 = ˆ1/8 and δ ˆ1/8 = ˆ5/8.

(4) Δ ◁ Γ, there exist an odd prime number p and a divisor r > 1 of p − 1 such that ˆ1/pr ∈ A^{Δ}, ˆ1/p ∉ A^{Γ}, ˆ1/l ∈ A^{Γ} for all l ∈ P with l | r, and

Γ/Δ ≅ (ϵ_{p}^{⊥}/Δ) ⋊ (Γ/ϵ_{p}^{⊥}), Γ/ϵ_{p}^{⊥} ≅ ℤ/rℤ, ϵ_{p}^{⊥}/Δ ≅ ℤ/pℤ.

[For n ∈ ℕ, ϵ_{n} denotes the coboundary γ ↦ (γ − 1)ˆ1/n, while P stands for the set of all prime natural numbers where 2 is replaced by 4.]

The field theoretic notions which are required for stating the field theoretic analogue of Proposition 7 above are briefly recalled in the following.

Let E/F be an algebraic field extension. We denote by T (E/F) the multiplicative group of all elements x ∈ E^{∗} which are radicals over F, i.e. x^{n} ∈ F^{∗} for some n ∈ ℕ \ { 0 }. The co-Galois group Cog(E/F) of the field extension E/F is the factor group T (E/F)/F^{∗}, i.e. the torsion subgroup of the factor group E^{∗}/F^{∗}. By Hilbert’s Satz 90, Cog(E/F) is canonically isomorphic with Z^{1}(Γ, μ) whenever E/F is Galois with Galois group Γ and μ is the multiplicative quasicyclic group of all roots of unity contained in E. A dictionary relating the group theoretic notions and the the corresponding field theoretic ones is transparent.

Definition 3 A finite separable field extension E/F is Kneser if E = F (G) for some subgroup G ⊆ T (E/F) lying over F^{∗} such that [E : F] = (G : F^{∗}). If, in addition, the lattice 𝕃(E/F) of all intermediate fields between F and E and the lattice 𝕃(G | F^{∗}) are canonically isomorphic, we say that E/F is co-Galois. [In the latter case, the group G of radical elements is uniquely determined.]

An algebraic separable field extension E/F is hereditarily Kneser if each finite subextension L/F is Kneser. E/F is co-Galois if each finite subextension L/F is co-Galois.

An algebraic separable field extension E/F is called quasi-co-Galois if it is minimal with the property that it is hereditarily Kneser but not co-Galois.

It turns out that the quasi-co-Galois field extensions are exactly those separable finite field extensions E/F which are Kneser, not co-Galois, but each subextension L/F, L ≠ E is co-Galois. As a consequence of Proposition 7, we obtain a precise description of the quasi-co-Galois field extensions as follows. Proposition 5 [5] Let E/F be an algebraic separable field extension. Then, the necessary and sufficient condition for E/F to be quasi-co-Galois is that one of the following situations hold :

(1) char(F) ≠ 2, ζ_{4} ∉ F, and E = F (ζ_{4}, θ) for some θ ∈ E satisfying θ^{4} = a, where a ∈ F \ ±F^{2}.

(2) char(F) ≠ 2, 2 ∈ (−F^{2}) \ F^{2}, and E = F (θ) for some θ ∈ E such that a := θ^{8} ∈ (−F^{2}) \ (−F^{4}).

(3) char(F) ≠ 2, 2 ∉ ±F^{2}, ζ_{4} ∉ F, and E = F (θ) for some θ ∈ E such that a := θ^{8} ∈ (−F^{2}) \ (−F^{4} ∪ −4F^{4}).

(4) there exist a prime number p ≠ 2 and 1 < r | (p − 1) such that the characteristic exponent of F is prime to pr, ζ_{p} ∈ E \ F, ζ_{r} ∈ F (ζ_{p}), ζ_{l} ∈ F for all l ∈ P, l | r, [F (ζ_{p}) : F] = r, and E = F (ζ_{p}, θ) for some θ ∈ E satisfying θ^{p} = a, where 0 ≠ a ∈ F \ F^{p}.

[For n ∈ ℕ \ { 0 }, ζ_{n} denotes a primitive nth root of unity.]

From strongly co-Galois subgroups to strongly co-Galois field extensions

Proposition 6 [5, Lemma 2.8] Let Δ be a radical subgroup of Γ, i.e. Δ = Δ^{⊥⊥}. Then the following statements are equivalent

(1) Δ^{⊥} is a co-Galois subgroup of Z^{1}(Γ, A).

(2) Δ^{⊥} is a Kneser subgroup of Z^{1}(Γ, A).

(3) A^{Γ}[p] = A^{Δ}[p] for all p ∈ P. Call such a subgroup Δ ⊆ Γ strongly co-Galois The field theoretic analogue of Proposition 9 due to Greither and Harrison [7] reads as follows. Theorem 1 The following statements are equivalent for a separable radical extension E/F.

(1) E/F is T (E/F)-Kneser.

(2) E/F is T (E/F)-co-Galois.

(3) μ_{p}(E) ⊂ F for all p ∈ P. Call such an extension E/F strongly co-Galois. The following class of strongly co-Galois extensions plays a key role in the model theory of Henselian fields and p-adically closed fields.

Theorem 2 [8, Theorem 3.8], [2, Proposition 1.1] Let (F, v) be a Henselian valued field of characteristic zero. An algebraic field extension E of F is strongly co-Galois whenever E is a core-dense extension of F.

In particular, if F is a p-valued Henselian field, E/F is strongly co-Galois whenever E is of the same p-rank as F.

Note that the radical structure theorem above is a main ingredient in the proof of an isomorphism theorem for algebraic extensions [8, Corollary 3.11], [6], and of a (relative) quantifier elimination theorem [8, Theorem 5.6], [2, Theorems A, B].

References

[1] T. Albu, Ş.A. Basarab, An abstract Cogalois Theory for profinite groups, J. Pure Appl. Algebra 200 (2005), 227–250.

[2] Ş.A. Basarab, Relative elimination of quantifiers for Henselian valued fields, Annals of Pure and Applied Logic 53 (1991), 51-74.

[3] Ş.A. Basarab, Kneser and hereditarily Kneser subgroups of a profinite group, Serdica Math. J. 30 (2004), 325–348.

[4] Ş.A. Basarab, A new setting for co-Galois Theory, I: Universal cocycles and Kneser structures, preprint, 2006.

[5] Ş.A. Basarab, From cyclotomic abstract co-Galois theory to field theoretic co-Galois theory, in preparation.

[6] Ş.A. Basarab and F.-V. Kuhlmann, An isomorphism theorem for Henselian algebraic extensions of valued fields, Manuscripta Math. 77 (1992), 113-126; 84 (1994), 113-114.

[7] C. Greither and D.K. Harrison, A Galois correspondence for radical extensions of fields, J. Pure Appl. Algebra 43 (1986), 257–270.

[8] A. Prestel and P. Roquette, “Formally p-adic fields”, Lectures Notes in Math. 1050, Springer-Verlag, 1984.

Institute of Mathematics “Simion Stoilow” of the Romanian Academy

mailto:serban.basarab@imar.ro

http://www.imar.ro/~sbasarab/

Ratliff-Rush Monomial Ideals

Veronica Crispin Quiñonez

Let R be a Noetherian ring and let an ideal I in it be regular, that is, let I contain a nonzerodivisor. Then the ideals (I^{l+1} : I^{l}), l ⩾ 1, increase with l. The union Ĩ = ⋃_{l⩾1}^{∞}(I^{l+1} : I^{l}) was first studied by Ratliff and Rush in [1]. They show that (Ĩ)^{l} = I^{l} for sufficiently large l and that Ĩ is the largest ideal with this property. Hence, ˜ Ĩ = Ĩ. Moreover, they show that Ĩ^{l} = I^{l} for sufficiently large l. We call Ĩ the Ratliff-Rush ideal associated to I, and an ideal such that Ĩ = I a Ratliff-Rush ideal. The Ratliff-Rush reduction number of I is defined as r(I) = min {l ∈ ℤ_{⩾0} |Ĩ = (I^{l+1} : I^{l})}. The operation˜ cannot be considered as a closure operation in the usual sense, since J ⊆ I does not generally imply J̃ ⊆ Ĩ. An example from [2] shows this: let J = ⟨y^{4}, xy^{3}, x^{3}y, x^{4}⟩ ⊂ I = ⟨y^{3}, x^{3}⟩ ⊂ k[x, y], then I is Ratliff-Rush but x^{2}y^{2} ∈ J̃\Ĩ.

One of the reasons to study Ratliff-Rush ideals is the following. Let I be a regular 𝔪-primary ideal in a local ring (R, 𝔪, k). The Hilbert function H_{I}(l) = dim_{k}(R/I^{l}) is a polynomial P_{I}(l) called the Hilbert polynomial of I for all large l. Then Ĩ can be defined as the unique largest ideal containing I and having the same Hilbert polynomial as I.

Ratliff-Rush ideals associated to monomial ideals are monomial by definition, which makes the computations easier. There is always a positive integer L such that Ĩ = I^{L+1} : I^{L}, but it is not clear how big that L is (see Example 1.8 in [2]). If I is a monomial ideal and m is some monomial, then for all l ⩾ 0 we have

(mI)^{l+1} : (mI)^{l} = (m^{l+1}I^{l+1}) : (m^{l}I^{l}) = m(I^{l+1} : I^{l}).  (†)

Principal ideals are trivially Ratliff-Rush. Any non-principal monomial ideal J in the rings k[x, y] and k[[x, y]] can be written as J = mI, where m is a monomial and I is an ⟨x, y⟩-primary ideal; hence it suffices to consider ⟨x, y⟩-primary monomial ideals. Moreover, (†) shows that the Ratliff-Rush reduction numbers of I and mI are the same.

In this talk we show how to compute the Ratliff-Rush ideal associated to a monomial ideal in a certain class in the rings k[x, y] and k[[x, y]] and find an upper bound for the Ratliff-Rash reduction number for such an ideal. We start by giving some results about numerical semigroups that are crucial for our later work. We conclude by duscussing several useful examples.

References

[1] L. J. Ratliff, Jr and D. E. Rush, Two Notes on Reductions of Ideals, Indiana Univ. Math. J. 27 (1978), no. 6, 929-934.

[2] M. E. Rossi and I. Swanson, Notes on the Behavior of the Ratliff-Rush Filtration, Commutative Algebra (Grenoble/Lyon, 2001), 313-328, Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003.

KTH Matematik, Royal Institute of Technology, Stockholm, Sweden

mailto:vcrispin@kth.se

http://www.math.kth.se/~vcrispin

On the Cycle Structure of Permutation Polynomials of Finite Fields

Ayça Çeşmelioğlu

This is joint work with Wilfried Meidl and Alev Topuzoğlu. Permutation polynomials over 𝔽_{q} of degree ⩽ q − 1 form a group under composition and subsequent reduction modulo x^{q} − x which is isomorphic to S_{q}. Leonard Carlitz observed in his 1953 article [1] that any transposition (0 a), a ∈ 𝔽_{q} can be represented by the polynomial

P_{a}(x) = −a^{2}(((x − a)^{q−2} + a^{−1})^{q−2} − a)^{q−2}.

His result shows that the group of permutation polynomials over 𝔽_{q} is generated by x^{q−2} and ax + b, a, b ∈ 𝔽_{q}, a ≠ 0. The above mentioned isomorphism motivates the study of the cycle structure of permutation polynomials of 𝔽_{q}. See [3], [4] for work on cycle structure of some specific polynomials of interest.

In this work we study the cycle structure of the polynomials

P_{n}(x) = (((a_{0}x + a_{1})^{q−2} + a_{2})^{q−2} + … + a_{n})^{q−2} + a_{n+1}

over 𝔽_{q} with a_{0} ≠ 0 and note that any permutation polynomial of 𝔽_{q} can be expressed in this form for some n ⩾ 0, with P_{0}(x) = a_{0}x + a_{1}. We define a rational function corresponding to P_{n}(x) and by using the known structure of P_{1}(x) or the cycle structure of rational linear transformations we determine the cycle structure of P_{2}(x) and P_{3}(x) completely . With the help of [2] we also give constructions for P_{n}(x) with full cycle for arbitrary n ⩾ 1.

References

[1] L. Carlitz, Permutations in a finite field, Proc. Amer. Math. Soc. 4 (1953) 538.

[2] I. Beck, Cycle decomposition by transpositions. J. Combinatorial Theory Ser. A 23 no. 2 (1977), 198–207.

[3] R. Lidl, G.L. Mullen, Cycle structure of Dickson permutation polynomials, Math. J. Okayama Univ. 33 (1991) 1–11.

[4] W.-S. Chou, I.E. Shparlinski, On the cycle structure of repeated exponentiation modulo a prime, J. Number Theory 107 no. 2 (2004), 345–356.

Sabancı University, Istanbul

mailto:cesmelioglu@su.sabanciuniv.edu

Locally solvable^{∘} groups of finite Morley rank

Adrien Deloro

This is joint work with Eric Jaligot. The main conjecture concerning groups of finite Morley rank is the Cherlin–Zilber one, stating that any simple infinite group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field. A proof of (some reasonable part of) this conjecture is expected to run parallel to the classification of finite simple groups, and vice-versa.

As there is no Feit–Thompson theorem for groups of finite Morley rank, one has to assume there are involutions. Now as in the finite case, it is natural to consider at some point “locally solvable groups”. This amounts to transfering results obtained on so-called “minimal connected simple groups” to a larger class. Locally here does not mean finitely generated, but comes from the following.

Definition 1 Let G be a group. A subgroup H ⩽ G is said to be local if there is some non-trivial abelian subgroup A ⩽ G such that H ⩽ N_{G}(A).

Definition 2 Let G be a group of finite Morley rank. G is said to be locally solvable if for any non-trivial abelian subgroup A ⩽ G, N_{G}(A) is solvable.

As people usually work with connected components (denoted by a ^{∘}), our definition is the following.

Definition 3 Let G be a group of finite Morley rank. G is said to be locally solvable^{∘} if for any non-trivial abelian subgroup A ⩽ G, N_{G}^{∘}(A) is solvable.

The only non-solvable locally solvable^{∘} algebraic group is PSL_{2}(K), which does have involutions.

Our main tools rely on a notion of unipotence in characteristic 0 developed by Jeffrey Burdges in his PhD [Bur04]. Namely one has to generalize the notion of a characteristic in the following way.

Definition 4 A unipotence parameter is a pair q̃ = (p, d), where p is either prime or ∞, d ∈ ℕ ∪ {∞}, such that p ≠ ∞ ⇔ d = ∞.

So if p is a real prime number, this is just indicating the characteristic, but if p = ∞ the notion of unipotence becomes graded (d is called the unipotence degree). In the latter case Burdges defined q̃-unipotent subgroups, and q̃-Sylow subgroups. As far as results and not definitions are concerned, both these notions provide good analogs of the prime characteristic ones.

There is an important uniqueness lemma that was first proved in the minimal connected simple framework [Del07a].

Lemma 1 (Uniqueness Lemma) Let G be a locally solvable^{∘} group if finite Morley rank. Let q̃ = (p, d) be some unipotence parameter with d > 0, and let U be some q̃-Sylow subgroup of G.

Assume U contains some q̃-unipotent subgroup U_{1}, and that U_{1} contains some subset X such that the p-unipotence degree of C^{∘}(X) is not bigger than d.

Then U is the only q̃-Sylow subgroup of G containing U_{1}.

This lemma is strong enough to provide a good intersection control, allowing us to prove the following.

Theorem 1 (Mixed type theorem) Let G be a locally solvable^{∘} group of finite Morley rank. Assume G has mixed type. Then G^{∘} is solvable

Theorem 2 (Even type theorem) Let G be a locally solvable^{∘} group of finite Morley rank. Assume G has even type. Then either G^{∘} is solvable or G ≃ PSL_{2}(K) for some algebraically closed field K of characteristic 2.

Our results in odd type are not as positive yet.

Theorem 3 (Odd type algebraic theorem) Let G be a locally solvable^{∘} group of finite Morley rank. Assume G has odd type and Pruefer-rank 1. Assume also that there is a toral involution i such that C^{∘}(i) is not a Borel subgroup. Then either G^{∘} is solvable or G ≃ PSL_{2}(K) for some algebraically closed field K of characteristic not 2.

Theorem 4 (Odd type non-algebraic theorem) Let G be a non-solvable locally solvable^{∘} group of finite Morley rank. Assume G has odd type and Pruefer- -rank ⩾ 2. Then G has Pruefer-rank equal to 2, its Sylow subgroup is connected, and involutions are conjugate. Furthermore for any involution i of G, C^{∘}(i) is a Borel subgroup.

References

[Bur04] Jeffrey Burdges. Simple Groups of Finite Morley Rank of Odd and Degenerate Type. PhD thesis, Rutgers University, New Brunswick, New Jersey, 2004.

[Bur07] Jeffrey Burdges. The Bender method in groups of finite Morley rank. J. Algebra, 2007. To appear.

[Del07a] Adrien Deloro. Groupes simples connexes minimaux algébriques de type impair. Journal of Algebra, 2007. To appear.

[Del07b] Adrien Deloro. Groupes simples connexes minimaux non-algébriques de type impair. Submitted, 2007.

Paris 7 University

mailto:adeloro@logique.jussieu.fr

Algebras with skew-symmetric identity of degree 3

A. S. Dzhumadil’daev

Algebras with one of the following identities are considered:

[[t_{1}, t_{2}], t_{3}] + [[t_{2}, t_{3}], t_{1}] + [[t_{3}, t_{1}], t_{2}] = 0, (Lie-Admissible)
[t_{1}, t_{2}]t_{3} + [t_{2}, t_{3}]t_{1} + [t_{3}, t_{1}]t_{2} = 0, (0-Lie-Admissible (shortly 0-Alia))

{[t_{1}, t_{2}], t_{3}} + {[t_{2}, t_{3}], t_{1}} + {[t_{3}, t_{1}], t_{2}} = 0, (1-Lie-admissible (shortly 1-Alia))

where [t_{1}, t_{2}] = t_{1}t_{2} − t_{2}t_{1} and {t_{1}, t_{2}} = t_{1}t_{2} + t_{2}t_{1}. For algebra A = (A, ∘) with multiplication ∘ denote by A^{(q)} an algebra with vector space A and multiplication a ∘_{q} b = a ∘ b + q b ∘ a.

Theorem 1 Any algebra with a skew-symmetric identity of degree 3 is (anti-) isomorphic to one of the following algebras:

• Lie-admissible algebra

• 0-Alia algebra

• 1-Alia algebra

• algebra of the form A^{(q)} for some left-Alia algebra A and q ∈ K, such that q^{2} ≠ 0, 1.

Example 1 (C[x], ∘) under multiplication a ∘ b = ∂(a)∂^{2}(b) is 1-Alia and simple.

Example 2 (C[x], ⋆), where a⋆b = ∂^{3}(a)b+4∂^{2}(a)∂(b)+5∂(a)∂^{2}(b)+2a ∂^{3}(b), is 0-Alia and simple.

References

[1] N-commutators, Comm.Math. Helvetici, 79(2004), No.3, p. 516-553. Preprint available math.RA/0203036

S. Demirel University, Almaty

mailto:askar@math.kz

The Extension Class of a Subset Complex

Aslı Güçlükan

This is a joint work with E. Yalçın. Let G be a finite group and X be a finite G-set. The subset complex Δ(X) of X is defined as the simplicial complex whose simplices are subsets of X. The oriented complex of Δ(X) in mod 2 coefficients gives the following 𝔽_{2}G-module extension

0 → 𝔽_{2} → C_{n−1}(Δ(X)) → ⋯ → C_{0}(Δ(X)) → 𝔽_{2} → 0.

Let ζ̄_{X} ∈ Ext_{𝔽_{2}G}^{n}(𝔽_{2}, 𝔽_{2}) denote the extension class for this extension. In the case X = G/1, the class ζ_{G/1} is an essential class which makes it interesting for group cohomology. This class is a mod 2 reduction of a class ζ_{X} which is introduced by Reiner and Webb [1]. They raised the question that for which finite groups ζ_{G/1} ≠ 0? We answer their question under mod 2 reduction:

Theorem 1 Let G be a finite group. Then ζ̄_{G/1} is non-zero if and only if G is an abelian 2-group isomorphic to (ℤ/2)^{n} ×(ℤ/4)^{m} for some n ⩾ 0 and m = 0, 1.

References

[1] V. Reiner and P. Webb, The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), no. 1-3, 291-327.

Bilkent University

mailto:guclukan@fen.bilkent.edu.tr

Standard tableaux and Klyachko’s Theorem on Lie representations

Marianne Johnson

Let n ∈ ℕ and let λ = (λ_{1}, …, λ_{k}) be a partition of n. That is, λ_{1} ⩾ ⋯ ⩾ λ_{k} > 0 and λ_{1} +⋯+λ_{k} = n. The Young diagram corresponding to λ is a collection of n boxes arranged in left justified rows with λ_{i} boxes in the ith row. A standard tableau of shape λ is a numbering of the Young diagram of λ with the numbers from {1, …, n} in such a way that the entries strictly increase along every row and down every column. We say that an entry i is a descent in a standard tableau T if the entry i + 1 occurs in any row below the row in which i appears. The sum of all descents in T is called the major index of T.

For example, λ = (5, 3, 2, 1) is a partition of 11. Shown below is a standard tableau T of shape λ. The descents of T are 2, 6, 9 and 10. Hence, the major index of T is 2 + 6 + 9 + 10 = 27.

1 2 5 6 9 3 4 10 7 8 11

We show that for all partitions λ of n ⩾ 3 (λ ≠ (2^{2}), (2^{3}), (1^{n}), (n)) there exists a standard tableau of shape λ with major index coprime to n [1]. By a result of Kraśkiewicz and Weyman [3], this provides a new purely combinatorial proof of Klyachko’s Theorem [2] on Lie representations of the general linear group.

References

[1] Marianne Johnson. Standard tableaux and Klyachko’s Theorem on Lie representations. Journal of Combinatorial Theory, Series A,114 (2007), no. 1, 151–158.

[2] A.A. Klyachko. Lie elements in the tensor algebra. Sibirsk Mat. Ž, 15 (1974) 1296Ű-1304 (Russian). English translation: Siberian J. Math. 15 (1974), 914–921.

[3] Witold Kraśkiewicz and Jerzy Weyman. Algebra of coinvariants and the action of a Coxeter element. Bayreuth. Math. Schr., 63 (2001), 265Ű-284. (Preprint, 1987).

University of Manchester

mailto:marianne.johnson@maths.man.ac.uk

http://personalpages.manchester.ac.uk/postgrad/M.Johnson-3/

Modules of uniform Jordan type

Semra Öztürk Kaptanoğlu

Let G be an abelian p-group, M be a finitely generated k[G]-module, where k is a field of characteristic p > 0. We say M is of uniform Jordan type if the restriction of M to the subalgebra k[⟨1 + x⟩] has the same Jordan type as the restriction of M to the subalgebra k[C_{p^{r}}] for some cyclic subgroup C_{p^{r}} of G for all x in the Jacobson radical of k[G] with ⟨1 + x⟩ ≅ C_{p^{r}}.

We prove that the class of modules of uniform Jordan type contains endotrivial modules and it is closed under taking direct sums and tensor products.

References

[1] Modules of constant Jordan type, J. Carlson, E. Friedlander, J. Pevtsova, Preprint

Orta Doğu Teknik Üniversitesi, Ankara, Turkey

mailto:sozkap@metu.edu.tr

http://www.math.metu.edu.tr/~semra/

Connections between pseudo-free groups and RSA

Bilal Khan

Rivest introduced the notion of a pseudo-free group and showed that if Z_{n} is pseudo-free then it also satisfies several cryptographic assumptions, e.g. the Strong RSA Assumption and the difficulty of computing discrete logarithms. Later, Daniele Micciancio showed that the Strong RSA Assumption implies that Z_{n} is pseudo-free, thereby establishing an equivalence. In this talk, we will give a survey of cryptographic assumptions and their connections to the property of pseudo-freeness. We will then present new results on non-cyclic groups that are also pseudo-free.

References

[1] R. Rivest, “On notion of pseudo-free groups,” In M. Naor, editor, Proceedings of the First Theory of Cryptography Conference - TCC 2004.

[2] Daniele Micciancio, The RSA group is pseudo-free. Advances in Cryptology—Eurocrypt 2005. Aarhus, Denmark. May 2005. LNCS 3494, Springer, pp. 387-403.

City University of New York

mailto:grouptheory@hotmail.com

http://www.bilalkhan.org

Finite Covers and Congruences

Elisabetta Pastori

This talk can be placed within the area of infinite permutation groups and their relations with model theory. For brief surveys about this area see for example [Ca] and [EMI].

After giving the definition of finite cover of a structure W (we give it in terms of the automorphism groups of the structures involved), we show a theorem which describes all the possible kernels of finite covers of a transitive structure W having all binding groups and fibre groups isomorphic to a simple non abelian regular group G, in terms of the Aut(W)-congruences on W. This result has its basis in the article [EH] . Then we take in exam the case of W = Ω^{(n)}, the set of ordered n-tuples with all distinct entries from an infinite set Ω, with Aut(W) = Sym(Ω) and we show a result of biinterpretability.

References

[Ca] P. Cameron, Oligomorphic permutation groups, London Mathematical Society Lecture Notes Series 152, Cambridge University Press, Cambridge, 1990.

[EH] D.M. Evans and E. Hrushovski, On the automorphism groups of finite covers, Annals of Pure and Applied Logic 62 (1993), 83–112.

[EMI] D. M. Evans, A. A. Ivanov and D. Macpherson, Finite covers, London Mathematical Society Lecture Notes Series 244, Cambridge University Press, Cambridge, 1977.

University of Florence, Italy

mailto:pastori@math.unifi.it

Modules whose Maximal Submodules are Supplements

Dilek Pusat-Yılmaz

(Joint work with Engin Büyükaşık)

Let R be a ring and M be a unital left R-module. M is called an MS − module if every maximal submodule of M is a supplement in M, and M is called an MD − module if every maximal submodule of M is a direct summand of M.

In this talk some properties and some characterizations of MS-modules will be given.

Proposition 1 Let M = ∑_{i∈I} M_{i} for some index set I. Suppose M_{i} is an MS-module for every i ∈ I. Then M is an MS-module.

Theorem 1 Let R be a noetherian ring and M an R-module. M is an MS-module if and only if M/D(M) has no maximal submodules where D(M) = ∑{Rm : m ∈ Λ} and Λ = {m ∈ M : Rm ∩ P ≪ P ∀P ⩽ M such thatRm ⊈ P }.

References

[1] Alizade, R., Büyükaşık, E. (2003). Cofinitely weak supplemented modules Comm. Algebra. 31(11):5377-5390.

[2] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, (Springer, New York, 1992).

[3] R. Wisbauer, Foundations of Modules and Rings, (Gordon and Breach, 1991).

[4] H. Zöschinger, Komplementierte moduln über Dedekindringen, Journal of Algebra. 29 (1974), 42-56.

Izmir Institute of Technology

mailto:dilekyilmaz@iyte.edu.tr

Generalized PC-Groups And Generalized CC-Groups Of Matrices

Francesco Russo

Let 𝔛 be a class of groups. An element x of a group G is said to be 𝔛C-central, or an XC-element, if G/C_{G}(x^{G}) satisfies 𝔛, where the symbol x^{G} denotes the normal closure of the subgroup < x > in G (see [10]). A 𝔛C-central group G is also called a group with 𝔛-conjugacy classes. An XC-element x of a group G is also called an element which has 𝔛-conjugacy classes. The factor group G/C_{G}(x^{G}) is a group of automorphisms of x^{G} and an old question of Group Theory is the knowledge of a group, starting from information on its group of automorphisms.

Following [10], a class 𝔛 of groups is said to be S_{n}-closed if every normal subgroup of an 𝔛-group is an 𝔛-group. A class 𝔛 of groups is said to be N-closed if the subgroup generated by any system of normal 𝔛-subgroups of a group is an 𝔛-group. A class 𝔛 of groups is said to be P-closed if every extension of an 𝔛-group is an 𝔛-group. From now the symbol 𝔛 will denote a class of groups which is S_{n}, N and P-closed.

It is possible to introduce some series which generalize the usual notion of upper central series of a group (see for instance [10, Chapter 2]) thanks to the following remark.

Let G be a group. If x and y are XC-elements of G, then both G/C_{G}(x^{G}) and G/C_{G}(y^{G}) satisfies 𝔛, so G/(C_{G}(x^{G})∩C_{G}(y^{G})) satisfies 𝔛. But the intersection of C_{G}(x^{G}) with C_{G}(y^{G}) lies in C_{G}((xy^{−1})^{G}), so that G/C_{G}((xy^{−1})^{G}) satisfies 𝔛 and xy^{−1} is an XC-element of G. Hence the XC-elements of G form a subgroup X(G) and it is easy to see that X(G) is characteristic in G.

This remark has been first noted by R. Baer for the class of F C-groups (see [10, Lemma 4.31]) and allows us to define the series

1 = X_{0} ◁ X_{1} ◁ … ◁ X_{α} ◁ X_{α+1} ◁ …,

where X_{1} = X(G), X_{α+1}/X_{α} = X(G/X_{α}) and

X_{λ} = ⋃_{α<λ} X_{α},

with α ordinal and λ limit ordinal. This series is a characteristic ascending series of G and it is called upper XC-central series of G. The last term of the upper XC-central series of G is called XC-hypercenter of G. If G = X_{β}, for some ordinal β, we say that G is an XC-hypercentral group of type at most β.

The XC-length of an XC-hypercentral group is defined to be the least ordinal β such that G = X_{β}, in particular when G = X_{c} for some positive integer c, we say that G is XC-nilpotent of length c. A group G is said to be an XC-group if all its elements are XC-elements, that is, if G has XC-length at most 1. The first term X(G) of the upper XC-central series of G is called XC-center of G and the α-th term of the upper XC-central series of G is called XC-center of length α of G. Roughly speaking, the upper XC-central series of G measures the distance of G to be an XC-group.

If 𝔛 is the class 𝔉 of finite groups, we have the notion of F C-hypercentral group and classical results can be found in [10, Chapter 4]. In this circumstance, we specialize the previous symbol X in the symbol F, for introducing XC-hypercentral series and related definitions. If 𝔛 is the class 𝔓𝔉 of polycyclic-by-finite groups, we have the notion of P C-hypercentral group and a complete description of such groups can be found in [2]. In this circumstance, we specialize the previous symbol X in the symbol P, for introducing P C-hypercentral series and related definitions. If 𝔛 is the class ℭ of Chernikov groups, we have the notion of CC-hypercentral group and a complete description of such groups can be found in [3]. In this circumstance, we specialize the previous symbol X in the symbol C, for introducing CC-hypercentral series and related definitions.

Our approach to generalized central series of a group has been introduced by D.H.McLain, obtaining classical results in Theory of Generalized F C-groups (see [10, Theorem 4.37, Theorem 4.38]). Also [4] obtain similar results with this approach. One of the most recent improvement of McLain Theorems has been furnished by [1, Theorem A, Theorem B, Theorem C] for P C-hypercentral groups and CC-hypercentral groups.

On the other hand hypercentral groups, F C-hypercentral groups, P C- hypercentral groups and CC-hypercentral groups can be much different between themselves as it is shown either by the consideration of the infinite dihedral group or by means of examples in [1] and [5].

Our results are devoted to groups of matrices of P C-hypercentral groups and CC-hypercentral groups. A group G is called linear of degree n over an arbitrary field K, or a group of n × n matrices over an arbitrary field K, or briefly linear, if G is a subgroup of the group GL(n, K), where n is a positive integer. Classical literature on linear groups is represented by [6], [9], [12], [13]. More recent is [8], where [8, Chapter 5] gives a survey on chains conditions in linear groups. These chains conditions have been first introduced in Russian literature (see [6], [9], [12]). We prove the following results, which extend [7, Theorem 2].

Theorem 1 Let G be a linear group of degree n over an arbitrary field K. Then the following conditions are equivalent

• G is P C-nilpotent;

• G is P C-hypercentral;

• G is nilpotent-by-polycyclic-by-finite.

Theorem 2 Let G be a linear group of degree n over an arbitrary field K. If char K = 0 and G/Z(G) does not contain subgroups of infinite exponent, then the following conditions are equivalent

• G is F C-nilpotent;

• G is F C-hypercentral;

• G is nilpotent-by-finite;

• G is CC-nilpotent;

• G is CC-hypercentral.

The construction in [11] will allow us to give examples of groups which satisfy Theorem A and Theorem B.

Key Words: Conjugacy classes; linear P C-groups; linear CC-groups; P C-hypercentral series; CC-hypercentral series.

MSC 2000: 20C07; 20D10; 20F24.

References

[1] Beidleman, J.C., Galoppo, A. and Manfredino, M. (1998). On P C-hypercentral and CC-hypercentral groups. Comm. Alg. 26, 3045-3055.

[2] Franciosi, S., de Giovanni, F. and Tomkinson, M. J.(1990). Groups with polycyclic-by-finite conjugacy classes. Boll.UMI 4B, 35-55.

[3] Franciosi, S., de Giovanni, F. and Tomkinson, M. J.(1991). Groups with Chernikov conjugacy classes. J. Austral. Math. Soc A50, 1-14.

[4] Garas̆c̆uk, M. S.(1960). On theory of generalized nilpotent linear groups. Dokl. Akad. Nauk BSSR [in Russian] 4 276-277.

[5] Landolfi, T. (1995). On generalized central series of groups. Ricerche Mat. XLIV, 337-347.

[6] Merzlyakov,Yu.I. Linear groups. In: Itogi Nauki i Tkhniki, Algebra Topologiya, Geometriya, vol.16, 1978, pp.35-89.

[7] Murach,M.M. (1976). Some generalized FC groups of matrices. Ukr. Math. Journal 28, 92-97.

[8] Pinnock,C.J.E. Supersolubility and finitary groups. Ph.D. Thesis. University of London, Queen Mary and Westfield College, 2000, London.

[9] Platonov,V.P.(1967). Linear groups with identical relations. Dokl.Akad. Nauk BSSR [in Russian] 11 581-582.

[10] Robinson,D.J.S. Finiteness conditions and generalized soluble groups, Springer-Verlag, 1972, Berlin.

[11] Robinson,D.J.S. and Wilson,J. (1984). Soluble groups with many polycyclic quotients. Proc. London Math.Soc. 48, 193-229.

[12] Suprunenko,D.A. Groups of Matrices. [in Russian] Nauka, 1972, Moscow.

[13] Wehrfritz,B. A. F. Infnite linear groups. Springer, 1973, Berlin.

Mathematics Department, University of Naples, Office n.17, via Cinthia 80126, Naples, Italy; Phone: +39-081-675682; Fax: +39-081-7662106

mailto:francesco.russo@dma.unina.it

Groups Elementary Equivalent to a Free 2-nilpotent Group of Finite Rank

Mahmood Sohrabi

In this talk I describe our joint work with Alexei Miasnikov on characterization of groups elementary equivalent to a free nilpotent group of arbitrary finite rank and class two. I’ll outline a proof that a characterization similar to the one Belegradek, O. V. [1] found for groups elementary equivalent to the group of 3×3 unitriangular matrices over the rings of integers, which is a free 2-nilpotent group of rank 2, can be given for groups elementary equivalent to a free 2-nilpotent group of finite rank.

References

[1] Belegradek, O. V. Model theory of unitriangular groups. Model theory and applications, 1-116, Amer. Math. Soc. Transl. Ser. 2, 195, Amer. Math. Soc., Providence, RI, 1999.

Carleton University

mailto:msohrabi@math.carleton.ca

The elementary theory of Thompson’s group F is undecidable

Vladimir Tolstykh

(joint work with Valery Bardakov)

M. Sapir [3] asked whether the elementary theory of Thompson’s group F is decidable (see [1] for the background information on Thompson’s groups.) Since the group F is riddled with copies of the restricted wreath product Z wr Z, he asked later whether there is a first-order definable (with parameters) copy of Z wr Z in F. Note that the elementary theory of the group Z wr Z is hereditarily undecidable, for the group Z wr Z first-order interprets the ring of integers Z [2].

In our talk we shall explicitly construct a subgroup of F which is isomorphic to Z wr Z and which is first-order definable in F with certain parameters, thereby showing that the elementary theory of Thompson’s group F is hereditarily undecidable.

References

[1] J. Cannon, W. Floyd, and W. Parry, Introductory notes to Richard Thompson’s groups, Enseign. Math. 42 (1996) 215–256.

[2] G. Noskov, The elementary theory of a finitely generated almost solvable group, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983) 498–517.

[3] Problem list of the workshop ‘Thompson’s Group at 40 Years’ (Jan. 11-14 2004, American Institute of Mathematics, Palo Alto, California), available at http://aimath.org/WWN/thompsonsgroup/thompsonsgroup.pdf

Yeditepe University, İstanbul

mailto:vtolstykh@yeditepe.edu.tr

On Conjugacy Classes of Finite Groups with a regard towards Groups of Squarefree Order

Anna Torstensson

Conjugacy classes are a vital tool in the analysis of groups. A condensed piece of information about a group is its class number, that is its number of conjugacy classes. Starting with small numbers mathematicians have tried to catalogise all groups having a given class number. In my talk I will give brief history of this problem. The starting point was when Landau in 1903 managed to show that there is only a finite number of non-isomorphic finite groups with class number k, and also derived an upper bound on the group size as a function of k. In [6] Poland extends the complete list of groups with class number k from k ⩽ 5 to k ⩽ 7. Using computerised methods this was later pushed forward to cover k ⩽ 9 (references can be found in [3]) and considering only simple groups further to k ⩽ 12 by Komissarcik in [3]. The reason to limit the search to simple groups was that for 5 ⩽ k ⩽ 9 the largest groups with a given class number are all simple. Hence one might guess that this is true also for larger k even though this seems to be unknown.

There has also been successful attempts to attack other types of groups. For example p-groups where it is known that a group of order p^{m} has (n + r(p − 1))(p^{2} − 1) + p^{e} conjugacy classes where r is some non-negative integer and m = 2n + e, e = 0 or 1 depending on the parity of m. This result can be found in [7].

For the rest of the talk we will go in the opposite direction and instead of p-groups consider groups of squarefree order. We will look at an explicit formula for the class numbers groups of a given squarefree order can have. As a by product of the analysis we will also see that a group of squarefree order having k conjugacy classes can have no more than k^{3} elements. There are bounds on the group order in the general case, but in contrast to this bound they are far from sharp. All these results are proved in detail in [1].

It is of course a substantial confinement to consider only groups of squarefree orders. It should be noted however that most group orders are covered, as lim_{n→∞} \frac{Q(n)}{n} = \frac{6}{π^{2}} , where Q(n) denotes the number of squarefree numbers ⩽ n. (See [2].) In addition the techniques developed here can hopefully be modified in order to cover other types of groups which have relatively simple presentations.

References

[1] A. Torstensson, On Conjugacy Classes of Groups of Squarefree Order, preprint Lund University 2006. Submitted for journal publication.

[2] G. H. Hardy and E. M. Wright, The Number of Squarefree Numbers in An introduction to the theory of Numbers 4th (corrected) ed. Clarendon Press (Oxford) 1962.

[3] E.A. Komissarcik, Simple Groups with fewer than thirteen conjugacy classes, Soviet Math. Dokl. 18(6) (1977), 1536–1539.

[4] M.R. Murty and V.K. Murty, On Groups of Squarefree Order, Math. Ann. 267 (1984), 299–309.

[5] M. Newman, A bound for the number of conjugacy classes in a group, J. London Math. Soc. 43 (1968), 108–110.

[6] J. Poland, Finite Groups with a given number of Conjugate Classes, Canad. J. Math. 20 (1968), 456–463.

[7] J. Poland, Two problems on Finite Groups with k Conjugate Classes, J. Austral. Math. Soc. 8 (1968), 49–55.

The Royal Institute of Technology, Stockholm, Sweden

mailto:annator@kth.se

http://www.math.kth.se/~annator/

Permutation invariance of binary codes and linearity of Reed-Muller codes over some rings

Bahattin Yıldız

By starting with an open conjecture from [2] in which the authors conjectured that the Reed-Muller codes RM(r, m) won’t be ℤ_{4}-linear for non-trivial cases, we try to answer the same question for the ring 𝔽_{2} + u𝔽_{2}. Linear codes over this ring have been studied in works like [3], [4] and [5]. In this work we will be proving that the Reed-Muller codes RM(r, m) are 𝔽_{2} + u𝔽_{2}-linear contrary to the case of ℤ_{4}. In fact we will generalize this further to some other rings.

The tools that we will be using are going to be relating 𝔽_{2} + u𝔽_{2}-linearity as well as linearity over these other rings that we will consider of binary codes with permutation invariance where we consider some suitable permutation group in each case. These will help us understand these linear codes better.

A linear code over a ring R of length n is just an R-submodule of R^{n} and a permutation acts on a linear code by switching the coordinates. In this aspect linear codes are algebraic objects and while proving our results we will just use these aspects of them.

May 22–27, 2007  4 POSTERS

The main rings to be used are going to be 𝔽_{2} + u𝔽_{2} which is constructed via u^{2} = 0, 𝔽_{2} + u𝔽_{2} + u^{2}𝔽_{2} + u^{3}𝔽_{2} which is constructed via u^{4} = 0 and its generalizations. We will also be considering the ring 𝔽_{2} + u𝔽_{2} + v𝔽_{2} + uv𝔽_{2} which is constructed via u^{2} = v^{2} = 0, uv = vu. We will demonstrate that being linear over the ring 𝔽_{2} + u𝔽_{2} + u^{2}𝔽_{2} + ⋯ + u^{2^{k}−1}𝔽_{2}of a binary code of length 2^{k}n is equivalent to being invariant under the permutation group ℤ_{2^{k}} and we will also show that linearity over the ring 𝔽_{2} +u𝔽_{2} +v𝔽_{2} +uv𝔽_{2} of a binary code of length 4n means being invariant under the permutation group K_{4}, where K_{4} denotes the Klein-4 group. After these results we will finally consider the case of Reed-Muller codes RM(r, m) and we will answer the question posed by the authors in [2] for the rings that we have considered above.

References

[1] B. Yildiz, Weight Enumerators and Gray maps of Linear codes over rings, Thesis, California Institute of Technology, (2006)

[2] A.R. Hammons, V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, The mathbbZ_{4}-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory,

[3] S.T. Dougherty, P. Gaborit, M. Harada, A. Munemasa and P. Solé, Type IV self-dual codes over rings, IEEE Trans. Inform. Theory, 45 (1999), 2345–2360.

[4] S.T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over 𝔽_{2} + u𝔽_{2}, IEEE Trans. Infrom. Theory, 45 (1999), 32–45.

[5] K. Betsumiya, S.Ling, F.R. Nemenzo, Type II Codes over 𝔽_{2^{m}} + u𝔽_{2^{m}}, Discrete Math. textbf275 (2004), 43–65.

Department of mathematics, Fatih University

mailto:byildiz@fatih.edu.tr

4 Posters

On the strong Hadamard exponential and logarithm functions

Haci Civciv

This is joint work with Ramazan Türkmen. The strong Hadamard product is considered a powerful matrix multiplication tool for Hadamard and other orthogonal matrices from combinatorial theory. Also, it is well known that the matrix versions of the familiar real and complex exponential and logarithm functions are fundamental for the study of many aspects of matrix group theory. In this note, we first introduce the strong Hadamard exponential and logarithm functions of some special matrices, and then investigate some properties of these functions for these matrices.

References

[1] J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. on circuit and systems, 25, 772-781, 1978.

4 POSTERS  AAD IX

[2] A. Baker, Matrix groups; An introduction to Lie group theory, Springer-Verlag London Ltd., France, 2002.

[3] A. K. Feyzioglu, A course on algebra, Boğaziçi University Publication, İstanbul, Turkey, 1990.

[4] A. Graham, Kronecker products and matrix calculus with applications, Ellis Horwood Ltd., Chichester, England, 1981.

[5] R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991.

[6] F. Zhang, Matrix Theory; Basic results and techniques, Sipringer-Verlag, New York, 1999.

[7] W. D. Launey and J. Seberry, The strong Kronecker product, Journal of Combinatorial Theory, Series A, 66(2), 192-213, 1994.

Department of Mathematics, Faculty of Art and Science, Selcuk University, Konya, Turkey

mailto:hacicivciv@selcuk.edu.tr

Rational endomorphisms of a nilpotent group

Bounabi Daoud

Let G be a group. A function f : G → G is said to be rational if there exist elements a_{1}, …, a_{r}, a_{r+1} ∈ G and integers h_{1}, …, h_{r} ∈ ℤ such that

f(x) = a_{1}x^{h_{1}}a_{2}x^{h_{2}} ⋯ a_{r}x^{h_{r}}a_{r+1} for all x ∈ G.

If the exponents h_{1}, …, h_{r} can be chosen to be positive, we say that f is polynomial; it is the case when for example the group G is of finite exponent. The set F (G) of all rational functions of G forms a monoid with the usual composition of functions, this monoid has two important submonoids, F(G) the set of rational functions f such that f(1) = 1, and End_{r}(G) the set of rational functions which are endomorphisms. It is obvious that End_{r}(G) ⊆ F(G). We denote by F^{∗}(G), F^{∗}(G), and End_{r}^{∗}(G), the group of invertible elements of the corresponding monoids. Note that a bijective rational function of a group G is not necessarily in F^{∗}(G) (e.g., consider G = (0, ∞) with multiplication and f : x ↦ x^{3}). On the other hand, if G is finite, then F^{∗}(G) = Sym(G) ∩ F (G), End_{r}^{∗}(G) = Sym(G)∩End_{r}(G), where Sym(G) is the set of all bijections on G.

Following [2], we denote by J(G) the set of integers m such that there exist t integers m_{1}, …, m_{t} (t ⩾ 1) and t elements c_{1}, …, c_{t} of G such that m = m_{1} + ⋯+m_{t} and c_{1}x^{m_{1}} ⋯ c_{t}x^{m_{t}} = 1 for all x ∈ G. The set J(G) is an ideal of the ring of integers ℤ (see [2, p. 431]). Let m ⩾ 0 be the integer such that J(G) = (m) and f ∈ F (G), where f(x) = a_{1}x^{h_{1}} ⋯ a_{r}x^{h_{r}}a_{r+1}. Put h = h_{1} + ⋯ + h_{r} and h_{0} ∈ {0, …, m − 1} is the unique integer such that h_{0} − h ∈ J(G). If h_{0} = h + qm for some integer q, then f(x) = f(x)(c_{1}x^{m_{1}} ⋯ c_{t}x^{m_{t}})^{q}. Thus we may assume that h ∈ {0, …, m − 1} and we call h̄ the degree of f. If f(1) = 1 (i.e., f ∈ F(G)), then

f(x) = x^{h} ∏_{i=1}^{r}[x^{h_{i}}, b_{i}][x^{h_{i}}, b_{i}, x^{h−(h_{1}+···+h_{i})}],

where b_{i} = (a_{1} ⋯ a_{i})^{−1} for all i ∈ {1, …, r}.

Endimioni [2] has proved that if G is nilpotent, F^{∗}(G) is a nilpotent-by-(finite abelian) group. In this paper we prove that

Theorem 1 If G is a group, we denote by F_{1}(G) the set of elements o F(G) of degree ¯1. Then, if G is nilpotent of class c ⩾ 2, we have

(i) F_{1}(G) is a normal subgroup of F^{∗}(G), and \frac{F^{∗}(G)}{F_{1}(G)} is isomorphic to the group of invertible elements of the ring \frac{ℤ}{J(G)}.

(ii) F_{1}(G) is nilpotent of class c − 1.

Schweigert [3, Satz 3.5, p.37] showed that if G is a finite group, then G is nilpotent if and only if End_{r}^{∗}(G) is nilpotent. Corsi Tani and Rinaldi Bonafede [1, Theorem 3.5, p. 288] improved Schweigert’s result by proving that if G is a finite group, then it is nilpotent of class c ⩾ 2 if and only if End_{r}^{∗}(G) is nilpotent of class c − 1. Note that if G is abelian, then End_{r}^{∗}(G) = F^{∗}(G) coincides with the group of universal power automorphisms which is contained in the centre of Aut(G), the automorphisms group of G. We generalize [1, Theorem 3.5, p. 288] for infinite groups, in fact we prove

Theorem 2 Let G be a group and c ⩾ 3 be an integer. Then G is nilpotent of class c if and only if End_{r}^{∗}(G) is nilpotent of class c − 1. Moreover, End_{r}^{∗}(G) is normal in Aut(G).

It was proved in [1, Theorem 5.2, p. 291] that if G is a finite group and r ⩾ 2 is a given integer, then the sum of any r inner automorphisms of G is an automorphism of G (i.e., the map x ↦ x^{a_{1}} ⋯ x^{a_{r}} is an automorphisms of G for any r elements a_{1}, …, a_{r} ∈ G) if and only if G is 2-Engel with the property that \binom{r}{2} ≡ 0 modG′ and gcd(r, exp(G)) = 1. Here we generalize this result to any group.

Let h be an integer. We say that a group G satisfies I(h) (E(h)) if the map x ⟼ (x^{a_{1}})^{h_{1}} … (x^{a_{r}})^{h_{r}} is an invertible rational endomorphism (epimorphism) of G for all a_{1}, a_{2}, …, a_{r} ∈ G and h_{1}, h_{2}, …, h_{r} ∈ ℤ such that h = h_{1} + ⋯ + h_{r}.

Theorem 3 Let G be a group and h an integer different of −1, 0, 1. Then G satisfies I(h) if and only if G is 2-Engel with conditions:

1) exp(G) is finite and gcd(h,exp(G)) = 1.

2) exp(G′) divides h(h − 1)/2.

If G satisfies E(h), then G is a 2-Engel group with the only property that exp(G′) is finite and divides h(h − 1)/2.

References

[1] G. Corsi Tani and M.F. Rinaldi Bonafede, Polynomial automorphisms in nilpotent finite groups, Boll. Un. Mat. Ital. A (6) 5 (1986), no. 2, 285-292.

[2] G. Endimioni, Applications rationnelles d’un groupe nilpotent, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 431-434.

[3] D. Schweigert, Polynomautomorphismen auf endlichen gruppen, Arch. Math. (Basel) 29 (1977), no. 1, 34-38.

University of Setif, Algeria

mailto:boun_daoud@yahoo.fr

On Cofinitely Weak Supplemented Modules

Yılmaz Mehmet Demirci

An R-module M is called weakly supplemented, if every submodule N of M has a weak supplement, i.e. a submodule K satisfying N + K = M and N ∩ K ≪ M. Semisimple modules and artinian modules are weakly supplemented. A submodule N of a module M is said to be cofinite if M/N is finitely generated. M is called a cofinitely weak supplemented (briefly cws) module if every cofinite submodule of M has a weak supplement. An R-module M is called locally noetherian if every finitely generated submodule of M is noetherian.

Theorem 1 Let M be a locally noetherian module with Rad M ≪ M. The following statements are equivalent.

(i) M is weakly supplemented.

(ii) Every cyclic submodule of M has (is) a weak supplement.

(iii) Every finitely generated submodule of M/ Rad M is a direct summand.

Theorem 2 For a locally noetherian module M, the following are equivalent.

(i) M is a cws-module.

(ii) M/X is a cws-module for some X ⊆ Rad M.

(iii) M/ Rad M is a cws-module.

This is joint work with Engin Büyükaşık, İzmir Institute of Technology, enginbuyukasik@iyte.edu.tr.

References

[1] R. Alizade, G. Bilhan, P.F. Smith, “Modules whose maximal submodules have supplements", Comm. Algebra, 29 (2001), pp. 2389-2405.

[2] R. Alizade, E. Büyükaşık, “Cofinitely weak supplemented modules”, Comm. Algebra, 31 (2003), pp. 5377-5390.

[3] F. W. Anderson, and K. R. Fuller, Rings and Categories of Modules. New York: Springer-Verlag, 1992.

[4] T.Y. Lam, A First Course in noncommutative rings. New York:Springer-Verlag, 2001.

[5] C. Lomp, “On semilocal modules and rings”, Comm. Algebra, 27 (1999), pp. 1921-1935.

[6] E. Matlis, “Decomposable modules”, Trans. Am. Math. Soc., 125 (1966), pp. 147-179.

[7] D.W. Sharpe, P. Vamos, Injective Modules. Cambridge University Press, 1972.

[8] R. Wisbauer, Foundations of Modules and Rings. London: Gordon and Breach Science Publishers, 1991.

[9] H. Zöschinger, “Invarianten wesentlicher Überdeckungen”, Math. Annalen, 237 (1978), pp. 193-202.

İzmir Institute of Technology, Department of Mathematics, Gülbahçe Köyü, İzmir

mailto:yilmazdemirci@iyte.edu.tr

http://www.iyte.edu.tr/~yilmazdemirci

On the Singular Elliptic Curves over Finite Fields

Betul Gezer

This is joint work with Osman Bizim and Ahmet Tekcan. We consider the rational points on singular elliptic curves over finite fields 𝔽_{p}. We give results concerning the number of the points on the singular elliptic curves, y^{2} = x^{3} and y^{2} = x^{3} + ax^{2} over 𝔽_{p} where p is a prime. We give the number of solutions to y^{2} = x^{3} and y^{2} = x^{3} + ax^{2} over 𝔽_{p} by means of quadratic residue character, and also for the curve y^{2} = x^{3} we use the cubic residue character. Also some results are given on the sum of x-and y-coordinates of the points (x, y) on these curves. We determine the structure of the group of the rational points and torsion points on these curves and finally we generalize the results concerning the number of points in 𝔽_{p} to 𝔽_{p^{n}}.

References

[1] T. L. Heath, Diophantus of Alexandria, Cambridge University Press, New York, 1910.

[2] A.W. Knapp, Elliptic curves, volume 40 of Mathematical Notes. Princeton Universty Press, Princeton, NJ, 1992.

[3] F. Lemmermeyer, Reciprocity Laws. From Euler to Eisenstein. Springer Monogr. in Math., Springer-Verlag, 2000.

[4] R. A. Mollin, Fundamental Number Theory with Applications. CRC Press, 1998.

[5] S. Schmitt and H. Zimmer, Elliptic Curves A Computational Approach, Walter de Gruyter, Berlin, 2003.

[6] J.H.Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.

[7] J.H.Silverman, and J.Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.

[8] L.C.Washington, Elliptic Curves, Number Theory and Cryptography. Chapman&Hall/CRC, 2003.

Uludag University, Faculty of Science, Department of Mathematics, Görükle 16059. Bursa-TURKEY.

mailto:betulgezer@uludag.edu.tr

Characters and Quasi-Permutation Representations of 2-Groups with |Ω_{2}(G)| = 16

Ghodrat Ghaffarzadeh

By a quasi-permutation matrix we mean a square matrix over the complex field ℂ with non-negative integral trace. Thus every permutation matrix over ℂ is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field ℚ, and let c(G) denote the minimal degree of a faithful representation of G by complex quasi-permutation matrices. It is easy to see that

c(G) ⩽ q(G) ⩽ p(G)

where G is a finite group. In [3], finite 2−groups G have been determined with the property | Ω_{2}(G) |= 16. We recall that Ω_{2}(G) =< x ∈ G : x^{4} = 1 >. In this paper, we study characters and quasi-permutation representations of this groups which are cyclic center and have the additional property that G is non-abelian and has a normal elementary abelian subgroup E of order 8. In [3], these groups have been determined in terms of generators and relations. Also, it is shown that G/E is either cyclic or a generalized quaternion Q_{2^{n}} of order 2^{n}, n ⩾ 3. If G/E is cyclic, then there exists one class of 2−groups [3, theorem 3.1], and if G/E is generalized quaternion group, then there exist two classes of 2−groups [3, theorems 3.3 and 3.4].

References

[1] H. BEHRAVESH. Quasi-permutation representations of p-groups of class 2. J.London Math. Soc. (2) 55 (1997) 251-260.

[2] C. W. CURTIS AND I. REINER. Representation theory of finite groups and associative algebras. Wiley-Interscience, New York, 1962.

[3] ZVONIMIR JANKO. Finite 2−Groups G with Ω_{2}(G) = 16. Glasnic J. Math., 2004.

University of Urmia, Urmia, Iran

mailto:q_gafarzadeh@yahoo.com

Bernoulli–Frey Elliptic Curves

İlker İnam

This is joint work with Aysun Yurttaş, Osman Bizim, and İ. Naci Cangül.

In this work, authors establish some connections between Bernoulli numbers and Elliptic curves. They form a new class of elliptic curves of Frey type by using Bernoulli numbers as coefficients and call this class Bernoulli-Frey elliptic curves. In the first part, they give a summary about Bernoulli numbers, then in the second part, they introduce elliptic curve theory which became more famous with the solution of Fermat’s Last Theorem. Finally, they give a definition of a class of elliptic curves called "Bernoulli-Frey Elliptic Curves". Also, they give a classification of rational points on Bernoulli-Frey elliptic curves and they give some properties of them.

References

[1] Carlitz, L. A Property of the Bernoulli Numbers, Amer. Math. Monthly, 67 (1960), no. 10, 1011-1012.

[2] Carlitz, L., The Staudt-Clausen theorem, Math. Mag., 34 (1961), 131-146.

[3] Silverman, J. H., The Arithmetic of Elliptic Curves, Springer-Verlag, (1986), ISBN 0-387-96203-4.

[4] Schoof, R., Counting points on elliptic curves over finite fields, Journal de Théorie des Nombres de Bordeaux, 7 (1995), 219-254.

[5] Merel, L., Arithmetic of Elliptic Curves and Diophantine Equations, Les XXemes Journees Arithmetiques (Limoges, 1997), J. Theor. Nombres Bordeaux 11 (1999), no. 1, 173-200.

[6] Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331.

[7] Kim, M. S., Kim, T., An Explicit Formula on the Generalized Bernoulli Number with order n. Indian J. Pure Appl. Math. 31 (2000), no. 11, 1455-1461.

[8] Kim, M. S., Son, J. W., On Bernoulli Numbers. J. Korean Math. Soc. 37 (2000), no. 3, 391-410.

[9] Koblitz, N., Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984.

[10] Ono, K., Euler’s concordant forms, Acta Arith. 78 (1996), no. 2, 101-123.

[11] Parshin, A. N., The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications, Seminaire de Theorie des Nombres, Paris, 1986-87, 299-312, Progr. Math., 75, Birkhauser Boston, MA, 1998.

[12] Kamienny, S., Some remarks on torsion in elliptic curves, Comm. Alg. 23 (1995), no. 6, 2167-2169.

[13] Carlitz, L., Kummer’s congruences for the Bernoulli numbers, Portug. Math., 19 (1960), 203-210.

[14] Washington, L. C., Elliptic Curves, Chapman & Hall/CRC, 2003.

[15] Inam, I., Bizim, O., Cangul, I., N., Rational Points on Frey Elliptic Curves y^{2} = x^{3} − n^{2}x, Adv. Studies in Contemporary Maths. 14 (2007), No:1, 69-76.

Uludağ University, Faculty of Science, Department of Mathematics, Görükle 16059, Bursa, TURKEY

mailto:inam@uludag.edu.tr

Application of polytope method to irreducibility of polynomials over ℤ_{p^{k}}

Fatih Koyuncu

Ostrowski observed that there is a relation between the factorization of a polynomial f ∈ F [x_{1}, …, x_{n}] for any field F and the integral decomposition, with respect to Minkowski sum, of the Newton polytope of f. Here, extending this result to some families of polynomials in R[x_{1}, …, x_{n}] for an arbitrary ring R, we have found infinitely many classes of multivariate polynomials which are irreducible over ℤ_{n} for any positive integer n, in particular, over ℤ_{p_{i}^{k}} for any prime number p_{i} and arbitrary positive integer k.

References

[1] Gao S. Absolute irreducibility of polynomials via Newton polytopes, Journal of Algebra 237 (2001), No.2 501-520.

[2] Ostrowski A.M. On multiplication and factorization of polynomials, I. Lexicographic orderings and extreme aggregates of terms, Aequationes Math. 13 (1975), 201-228.

Fatih Koyuncu

Muğla University, Department of Mathematics

mailto:fatih@mu.edu.tr

Centralizer of Engel elements in a group

Carmela Sica

This is joint work with G. Endimioni (Université de Provence, Marseille (Fr)).

Let G be a group and H a subgroup. Given some information about the centralizer C_{G}(H), what can we deduce about G? This type of problem is well-known and has been studied by a number of people, see for instance [1, 3, 5]. We are specially interested in the following result of Onishchuk and Zaĭtsev ([4]).

Proposition 1 Let 𝔛 denote the class of polycyclic groups, or the class of Černikov groups, or the class of minimax groups, or the class of groups of finite rank (in the sense of Prüfer). Let G be a nilpotent group, H a finitely generated subgroup. Then C_{G}(H) belongs to 𝔛 if and only if G belongs to 𝔛.

From now on, the rank of a group is defined in the sense of Prüfer. Thus a group G has finite rank if there exists an integer r such that every finitely generated subgroup of G can be generated with at most r elements; in this case, the least integer r satisfying this condition is the rank of G.

An example given by the authors shows that their result can be false if we consider a locally nilpotent group G. Let A = ⟨a_{1}⟩ × ⟨a_{2}⟩ × … × ⟨a_{n}⟩ × … be an infinite elementary abelian p-group and B = ⟨b⟩ be an infinite cyclic group. Consider the semidirect product G = A ⋊ B, with a_{1}^{b} = a_{1} and a_{i}^{b} = a_{n−1}a_{n} for n > 1. Then G is a metabelian locally nilpotent group, with an element, b, whose centralizer is polycyclic (thus minimax and of finite rank), and which is of infinite rank (so it is neither polycyclic, nor minimax). Notice that this example does not show that Proposition 1 fails for the class of Čhernikov groups.

In the paper [2], we aimed at obtaining similar results, for instance when G is soluble. In particular, we strengthen the theorem of Onishchuk and Zaĭtsev when 𝔛 is the class of polycyclic groups or the class of Čhernikov groups. Obviously, if we just assume that G is soluble, we must formulate some extra hypothesis about H. We can expect reasonably that C_{G}(H) exercises a strong influence on G when the elements of H are not very far (in some sense) from the central elements. That leads to the notion of Engel element, that can be considered as a generalization of the notion of central element. We recall that in a group G, an element y is said to be a left Engel element if for all x in G, there exists an integer m ⩾ 0 such that [x,_{m} y] = 1 (as usual, [x,_{m} y] is inductively defined by [x,_{0} y] = x and [x,_{m} y] = [x,_{m−1} y]^{−1}y^{−1}[x,_{m−1} y]y when m > 0). If m can be chosen independently of x, one says that y is a bounded left Engel element. We shall write L(G) for the set of left Engel elements of G, and \overline{L}(G) for the set of bounded left Engel elements.

Our first result generalizes Proposition 1 when 𝔛 is the class of polycyclic groups.

Theorem 1 Let G be a soluble group. Let H be a finitely generated subgroup contained in \overline{L}(G) and suppose that C_{G}(H) is polycyclic. Then G is polycyclic.

The example in [4] shows also that Theorem 2 fails if we replace \overline{L}(G) by L(G) in its statement. As we noticed above, Onishchuk and Zaĭtsev did not say if Proposition 1 is still true for locally nilpotent groups in which the centralizer is a Čhernikov group. Our next theorem gives a positive answer. Before stating this result, recall also that a radical group is a group having an ascending series with locally nilpotent quotients. The class of radical groups is very large, containing for instance all soluble groups and all locally nilpotent groups.

Theorem 2 Let G be a radical group. Let H be a finitely generated subgroup contained in L(G) and suppose that C_{G}(H) is a Čhernikov group. Then G is a soluble Čhernikov group.

Finally, we have proved that Proposition 1 holds for a periodic locally nilpotent group G when 𝔛 is the class of groups of finite rank (with this choice of 𝔛, we have seen that Proposition 1 fails for an arbitrary locally nilpotent group).

Theorem 3 Let G be a periodic locally nilpotent group. Let H be a finitely generated subgroup such that C_{G}(H) has finite rank. Then G has finite rank.

References

[1] G. Endimioni, Groupes dont le centralisateur d’un sous-groupe est d’ordre borné. Math. Ann. 291 (1991), 475–479.

[2] G. Endimioni and C. Sica, Centralizer of Engel elements in a group.

[3] E.I. Khukhro and V.D. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank. J. Algebra 301 (2006), 474–492.

[4] V.A Onishchuk and D.I. Zaĭtsev, On locally nilpotent groups with a centralizer that satisfies a finiteness condition. Ukrainian Math. J. 43 (1991), 1018–1021;

[5] V.P. Šunkov, On periodic groups with an almost regular involution. Algebra and Logic 11 (1972), 260–272.

Università di Salerno (Italy)

mailto:csica@unisa.it

The Connection Between Principal Indefinite Quadratic Forms and Elliptic Curves

Ahmet Tekcan

This is joint work with Betül Gezer and Osman Bizim. Let p be a prime number such that p ≡ 1(mod4), let 𝔽_{p} be a finite filed. Let F (x, y) = x^{2} + xy + \frac{1−p}{4}y^{2} be the principal indefinite binary quadratic form of discriminant Δ = p and let E : y^{2} = x^{3}+x^{2}+ \frac{1−p}{4}x be the corresponding elliptic curve over 𝔽_{p}. In this paper we prove that the order of E over 𝔽_{p} is p if p ≡ 1(mod8) or p+2 if p ≡ 5(mod8). Further we obtain some formulas on the sum of x−and y−coordinates of the points (x, y) on E. Later, we consider the reduction of F. We prove that the reduction of F is R^{2}(F) = (1, 1 + 2j, j^{2} + j − 2k) if p ≡ 1(mod8) or R^{2}(F) = (1, 1 + 2j, j^{2} + j − 1 − 2k) if p ≡ 5(mod8), where k = \frac{p−1}{8} or \frac{p−5}{8}, respectively and j is any positive integer such that k ∈ A_{j}.

References

[1] Atkin A.O.L. and Moralin, F. Elliptic Curves and Primality Proving. Math. Comp. no: 203, 61(1993), 29–68.

[2] Buchmann, J. Algorithms for Binary Quadratic Forms. In preparation, accepted by Springer-Verlag.

[3] Buell, D.A. Binary Quadratic Forms, Clasical Theory and Modern Computations Springer-Verlag, New York, 1989.

[4] Flath, D.E. Introduction To Number Theory. Wiley, 1989.

[5] Goldwasser, S and Kilian, J. Almost all Primes Can be Quickly Certified. In Proc. 18th STOC (Berkeley, May 28-30, 1986), ACM, New York, (1986), 316-329.

[6] Koblitz, N. A Course in Number Theory and Cryptography, Springer-Verlag, 1994.

[7] Lenstra, H.W.Jr. Factoring Integers with Elliptic Curves. Ann. of Math. 3, 126(1987), 649-673.

[8] Miller, V.S. Use of Elliptic Curves in Cryptography, in Advances in cryptology-CRYPTO 85, Lect. Notes in Comp. Sci. 218, Springer-Verlag, Berlin (1986), 417-426.

[9] Mollin, R.A. An Introduction to Cryptography, Chapman&Hall/CRC, 2001.

[10] Mordell, L.J. On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees. Proc. Cambridge Philos. Soc. 21(1922), 179-192.

[11] Schoof, R. Counting Points on Elliptic Curves Over Finite Fields. Journal de Theorie des Nombres de Bordeaux, 7(1995), 219-254.

[12] Silverman, J.H. The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.

[13] Silverman, J.H. and Tate, J. Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.

[14] Washington, L.C. Elliptic Curves, Number Theory and Cryptography. Chapman&Hall/CRC, Boca London, New York, Washington DC, 2003.

[15] Wiles, A. Modular Elliptic Curves and Fermat’s Last Theorem. Ann. of Math., 141(3)(1995), 443-551.

Address: Uludag University, Faculty of Science, Department of Mathematics, Görükle 16059. Bursa-TURKEY.

mailto:tekcan@uludag.edu.tr

Weakly Supplemented Lattices

S. Eylem Toksoy

L will mean a complete modular lattice with greatest element 1. An element a in a lattice with zero is called an atom if 0 ≺ a, i.e. there is no element b ∈ L such that 0 < b < a. A lattice L is called semiatomic if 1 is a join of atoms in L. An element c of a complete lattice L is said to be compact if for every subset X of L with c ⩽ ⋁ X there exists a finite subset F of X such that c ⩽ ⋁ F. A lattice L is called a compact lattice if 1 is compact and a compactly generated lattice if each of its elements is a join of compact elements. We call an element a of L a supplement of an element b if a ∨ b = 1 and a is minimal with respect to this property. Minimality of a is equivalent to a ∧ b ≪ a/0. Reducing this equivalent condition to a ∧ b ≪ L, we get the definition of weak supplement. L is said to be supplemented ( respectively, weakly supplemented) if every element a of L has a supplement (respectively, weak supplement) in L. In a lattice L the intersection of all maximal (≠ 1) elements is called the radical of L, denoted by r(L).

Theorem 1 If a compactly generated lattice L is weakly supplemented then 1/r(L) is semiatomic.

Corollary 1 A compactly generated lattice L with r(L) = 0 is weakly supplemented if and only if it is semiatomic.

Corollary 2 Let L be a compactly generated lattice with small radical. Then L is weakly supplemented if and only if 1/r(L) is semiatomic.

Joint work with: Refail Alizade, İzmir Institute of Technology

mailto:rafailalizade@iyte.edu.tr

References

[1] Alizade, R., Büyükaşık, E.: Cofinitely supplemented modules, Comm. Algebra, Vol. 31, No. 11, pp. 5377-5390, (2003).

[2] Büyükaşık, E.: Weakly cofinitely supplemented modules. Master’s Thesis, DEÜ Fen Bilimleri Enstitüsü, İzmir, (2001).

[3] Büyükaşık, E.: Weakly and cofinitely weak supplemented modules over Dedekind rings. Ph.D. Thesis, DEÜ Fen Bilimleri Enstitüsü, İzmir, (2005).

[4] Cǎlugǎreanu, G.: Lattice Concepts of Module Theory, Kluwer Academic Publishers, Dordrecht, Boston, London (2000).

[5] Clark, J., Lomp. C., Vanaja, N., Wisbauer, R.: Lifting Modules. Frontiers in Mathematics, Birkhäuser, (2006).

[6] Lomp, C.: On dual Goldie dimension. Diplomarbeit (M.Sc. Thesis), Mathematischen Institut der Heinrich-Heine Universität, Düsseldorf. Revised version (2000).

İzmir Institute of Technology

mailto:eylemtoksoy@iyte.edu.tr

http://www.iyte.edu.tr/~eylemtoksoy

Locally Graded Quotients of Locally Graded Groups

Maria Tota

The presentation will be about a joint work with Akbar Rhemtulla (University of Alberta, Canada), on locally graded groups.

A group G is said to be locally graded if every nontrivial, finitely generated subgroup of G has a nontrivial finite image. The class L of locally graded groups is quite wide (locally soluble groups and locally residually finite groups belong to L) and it frequently appears in literature ([1], [2], [3], [4], and the papers cited in these) mainly in the study of groups that do not have infinite, finitely generated simple subgroups.

It can be easily seen that a subgroup of an L-group is an L-group but the class L is not closed under forming quotients, for free groups are locally graded thus every group can occur as a quotient of an L-group.

Hence, naturally arises the problem of looking for L^{QS}, the {Q-S}-closed interior of L, that is the largest subclass of L which is closed with respect to forming sections.

If we denote by L_{1} the class of all groups in which every finitely generated simple section is finite, it turns out that L_{1} = L^{QS}.

Moreover, we give conditions under which a quotient of a locally graded group is locally graded.

In fact, we generalize the result proved in [3], stating that if G belongs to L and N is a locally nilpotent, normal subgroup of G, then the factor group G/N still belongs to L.

Then, we focus on linear groups which are in particular locally graded and we get more precise conditions for a quotient of a linear group to be locally graded.

References

[1] N.S. Chernikov, D.Ya. Trebenko, Quotient groups of locally graded groups and of groups of some Kuros-Chernikov classes. Ukrainian Math. J. 50 (1998), no.11, 1765-1773 (1999).

[2] Y. Kim and A.H. Rhemtulla, On locally graded groups. Groups-Korea ’94 (Pusan), de Gruyter, Berlin (1995), 189-197.

[3] P. Longobardi, M. Maj, H. Smith, A Note on Locally Graded Groups. Rend. Sem. Mat. Univ. Padova, 94 (1995), 275-277.

[4] H. Smith, On homomorphic images of locally graded groups. Rend. Sem. Mat. Univ. Padova, 91 (1994), 53-60.

Università di Salerno (Italy)

mailto:mtota@unisa.it

http://www.dmi.unisa.it/people/tota/www/

On minimal non torsion-by-nilpotent and non (locally finite)-by-nilpotent groups

Nadir Trabelsi

Denote by N (respectively, N_{c}, F) the class of nilpotent (respectively, nilpotent of class at most c, finite) groups and let X stands for the class of torsion groups or the class of locally finite groups. If Ω is a class of groups, then a group is said to be minimal non-Ω if it is not an Ω-group, while all its proper subgroups belong to Ω. Many results have been obtained on minimal non-Ω, for various classes of groups Ω. In particular, in [1] it is proved that if G is an infinite finitely generated minimal non-N group, then G/F rat(G) is an infinite simple group, where F rat(G) stands for the Frattini subgroup of G. Also in [3] it is proved that if G is a finitely generated minimal non-FN group, then G is a perfect group which has no proper subgroup of finite index and G/F rat(G) is an infinite simple group. In this note we obtain a similar result on minimal non-X N groups and we prove that there are no minimal non-X N groups which are not finitely generated. More precisely, we prove the following result:

Theorem 1 If G is a minimal non-X N group, then G is a finitely generated perfect group which has no proper subgroup of finite index and G/F rat(G) is an infinite simple group.

Using a well known result of Zaïcev [4] which asserts that an infinite nilpotent group whose proper subgroups are in the class N_{c} is itself in N_{c}, we can see that Theorem 1 has the following consequence:

Corollary 1 If G is a minimal non-X N_{c} group, then G is a finitely generated perfect group which has no proper subgroup of finite index and G/F rat(G) is an infinite simple group.

Among minimal non-X N groups and minimal non-X N_{c} groups we can cite the example constructed by Ol’shanskii [2] of an infinite simple torsion-free finitely generated group whose proper subgroups are cyclic.

References

[1] M. F. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241-250.

[2] A. Y. Ol’shanskii, An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 1328-1393.

[3] M. Xu, Groups whose proper subgroups are finite-by-nilpotent, Arch. Math. 66 (1996), 353-359.

[4] D. I. Zaicev, Stably nilpotent groups, Mat. Zametki 2 (1967), 337-346.

University of Setif - Algeria

mailto:nadir_trabelsi@yahoo.fr

Radical supplemented modules

E. Türkmen

This is joint work with A. Pancar, and F.Yüzbaşı. Let R be a ring , M a left R - module and U a submodule of M. It is known that V a submodule of M is called a supplement of U in M if U cap V << V with U + V = M. We define that V a submodule of M is called a Radical supplement of U in M if U cap V ⊆ Rad V with U + V = M. In this paper equality of the radical supplement is shown and various properties of both the radical supplement and the submodule of M has a radical supplement in M are given.Similary, we define that M is called radical supplemented if every submodule U of M has a radical supplement in M. Several properties of the radical supplemented modules are studied.

Theorem 1 Let M be an R - module and U, V ⩽ M. V is a Rad - supplement of U if and only if U + V = M and < m >= Rm << V, for each m ∈ U ∩ V.

Theorem 2 Let M be an R- modul and U ⩽ M . If U << M, then M is a Rad - supplement of U.

Lemma 1 Let M be an R- module and V be a Rad - supplement of U in M. If K << M, then K ∩ V ⊂ RadV.

Theorem 3 Let M be an R- module and V be a Rad - supplement of U in M. If RadM << M, RadV = V ∩ RadM.

Theorem 4 Let M be an R- module and V be a Rad - supplement of U in M. If L ⊂ U , then (V + L)/L is a Rad - supplement of U/L in M/L.

Theorem 5 Let M be an R - module and M_{1}, M_{2} ⩽ M. If M_{1} is Rad supplemented and M_{2} is hollow , then M_{1} + M_{2} has a Rad - supplement in M

Theorem 6 Let M be an R- module and K < L < M.If L is local and L/K is a Rad - supplement in M/K, then L is a Rad - supplement in M.

References

[1] R.Alizade ve A. Pancar. 1999 Homoloji Cebire Giriş, Ondokuz Mayi s Üniversitesi, Fen-Edebiyat Fakültesi, Samsun.

[2] R. Wisbauer. 1991.Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia.

[3] J. Clark , C. Lomp , N. Vajana , R. Wisbauer. 2006. Lifting Modules , Birkhauser Verlag Basel , Boston - Berlin.

[4] R. Wisbauer. 1996. Modules and Ring, Henrich Heine Universität Düsseldorf.

[5] H. Çalışıcı and A.Pancar. ⊕ - Cofinitely Supplemented Modules, Czechoslovak Mathematical Journal, 54 (129) (2004), 1083 - 1088.

[6] C. Nebiyev. and A. Pancar. On Π - Projective Modules, International Journal of Applied Mathematics, Volume 12 No.1 2003, 51-57.

[7] C. Nebiyev. and A. Pancar. On Amply Supplemented Modules, International Journal of Applied Mathematics Volume 12 No.3 2003, 213-220.

[8] C. Nebiyev. and A. Pancar. Strongly ⊕ - Supplemented Modules, International Journal of Computational Cognition Volume 2, Number 3, Pages 57-61.

[9] H. Çalışıcı and A.Pancar. Cofinitely Semiperfect Modules, Siberian Mathematical Journal Vol.46, No.2, pp. 359-363, 2005.

[10] H. Çalışıcı and A.Pancar. Finitely ⊕− Supplemented Modules, International Journal of Pure and Applied Mathematics, Volume 21 No.1 2005, 55-61.

[11] C. Lomp. On semilocal Modules and rings, Communications in Algebra, 27, No.4 (1999), 1921-1935

Ondokuz Mayıs University, Faculty of Sciences and Arts, Department of Mathematics, 55139 Kurupelit, Samsun, TURKEY

mailto:ergulturkmen@hotmail.com.tr

Some special matrix groups under the strong Hadamard product

Ramazan Türkmen

This is joint work with Haci Civciv. Matrix groups are central in many parts of mathematics and its applications, and the theory of matrix groups is ideal as an introduction to mathematics. On the one hand it is easy to calculate and understand examples, and on the other hand the examples lead to an understanding of the general theoretical framework that incorporates the matrix groups. However, the strong Hadamard product has a ring and very pleasing algebra that supports a wide range of fast, elegant and practical algorithms. Several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future. First, these product include signal processing, image processing, and quantum computing. Second, the strong Hadamard product is proving to be a very effective way to look at fast linear transforms. In this note, we examine some properties some specific matrix groups under the strong Hadamard multiplication.

References

[1] J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. on circuit and systems, 25, 772-781, 1978.

[2] A. Baker, Matrix groups; An introduction to Lie group theory, Springer-Verlag London Ltd., France, 2002.

[3] A. K. Feyzioglu, A course on algebra, Boğaziçi University Publication, İstanbul, Turkey, 1990.

[4] A. Graham, Kronecker products and matrix calculus with applications, Ellis Horwood Ltd., Chichester, England, 1981.

[5] R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991.

[6] W. H. Steeb, Matrix calculus and Kronecker product with applications and C + + programs, World Scientific Publishing, Co., Signapure.

[7] F. Zhang, Matrix Theory; Basic results and techniques, Sipringer-Verlag, New York, 1999.

Department of Mathematics, Faculty of Art and Science, Selcuk University, Konya, Turkey

mailto:rturkmen@selcuk.edu.tr

5 PARTICIPANTS  AAD IX

5 Participants

1. Mohammad Abbaspour mh_abbaspour@yahoo.com 2. Esen Aksoy eaksoy@su.sabanciuniv.edu 3. Hacı Aktaş haktas@gop.edu.tr 4. Emine Albaş emine.albas@ege.edu.tr 5. Rafail Alizade rafailalizade@iyte.edu.tr

6. Fatma Altunbulak fatma@fen.bilkent.edu.tr

7. Murat Altunbulak murat@fen.bilkent.edu.tr

8. Bernard Amberg Amberg@Mathematik.Uni-Mainz.DE 9. Nurcan Argac

10. Ahmet Arıkan arikan@gazi.edu.tr 11. Aynur Arıkan yalincak@gazi.edu.tr 12. Aykut Arslan aykutmath@yahoo.com 13. Okan Arslan oarslan@adu.edu.tr 14. A. O. Asar aliasar@gazi.edu.tr

15. Fırat Ateş firat@balikesir.edu.tr

16. Ela Aydın eaydin@cu.edu.tr 17. Gonca Ayık agonca@cu.edu.tr 18. Hayrullah Ayık hayik@cu.edu.tr 19. Selim Bahadır sbahadir@ug.bilkent.edu.tr

20. Alp Bassa bassa@sabanciuniv.edu 21. Serban Başarab Serban.Basarab@imar.ro 22. Houshang Behravesh hbehravesh@yahoo.com 23. M. Gökhan Benli gokhan@metu.edu.tr 24. Ayşe Berkman aberkman@metu.edu.tr 25. Cansu Betin cbetin@metu.edu.tr 26. Özlem Beyarslan ozlem@logique.jussieu.fr

27. Mehpare Bilhan bilhanm@metu.edu.tr 28. Osman Bizim obizim@uludag.edu.tr 29. Alexandre Borovik alexandre.borovik@manchester.ac.uk

30. Cathrine Barber Brown catherine.barber-brown@britannia.co.uk

31. Ken Brown kab@maths.gla.ac.uk 32. Roger Bryant roger.bryant@manchester.ac.uk 33. Jeff Burdges burdges@math.univ-lyonl.fr 34. Sermin Cam

35. Zoe Chatzidakis zoe@logique.jussieu.fr

36. Hacı Civciv hacicivciv@selcuk.edu.tr

37. İlke Çanakcı icanakci@dogus.edu.tr 38. Münevver Çelik e127417@metu.edu.tr 39. Ayça Çeşmelioğlu cesmelioglu@su.sabanciuniv.edu

40. A Sinan Çevik scevik@balikesir.edu.tr

41. Bounabi Daoud boun_daoud@yahoo.fr 42. Adrien Deloro adeloro@logique.jussieu.fr

43. Yılmaz Demirci yilmazdemirci@iyte.edu.tr 44. Yasha Diasamide diasamidze_ya@mail.ru 45. Semra Doğruöz sdogruoz@adu.edu.tr 46. Papatya Duman

May 22–27, 2007  5 PARTICIPANTS

47. Askar Dzhumadil’daev askar@math.kz 48. Şükrü Uğur Efem

49. Esra Egici

50. Naime Ekici nekici@cu.edu.tr 51. Gülin Ercan ercan@metu.edu.tr 52. Selami Ercan ercans@gazi.edu.tr 53. Ali Erdoğan alier@hacettepe.edu.tr

54. Sultan Erdoğan erdogan@fen.bilkent.edu.tr

55. Berna Ersoy

56. Kıvanç Ersoy kersoy@metu.edu.tr 57. Zerrin Esmerligil ezerrin@cu.edu.tr 58. Betül Gezer

59. Niyazi Anıl Gezer agezer@metu.edu.tr 60. Ghodrat Ghaffarzadeh q_gafarzadeh@yahoo.com 61. Haydar Göral

62. Rostislav Grigorchuk grigorch@math.tamu.edu 63. Aslı Güçlükan guclukan@fen.bilkent.edu.tr 64. Doğa Güçtenkorkmaz

65. İsmail Güloğlu

66. Cem Güneri guneri@sabaciuniv.edu 67. Derya Çıray Güven

68. Eylem Güzel eguzel@balikesir.edu.tr

69. Jonathan I Hall jhall@math.msu.edu 70. İlker İnam inam@uludag.edu.tr 71. Katarzyna Jachim kjachim@mimuw.edu.pl 72. Eric Jaligot jaligot@math.univ-lyon1.fr

73. Joanna Jaszunska joasiaj@mimuw.edu.pl 74. Marianne Johnson Marianne.Johnson@maths.manchester.ac.uk 75. Dilek Kahyalar dilekah@cu.edu.tr 76. Müge Kanuni muge.kanuni@boun.edu.tr 77. Semra Öztürk Kaptanoğlu sozkap@metu.edu.tr 78. Z. Yalçın Karataş zykaratas@hotmail.com 79. Özcan Kasal ozcankasal@yahoo.com 80. Canan Kaşıkcı canank@su.sabanciuniv.edu 81. Otto Kegel Otto.H.Kegel@t-online.de 82. Barış Kendirli bkendirli@fatih.edu.tr

83. Bilal Khan bkhan@jjay.cuny.edu 84. Melek Kılıç

85. Nayil Kılıç nayilkilic_61@yahoo.co.uk 86. Fatih Koyuncu fatih@mu.edu.tr 87. Feride Kuzucuoğlu feridek@hacettepe.edu.tr

88. Mahmut Kuzucuoğlu matmah@metu.edu.tr 89. Sandro Lashki lashkhi@spider.gtu.edu.ge

90. Felix Leinen Leinen@Mathematik.Uni-Mainz.DE 91. Vladimir Levchuk levchuk@lan.krasu.ru 92. James Lewis lewisjd@ualberta.ca 93. Annamaria Lucibello annamaria.lucibello@libero.it

94. Natalia Makarenko natalia_makarenko@yahoo.fr 95. Shota Makharadze

96. Wilfried Meidl wmeidl@sabanciuniv.edu 97. Ulrich Meierfrankenfeld meier@math.msu.edu 98. Ali Nesin anesin@bilgi.edu.tr 99. Aslı Nesin

100. Alexander Olshanskii aolshanskiy@gmail.com 101. Neslihan Ös neslihanos@yahoo.com 102. Hakan Özadam ozhakan@metu.edu.tr 103. Furuzan Özbek e132724@metu.edu.tr 104. Salahattin Özdemir salahattin.ozdemir@deu.edu.tr

105. Meltem Özgül meltemfinachine@yahoo.com 106. Zeynep Özkurt zyapti@mail.cu.edu.tr 107. Özer Öztürk ozero@metu.edu.tr 108. Erdal Özyurt eozyurt@adu.edu.tr 109. Ali Pancar

110. Athanassios Papistas apapist@math.auth.gr 111. Donald Passman passman@math.wisc.edu 112. Elisabetta Pastori pastori@math.unifi.it

113. Saliha Pehlivan saliha@su.sabanciuniv.edu 114. David Pierce dpierce@metu.edu.tr 115. Piotr Pragacz P.Pragacz@impan.gov.pl 116. Mike Prest mprest@maths.manchester.ac.uk 117. Stephen Pride sjp@maths.gla.ac.uk 118. Veronica C Quinonez vcrispin@kth.se 119. Peter Roquette roquette@uni-hd.de 120. Peter Rowley peter.j.rowley@manchester.ac.uk

121. Francesco Russo francesco.russo@dma.unina.it

122. Öznur Mut Sağdıçoğlu oznurmut@hotmail.com 123. Ali Sinan Sertöz sertoz@bilkent.edu.tr

124. Sezgin Sezer sezgin@cankaya.edu.tr 125. Anev Shalev shalev@math.huji.ac.il

126. Pavel Shumyatsky pavel@unb.br 127. Carmela Sica csica@unisa.it 128. Bilge Sipal

129. Howard Smith howsmith@bucknell.edu 130. Patrick Smith pfs@maths.gla.ac.uk 131. Mahmood Sohrabi msohrabi@math.carleton.ca 132. Ebru Solak ebrusolak@hotmail.com 133. Gökhan Soydan gsoydan@uludag.edu.tr 134. Henning Stichtenoth henning@sabanciuniv.edu 135. Ralph Stöhr Ralph.Stohr@manchester.ac.uk 136. Galina Suleymanova suleymanova@list.ru 137. Serkan Sütlü ssutlu@dogus.edu.tr 138. Nil Şahin nilsahin16@hotmail.com 139. Zekiye Şahin

140. Ahmet Tekcan tekcan@uludag.edu.tr 141. Ahmet Temizyürek tahmet@cu.edu.tr 142. Simon Thomas sthomas@math.rutgers.edu 143. S. Eylem Toksoy eylem.toksoy@gmail.com 144. Vladimir Tolstykh tvlaa@rambler.ru 145. Alev Topuzoğlu alev@sabanciuniv.edu 146. Anna Torstensson annator@kth.se 147. Antonio Tortora antortora@unisa.it 148. Maria Tota mtota@unisa.it 149. Nadir Trabelsi

150. Erkan Murat Türkan emturkan@gmail.com 151. Ergül Türkmen ergulturkmen@hotmail.com 152. İnan Utku Türkmen turkmen@fen.bilkent.edu.tr 153. Ramazan Türkmen

154. Pınar Uğurlu pugurlu@yahoo.com 155. Mecit Kerem Uzun mecituzun@yahoo.com 156. Yusuf Ünlü yusuf@cu.edu.tr 157. Şükrü Yalçınkaya ysukru@metu.edu.tr 158. Ergun Yalçın yalcine@fen.bilkent.edu.tr

159. Özcan Yazıcı ozcanyzc@gmail.com 160. Dilek Pusat Yılmaz dilekyilmaz@iyte.edu.tr 161. Babak Yousefzadeh babakyoosefzadeh@yahoo.com 162. Figen Yüzbaşı fyuzbasi@omu.edu.tr 163. Alexandre Zalesski alexandre@azalesski.wanadoo.co.uk

164. Efim Zelmanov ezelmano@tmo.blackberry.net

Scientific committee: E. Akyıldız, A. Borovik, R. Bryant, J. Hall, O. Kegel, C. Koç, M. Kuzucuoģlu, F. Leinen, V. Mazurov, U. Meierfrankenfeld, A. Nesin, D. Passman, P. Rowley, S. Sertöz, P. Shumyatsky, P. Smith, R. Stohr, S. Thomas, A. Topuzoğlu, A. Zalesskii

Organizing committee: Ayşe Berkman, Otto H. Kegel, M. Kuzucuoğlu, Victor D. Mazurov, David Pierce, Alexandre Zalesskii

Hospitality: Tivrona Tours

Financial support: TÜBİTAK (the Scientific and Technological Research Council of Turkey), the Turkish Mathematical Association, Bilgi University, and Middle East Technical University

TIMETABLE  AAD IX

Tue 22  Wed 23 Thu 24  Fri 25  Sat 26  Sun 27
9.00  Kegel  Stöhr  Hall  Thomas  Passman Meierfrankenfeld
10.00 P. Smith Olshanskii Borovik Grigorchuk  Lashki  Pragacz
11.00  coffee
11.30 Leinen Brown Rowley Stichtenoth Shumyatsky H. Smith
12.30  lunch
14.30 Kuzucuoğlu  Lewis  Shalev  Zelmanov
15.30  Bryant  Zalesski  Roquette  Pride
16.00  Papistas  Makarenko
16.30  coffee  coffee
17.00  Amberg  ∥ talks  ∥ talks  ∥ talks
17.30 Behravesh
18.00 reception
18.30  ∗

Moderators:

9.00 Ali Nesin  P. Smith Bilhan Shumyatsky  Hall  Bryant
14.30  Rowley  Sertöz  Stichtenoth  Zalesski
17.00 A  Asar  Stöhr  H. Smith  Güloğlu
17.00 B  Ünlü  Topuzoğlu  Alizade

∗ Friday, after dinner: memories of Brian Hartley, with his daughter, C. Barber-Brown

Parallel talks

17.00  17.30  18.00  18.30
Wed A  Yalçın  Yalçınkaya  †
B  Alizade  Pusat-Yılmaz Crispin Q.  Kaptanoğlu
Fri A  Asar  Suleymanova Diasamidze
B Çeşmelioğlu  Khan  Yıldız
Sat A  Levchuk  Ercan  ‡
B  Basarab  Güçlükan  Altunbulak Dzhumadil’daev

† Johnson, Arıkan, Torstensson, Russo ‡ Tolstykh, Deloro, Pastori,