\titleofpresentation{Barely Transitive Groups}

\presenter{Mahmut Kuzucuo\u{g}lu}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

%  [Body of abstract goes here.]



\noindent

  A group $G$ has a  barely transitive representation if $G$ acts on an
infinite set $\Omega$ 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 $\left|G:H\right|$ is infinite,
$\bigcap_{x\in G}H^x= \{1\} $ and for every proper subgroup $K$ of
$G$, $\left|K:K\cap H\right|<\infty $: here $H$ denotes the stabilizer of a
point.


 The concept of a barely transitive
permutation group was introduced by Hartley in \cite{kuzucuogluhartley}, 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 $C_{p^\infty}$, 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 \cite{kuzucuogluhartley} see also
\cite[Question 15.20]{kuzucuoglukaurovka} is still not answered. Basic
properties of locally finite barely transitive groups are studied
in \cite{kuzucuoglu} We will give a survey on barely transitive
groups.



\begin{ourbiblio}{99}


\bibitem{kuzucuogluhartley}  B. Hartley,  The normalizer condition and barely-transitive permutation
groups,¡ Algebra Logika, 13, No. 5, 539-602 (1974).

\bibitem{kuzucuoglu} M. Kuzucuo\u{g}lu, {\em Barely transitive permutation
groups, } Arch. Math. 55, 521-532 (1990) .

\bibitem{kuzucuoglukaurovka}  Unsolved Problems in Group Theory
The Kourovka Notebook, 15th edition, Novosibirsk 2002;
\end{ourbiblio}

\presenteraddress{Department of Mathematics, Middle East Technical
University,
Ankara, 06531 Turkey}  % The name of a university is fine.

\email{matmah@metu.edu.tr}

\web{www.math.metu.edu.tr/people/mahmut_kuzucuoglu.shtml}  % Those without homepages should delete this.
