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\title{ Subgroups of Finitary Permutation Groups }


\author{{\large A. O. Asar}}



\begin{document}

\maketitle

\abstract {}This work continues the study of the subgroups of
totally imprimitive groups of finitary permutations which was
started in [1]. Some characterizations of non $FC$-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.

\section{Introduction.} Let $G$ be a totally imprimitive subgroup of
$FSym(\Omega)$, where $\Omega$ is infinite. In this work we
continue the study of the following problem which was started in
[1]: Can $G$ contain a proper non $FC$-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 $FC$-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
$G^{\prime}$ is a minimal non $FC$-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 $FC$-group and in particular, $G$ is
a minimal non $FC$-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 $FC$-($p$-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 $G\leq Sym(\Omega)$. For
any non-trivial block $\Delta$ for  $G$ let

\begin{center}

 $S_G(\Delta) = \{x\in G : supp(x)\subseteq \Delta\}$.

\en{center} It is easy to see that $S_G(\Delta)$ is a normal
subgroup of $G_{\{\Delta\}}$ (See[1, Lemma 2.3(a)]). If
$S_G(\Delta)$ is not contained in $G_{\{\Pi\}}$ for any block
$\Pi\subset \Delta$, then $\Delta$ will be called a
$\bf{dominant}$ $\bf{block}$ for $G$.  Note that if $S_G(\Delta)$
is transitive on $\Delta$ then $\Delta$ is dominant. Let

\begin{center}

$\Delta_1\subset \Delta_2\subset\dots \subset\Delta_n\subset\dots$

\en{center} be an ascending chain of non-trivial blocks for $G$
whose union is $\Omega$. If each $\Delta_i$ is dominant, then this
chain is called a $\bf{dominant}$ $\bf{chain}$ for $G$. We also
let $Ker(\Delta)$ denote the kernel of the natural permutation
representation of $G$ with respect to $\Delta$(i.e, into
$FSym(\Sigma)$, where $\Sigma = \{x(\Delta): x\in G\}$).\medskip

The main results of this work are stated below.


\be{theorem}{1.1} Let $G$ be a totally imprimitive subgroup of
 $FSym(\Omega)$, where  $\Omega$ is infinite. Then the following
 hold:
\bl

\li a Let $W$ be a subgroup of $G$ and let $g\in G\setminus
N_G(W)$. Then $W\cap W^g$ is an $FC$-group. Thus if  $W$ is not an
$FC$-group and if $W\leq Y\leq G$, then $N_G(W)\leq N_G(Y)$;

\li b $G$ is a minimal non $FC$-group if and only if any two
proper subgroups of $G$ generate a proper subgroup of $G$.

 \el
\en{theorem}\medskip


\be{theorem}{1.2} Let $G$ be a totally imprimitive subgroup of
$FSym(\Omega)$, where  $\Omega$ is infinite. Let $\Delta$ be
proper block for $G$ and let $S$ be a subnormal subgroup of
$G_{\{\Delta\}}$. Then  $<S^x: x\in G>$ has finite exponent if one
of the following holds:

\bl

\li a $S$ is solvable;

\li b $S$ has finite exponent. \el

 \en{theorem}\medskip

Theorem 1.2(a) generalizes [1, Theorem 1.1].  If  $G$ is a minimal
non $FC$-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 $FC$-group then it is a $p$-group, and so $<S^x: x\in G>$ is
nilpotent of finite exponent by [8, Theorem 2.4(ii)] or [3, Lemma
8.3D(ii)].\medskip

\be{theorem}{1.3} Let $G$ be a totally imprimitive $p$ - subgroup
of $FSym(\Omega)$, where  $\Omega$ is infinite. Let $\Delta$ be a
finite block for $G$ and let M be the kernel of the representation
of $G$ with respect to $\Delta$. Then $Core_G(N_G{(G_{\{\Delta
\}})}\neq M$. In particular if $\Delta$ is non-trivial then
$Core_G(G_{\{\Delta \}})\neq 1$.

\en{theorem}(For more on the kernels the reader is referred to[7])


\be{theorem}{1.4} Let $G$ be a totally imprimitive $p$ - subgroup
of $FSym(\Omega)$, where  $\Omega$ is infinite. Suppose that $G$
has a non-trivial block  $\Delta$ such that any non-trivial block
for $G$ containing $\Delta$ is dominant. Then $G$ contains a
proper subgroup that has an infinite orbit.
 \en{theorem}
\medskip

\be{corollary}{1.5} Let $G$ be a totally imprimitive $p$ -
subgroup of $FSym(\Omega)$, where  $\Omega$ is infinite. Suppose
that  $G$ has an ascending chain of non-trivial blocks
\begin{center}

$\Delta = \Delta_o\subset \Delta_1\subset \dots
\subset\Delta_n\subset\dots$ \en{center}
\medskip whose union is equal to $ \Omega$ such that
for each $n\geq 0$ the following holds: $G_{\{\Delta_{n+1}\}}=
N_G(G_{\{\Delta_n\}})$ and there exists a cycle $y_{n+1}\in
G_{\{\Delta_{n+1}\}}$ such that $G_{\{\Delta_{n+1}\}}=
<y_{n+1}>G_{\{\Delta_n\}}$. Then $G$ contains a proper subgroup
that has an infinite orbit.

\en{corollary}

It can be shown that the hypotheses of Theorem 1.4 Corollary 1.5
are satisfied by the example given in [12].\medskip

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


\begin{thebibliography}{999}

\bibitem{1} {A.O. Asar, 'On finitary permutation groups'}, $Turk \
J. \ Mat. $  30 (2006)  101-116.

\bibitem{2} V. V. Belyaev, 'On the question of existence of minimal
non $FC$ - groups',  $Siberian Math. J.$ 39 (1998) 1093-1095.

\bibitem{3} J. D. Dixon and B. Moretimer, $Permutation \ Groups$ (Springer-Ferlag, New
York, 1996)

\bibitem{4} B. Hartley and M. Kuzucuoglu, 'Non-simplicity of
locally finite barely transitive groups', Proc. Edinburg Math.
Soc.(1997) 40, 483-490.
\bibitem{5} M Kuzucuoglu, 'Barely transitive permutation groups',
$Arch. \ Math.$ 55(1990) 521-532

\bibitem{6} F. Leinen, 'A reduction theorem for perfect locally
finite minimal non $FC$-groups', $Glasgow \ Math. \ J.$ 41 (1999)
81-83.

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

\bibitem{8} P. M.Neumann, 'The lawlessness of groups of finitary
permutations', $Arch. \ Math.$ 26 (1975) 561-566.

\bibitem{9} P. M.Neumann, 'The structure of finitary permutation
groups', $Arch.\ Math.$ 27 (1976) 3-17.

\bibitem{10} C.J.E Pinnock,  'Irreducible and transitive
locally-nilpotent by abelian groups', $Arch.\  Math.$ 74 (2000)
168-172.

\bibitem{11}D. J. S. Robinson,  $A \ Course \ in \ the \ Theory \ of \ Groups$, (Springer-Verlag,
New York, Heidelberg, Berlin, 1980)

\bibitem{12}J. Wiegold,  'Groups of finitary permutations', $Arch.\ Math.$ 25 (1974)
466-469(1974).

\end{thebibliography}

\medskip
A. O. Asar

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

e-mail: aliasar@gazi.edu.tr







\end{document}
