
\titleofpresentation{Subgroups of finitary permutation groups}


\presenter{A. O. Asar}

This work continues the study of the subgroups of
totally imprimitive groups of finitary permutations which was
started in~\cite{asar1}. 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.

%\subsection{Introduction.} 
Let $G$ be a totally imprimitive subgroup of
$\operatorname{FSym}(\Omega)$, where $\Omega$ is infinite. In this work we
continue the study of the following problem which was started in~\cite{asar1}: 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~\cite[Theorem 1]{asar12} or~\cite[Lemma
8.3D]{asar3}). In
\cite{asar1}, 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~\cite[Lemma 2.5]{asar1} 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~\cite{asar12}. 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~\cite{asar2} and
\cite{asar6} 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
\cite{asar3,asar8,asar9} and~\cite{asar10}. 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
\[S_G(\Delta) = \{x\in G : supp(x)\subseteq \Delta\}.\]
It is easy to see that $S_G(\Delta)$ is a normal
subgroup of $G_{\{\Delta\}}$ (see~\cite[Lemma 2.3(a)]{asar1}). 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
\[\Delta_1\subset \Delta_2\subset\dots \subset\Delta_n\subset\dots\]
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
$\operatorname{FSym}(\Sigma)$, where $\Sigma = \{x(\Delta): x\in G\}$).\medskip

The main results of this work are stated below.

\begin{theorem}\label{asar1.1}
Let $G$ be a totally imprimitive subgroup of
 $\operatorname{FSym}(\Omega)$, where  $\Omega$ is infinite. Then the following
 hold:
\begin{enumerate}
\item
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 $N_G(W)\leq N_G(Y)$;
\item
$G$ is a minimal non $FC$-group if and only if any two
proper subgroups of $G$ generate a proper subgroup of $G$.
\end{enumerate}
\end{theorem}

\begin{theorem}\label{asar1.2}
Let $G$ be a totally imprimitive subgroup of
$\operatorname{FSym}(\Omega)$, where  $\Omega$ is infinite. Let $\Delta$ be
proper block for $G$ and let $S$ be a subnormal subgroup of
$G_{\{\Delta\}}$. Then  $\langle S^x: x\in G\rangle$ has finite
exponent if one of the following holds:
\begin{enumerate}
\item\label{asar1.2a}
$S$ is solvable;
\item
$S$ has finite exponent.
\end{enumerate}
\end{theorem}

Theorem \ref{asar1.2} \eqref{asar1.2a} generalizes~\cite[Theorem
1.1]{asar1}.  If  $G$ is a minimal 
non $FC$-group, then subnormality is not needed when $\exp(S)$ is
finite by~\cite[Remark 1.10]{asar1}. 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
 $\langle S^x: x\in G\rangle$ is
nilpotent of finite exponent by~\cite[Theorem 2.4(ii)]{asar8} or
\cite[Lemma 8.3D(ii)]{asar3}. 

\begin{theorem}\label{asar1.3} 
Let $G$ be a totally imprimitive $p$-subgroup
of $\operatorname{FSym}(\Omega)$, 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 $\operatorname{Core}_G(N_G{(G_{\{\Delta
\}})}\neq M$. In particular if $\Delta$ is non-trivial then
$\operatorname{Core}_G(G_{\{\Delta \}})\neq 1$.
\end{theorem}

(For more on the kernels the reader is referred to~\cite{asar7}.)

\begin{theorem}\label{asar1.4} 
Let $G$ be a totally imprimitive $p$-subgroup
of $\operatorname{FSym}(\Omega)$, 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.
\end{theorem}


\begin{corollary}\label{asar1.5} 
Let $G$ be a totally imprimitive $p$-subgroup of
$\operatorname{FSym}(\Omega)$, where $\Omega$ is infinite. Suppose 
that  $G$ has an ascending chain of non-trivial blocks
\[\Delta = \Delta_o\subset \Delta_1\subset \dots
\subset\Delta_n\subset\dots\]
whose union is equal to $ \Omega$ such that
for each $n\geq 0$ the following holds: 
\begin{itemize}
\item
$G_{\{\Delta_{n+1}\}}=
N_G(G_{\{\Delta_n\}})$ and 
\item
there exists a cycle $y_{n+1}\in
G_{\{\Delta_{n+1}\}}$ such that $G_{\{\Delta_{n+1}\}}=
\langle y_{n+1}\rangle G_{\{\Delta_n\}}$. 
\end{itemize}
Then $G$ contains a proper subgroup
that has an infinite orbit.
\end{corollary}

It can be shown that the hypotheses of Theorem \ref{asar1.4}
Corollary~\ref{asar1.5}
are satisfied by the example given in~\cite{asar12}.

The notation and the definitions are standard and may be found in
\cite{asar3} and~\cite{asar11} (see also~\cite{asar1}).


\begin{ourbiblio}{999}

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\bibitem{asar12}J. Wiegold,  `Groups of finitary permutations',
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\end{ourbiblio}

\presenteraddress{Gazi University, Gazi Egitim Faculty, 06500
Teknikokullar, Ankara, Turkey}

\email{aliasar@gazi.edu.tr}

