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\begin{document}
\title{Generating infinite symmetric groups%
\thanks{2000 Mathematics Subject Classifications.
Primary: 20B30.
% sym_gps
Secondary: 20B07, 20E15.
% inf gps chns of sgps
\protect\\
Preprint versions of this paper: %%
/\protect\linebreak[0]%
{$\!\sim$}gbergman%
/papers/Sym\<\_\,Omega:1.\{tex,dvi,ps\}
and arXiv:math.GR\protect\linebreak[0]/0401304\,.
}}
\author{George M. Bergman}
\maketitle
\begin{abstract}
Let $S=\Sym(\Omega)$ be the group of all permutations of an infinite
set $\Omega\<.$
Extending an argument of Macpherson and Neumann, it is shown that
if $U$ is a generating set for $S$ as a group, respectively
as a monoid, then there exists a positive integer $n$ such that
every element of $S$ may be written as a group word, respectively a
monoid word, of length $\leq n$ in the elements of $U.$
Some related questions and recent results are noted,
and a brief proof is given of a result of Ore's on commutators
that is used in the proof of the above result.
\end{abstract}
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\section{Introduction, notation, and some
lemmas on full moieties.}\label{S.moieties}
In \cite[Theorem~1.1]{DM&PN} Macpherson and Neumann show that if
$\Omega$ is an infinite set, then the group $S=\Sym(\Omega)$ is not
the union of a chain of $\leq|\<\Omega\<|$ proper subgroups.
We will repeat the beautiful proof of that result, with modifications
that will allow us to obtain along with it the result stated in the
abstract.
The present section is devoted to obtaining strengthened versions of
the lemmas used in that proof.
Following the notation of~\cite{DM&PN},
for $\Omega$ an infinite set, $\Sym(\Omega),$ generally
abbreviated $S,$ will denote the group of all permutations of $\Omega,$
and such permutations will be written to the right of their arguments.
For subsets $\Sigma\subseteq\Omega$ and $U\subseteq S,$
the symbol $U_{(\Sigma)}$ will denote
the set of elements of $U$ that stabilize $\Sigma$ pointwise, and
$U_{\{\Sigma\}}$ the set $\{f\in U:\Sigma f=\Sigma\}.$
(In~\cite{DM&PN} this notation was used only for $U$ a subgroup.)
A subset $\Sigma\subseteq\Omega$ will be called {\em full} with respect
to $U\subseteq S$ if the set of permutations of $\Sigma$
induced by members of $U_{\{\Sigma\}}$ is all of $\Sym(\Sigma).$
The cardinality of a set $X$ will be written $|X|,$
and a subset $\Sigma\subseteq\Omega$ will be called
a {\em moiety} if $|\Sigma|=|\<\Omega\<|=|\<\Omega-\Sigma\<|.$
\vspace{6pt}
Suppose $\Sigma_1$ and $\Sigma_2$ are moieties of $\Omega$ whose
intersection is also a moiety, and whose union is all of $\Omega\<.$
Then \cite[Lemma~2.3]{DM&PN} says that if $G$ is a subgroup of
$S=\Sym(\Omega)$ such that
$\Sigma_1$ and $\Sigma_2$ are both full with respect to $G,$ then $G=S.$
To strengthen this result, we will consider subsets
$U, V\subseteq S,$ closed under inverses, such that $\Sigma_1$ is full
with respect to $U$ and $\Sigma_2$ with respect to $V.$
By the lemma cited, $\langle\