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\begin{document}
\title{Closed subgroups of the infinite symmetric group}
\subjclass[2000]{Primary: 20B07,
% inf perm gps
secondary: 22F50.}
% top_aut_gps (vs 22Fxx, "noncp tfmn gps"; 22F05 "gen th actns")
% 54E40 "special maps on metric spaces"
% 54H15 "tfn gps & semigps"
\keywords{full permutation group on a countably infinite set,
subgroups closed in the function topology,
equivalence relation on subgroups,
cardinalities of orbits of stabilizers of finite sets}
\thanks{
Second author's research supported by the United States--Israel
Binational Science Foundation.
\protect\\
\indent
A preprint version of this paper is readable at
\url{/~gbergman/papers/Sym_Omega:2.{tex,dvi}}\<,
at \url{http://shelah.logic.at}
as publication 823, and at \url{arXiv:math.GR/0401305}\,.
}
\author{George M. Bergman}
\address[G. Bergman]{University of California\\
Berkeley, CA 94720-3840, USA}
\email{gbergman@math.berkeley.edu}
\author{Saharon Shelah}
\address[S. Shelah]{University of Jerusalem\\
Einstein Institute of Mathematics\\
Jerusalem 91904, Israel\\
and
Hill Center-Busch Campus\\
Rutgers, The State University of New Jersey\\
Piscataway, NJ 08854-8019, USA}
\email{shelah@math.huji.ac.il}
\begin{abstract}
Let $S=\Sym(\Omega)$ be the group of all permutations of a countably
infinite set $\Omega,$ and for subgroups $G_1,\,G_2\leq S$ let us
write $G_1\approx G_2$ if there exists a finite set $U\subseteq S$
such that $\langle\_1}
Let $\Omega$ be an infinite set.
Then on subgroups of $S=\Sym(\Omega),$\\[2pt]
{\rm(i)} The binary relation
$\preq_{\strt\aleph_0}$ coincides with $\preq_{\strt\aleph_1}$
{\rm(}hence $\approx_{\strt\aleph_0}$ coincides with
$\approx_{\strt\aleph_1}).$\\[2pt]
{\rm(ii)} The unary relation $\approx_{\strt\aleph_0}S$
coincides with $\approx_{\strt|\<\Omega\<|^{^+}}S.$
\end{lemma}\begin{proof}
(i) follows from~\cite[Theorem~3.3]{FG}, which says that every countably
generated subgroup of $S$ is contained in a $2\!\<$-generator subgroup.
We claim that~(ii) is a consequence of \cite[Theorem~1.1]{DM&PN}
$(=$\cite[Theorem~5]{Sym_Omega:1}) which, as recalled
in \S\ref{S.Intro}, says that any chain of proper subgroups of
$S$ having $S$ as union must have $>|\<\Omega\<|$ terms.
For if $G\approx_{\strt|\<\Omega\<|^{^+}}S,$ then among
subsets $U\subseteq S$ of cardinality $\leq|\<\Omega\<|$ such that
$\langle\_1}(i)
lets us replace any countable $U$ by a set of cardinality~$2.$
Can we see this stronger assertion without calling on~\cite{FG}?
Yes, by a slight modification of the proof of our theorem:
We write the countable set $U$ as $\{f_1, f_2,\<\ldots\<\},$ and
replace each of the ``$\!1\!$''s in~(\ref{x.d(a,b)}) by a value
$N$ such that $\alpha_{i+1}=\alpha_i\0,$
\begin{xlist}\item\label{x.0-j.1,g-1}
$\{\<\eps_0,\ldots,\eps_{j-1\<}\}\,\cup\,
\{\<\eps_0\,g_{j-1}^{-1},\ldots,\eps_{j-1}\,g_{j-1}^{-1\<}\}\;
\subseteq\;\Gamma_j,$
\end{xlist}
and
\begin{xlist}\item\label{x.gjinS_*Gg}
$g_j\in S_{(\Gamma_j)}\;g_{j-1}.$
\end{xlist}
Then the sequence $(g_j)_{j=0,1,\ldots}$ converges in $S.$
\end{lemma}\begin{proof}
Let $i\geq 0.$
For all $j>i,$ the conditions $\eps_i\in\Gamma_j$
and~(\ref{x.gjinS_*Gg}) imply that $\eps_i\,g_j=\eps_i\,g_{j-1}.$
Thus, the sequence $(g_j)$ is eventually constant on $\eps_i.$
Likewise,~(\ref{x.gjinS_*Gg}) and the condition
$\eps_i\,g_{j-1}^{-1}\in\Gamma_j$ imply that
$\eps_i\,g_j^{-1}=\eps_i\,g_{j-1}^{-1};$ hence the element
of $\Omega$ carried to $\eps_i$ by $g_j$ is the same for all $j>i.$
Since the first conclusion holds for all $\eps_i\in\Omega,$
the sequence $(g_j)_{j=0,1,\ldots}$ converges to an element
of $\Omega^\Omega,$ which is one-to-one because all the $g_j$ are.
Applying the second conclusion, we see that each
$\eps_i$ is in the range of $g,$ so $g\in\Sym(\Omega).$
\end{proof}
We note some elementary
facts about closures of subgroups in the function topology.
\begin{lemma}\label{L.clG_*G}
Suppose $\Omega$ is a set and $G$ a subgroup of $S=\Sym(\Omega).$
Then
\\[2pt]
{\rm(i)}~ $\cl(G)$ is also a subgroup of $S.$
\\[2pt]
{\rm(ii)}~ $G$ and $\cl(G)$ have the same orbits in $\Omega\<.$
\\[2pt]
{\rm(iii)}~ If $\Gamma$ is a finite subset of $\Omega,$ then
$\cl(G)_{(\Gamma)}=\cl(G_{(\Gamma)}).$
\end{lemma}\begin{proof}
Statement~(i) is an immediate consequence of the continuity
of the group operations.
From the characterization of the closure of a
set in our topology, we see that for $\alpha,\,\beta\in\Omega,$
the set $\cl(G)$ will contain elements carrying $\alpha$
to $\beta$ if and only if $G$ does, from which~(ii) is clear.
The direction $\supseteq$ in~(iii) follows by applying~(ii) to the
orbits of elements of $\Gamma.$
(Finiteness of $\Gamma$ is not needed for this direction.)
To get $\subseteq,$ assume $f\in\cl(G)_{(\Gamma)}.$
Since $f\in\cl(G),$ every neighborhood of $f$ contains elements of $G.$
But as $f$ fixes all points of the finite set $\Gamma,$ every
sufficiently small neighborhood of $f$ consists of elements which do
the same, hence every such neighborhood contains points
of $G_{(\Gamma)};$ so $f\in\cl(G_{(\Gamma)}).$
\end{proof}
The above lemma has the consequence that once we show
(for $\Omega$ countable) that
the $\approx_{\aleph_0}\!$-class of a closed subgroup of $\Sym(\Omega)$
is determined by which of conditions~(i)--(iv) in our abstract
hold, we can also say for an arbitrary subgroup $G\leq\Sym(\Omega)$
that the $\approx_{\aleph_0}\!$-class of $\cl(G)$ is determined
in the same way by which of those conditions $G$ satisfies.
The subgroups of $\Sym(\Omega)$ closed in the function topology are
known to be precisely the automorphism groups of the finitary relational
structures on $\Omega\<.$
(Indeed, one may take the $n\!$-ary relations in such a
structure, for each $n,$ to be all orbits of
$n\!\<$-tuples of elements of $\Omega$ under the group.)
But we shall not make use of this fact here.
(Incidentally, $\Sym(\Omega)$ is also not {\em open} in $\Omega^\Omega.$
It is easy to give a sequence of non-injective or non-surjective
maps in which the failures of injectivity or surjectivity ``drift off
to infinity'', so that the limit is a bijection, e.g., the identity.)
\section{Infinite orbits.}\label{S.inf_orbs}
In this and the next three sections (and with minor exceptions,
in subsequent sections as well), we shall restrict attention to
the case of countable $\Omega\<.$
When an enumeration of its elements is required, we shall write
\begin{xlist}\item\label{x.enumerate}
$\Omega\ =\ \{\<\eps_i\;:\;i\in\omega\<\}.$
\end{xlist}
References to limits etc.\ in $S=\Sym(\Omega)$ will always refer to the
function topology; in particular, a closed subgroup
of $S$ will always mean one closed in $S$ under that topology.
The symbols $\preq$ and $\approx$ will mean $\preq_{\strt\aleph_0,S}$
and $\approx_{\strt\aleph_0,S}$ respectively.
We shall show in this section that if $G$ is a closed subgroup
of $S$ such that
\begin{xlist}\item\label{x.inf_orbs}
For every finite subset $\Gamma\subseteq\Omega,$ the subgroup
$G_{(\Gamma)}$ has at least one infinite orbit in $\Omega,$
\end{xlist}
then $G\approx S.$
Our proof will make use of the following result of
Macpherson and Neumann:
\begin{xlist}\item\label{x.M&N.full}
\cite[Lemma~2.4]{DM&PN} (cf.\ \cite[Lemma~3]{Sym_Omega:1}):~
Suppose $\Omega$ is an infinite set and $H$ a subgroup of
$\Sym(\Omega),$ and suppose there exists a
subset $\Sigma\subseteq\Omega$ of the same cardinality as $\Omega,$
such that $H_{\{\Sigma\}}$ (i.e., $\{\i$
in $\omega$ such that $g$ carries $\{0,\ldots,j\<{-}\<1\}$ to itself.
Then every permutation $f$ of $\omega$ is a product
$gh$ of two local permutations.
\end{lemma}\begin{proof}
Given $f\in\Sym(\omega),$ let us choose integers
$0=a(0)a(i{-}1)$ such that $\{0,\ldots,a(i{-}1){-}1\}\,f\linebreak[0]
\,\cup\,\{0,\ldots,a(i{-}1)\<{-}\<1\}\,f^{-1}
\subseteq\{0,\ldots,a(i)\<{-}\<1\}.$
Let $\Sigma_{-1}=\varnothing,$ and for $i\geq 0$ let
$\Sigma_i=\{a(i),\,a(i){+}1,\,\ldots,\,a(i{+}1){-}1\}.$
Thus the set $A=\{\Sigma_i:i\geq 0\}$ is a
partition of $\omega$ into finite subsets, such that for each $i\geq 0$
one has $\Sigma_i f\subseteq\Sigma_{i-1}\cup\Sigma_i\cup\Sigma_{i+1}.$
Note that for each $i\geq 0,$ the number of elements
which are moved by $f$ from $\Sigma_{i-1}$ into $\Sigma_i$ is equal to
the number that are moved from $\Sigma_i$ into $\Sigma_{i-1}$
(since these are the elements of $\omega$
that are moved ``past $a(i)-1/2$'' in the upward, respectively
the downward direction).
We shall now construct a permutation $g$ such that $g$ carries each set
$\Sigma_{2i}\<\cup\<\Sigma_{2i+1}$ $(i\geq 0)$ into itself, and
$g^{-1} f$ carries each set $\Sigma_{2i-1}\<\cup\<\Sigma_{2i}$
$(i\geq 0)$ into itself; thus each of these permutations will
be local, and they will have product $f,$ as required.
To do this let us, for each $i,$ pair elements $\alpha$ that $f$ carries
from $\Sigma_{2i}$ upward into $\Sigma_{2i+1}$ with elements $\beta$
that it carries from $\Sigma_{2i+1}$ downward into $\Sigma_{2i}$
(having seen that the numbers of such elements are equal), and let $g$
exchange the members of each such pair, while fixing other elements.
Clearly $g$ preserves the sets $\Sigma_{2i}\<\cup\<\Sigma_{2i+1}.$
It is not hard to verify that if we now look at $\omega$ as
partitioned the other way, into the intervals
$\Sigma_{2i-1}\<\cup\<\Sigma_{2i},$ then the $g$
we have constructed has the property that for every $\alpha\in\omega,$
the element $\alpha\0,$ we fix arbitrarily an order in which the
elements of $K_j$ are to be constructed.
When it is time to construct $g(k_0,k_1,\ldots,k_{r-1}),$
let us write $g'=g(k_0,k_1,\ldots,k_{r-2}),$ noting
that this is a member of $K_{j-k_{r-1}-1},$ hence already defined.
We will take for $g(k_0,k_1,\ldots,k_{r-1})$ the result of
left-multiplying $g'$ by a certain element $h\in G_{(\Gamma_{r-1})}.$
Note that whatever value in this group we choose for $h,$
the images of $\alpha_0,\ldots,\alpha_{r-2}$ under $h\_1}(i), and the final strict inequality by the last
sentence of Theorem~\ref{T.no_uncrwdd}.
Thus $G\not\approx S.$
\end{proof}
{\em Notes on the development of the above theorem:}
In the proof of Lemma~\ref{L.D_i}, and again in the arguments following
that proof, it might at first appear that our hypotheses
that certain subsets of $\Omega$ were infinite (namely,
in the former case, the set of ``next terms'' extending each member
of $D_i,$ and in the latter, at least one orbit of
$G_{(\Gamma)}$ for each finite set $\Gamma)$
could have been replaced by statements that those sets could be taken
to have large enough finite cardinalities, since at each
step, we had to make only finitely many choices from these sets,
and to avoid only finitely many elements of $\Omega\<.$
But closer inspection shows that we dipped into
these sets for additional elements infinitely many times.
In the proof of Lemma~\ref{L.D_i}, this is because
for fixed $n_0,\dots,n_{r-1}$ there are
infinitely many possibilities for $n_r>n_{r-1},$ and for {\em each}
of these, the construction of $E_{n_r}$ requires extending the elements
$e(n_0,\ldots,n_{r-1};\,\pi_1,\ldots,\pi_{r-1})\in D_{n_{r-1}}$ to
elements of $D_{n_{r-1}+1}.$
Likewise, in (\ref{x.g()}), note that $r\leq j,$ and
each value of $r$ comes up for infinitely many $j,$ so that for each
$r$ we must choose, in the long run, elements of $G_{(\Gamma_{r-1})}$
having infinitely many different effects on $\alpha_{r-1}.$
This spreading out of the choices we made from each infinite
set, into infinitely many clumps of finitely many choices each,
was necessary: If we had made infinitely many choices at one time
from one of our sets, we would have had infinitely many obstructions
to our choices from the next set, and could not have argued that
those choices could be carried out as required.
Could the two very similar recursive constructions just referred
to have been carried out simultaneously?
In an earlier draft of this note they were.
That arrangement was more efficient (if less transparent as to
what was being proved), and could be considered preferable if
one had no interest except in closed subgroups.
However, the present development yields the
intermediate result Lemma~\ref{L.D_i},
which can be used to show the $\approx\!\<$-equivalence to $S$
of many non-closed subgroups for which, so far as we can see,
Theorem~\ref{T.inf_orbs} is of no help.
For example, consider a partition $A$ of $\Omega$ into a countably
infinite family of countably infinite sets $\Sigma_i,$ and let
$G$ be the group of permutations of $\Omega$ that, for each $i,$ carry
$\Sigma_i,$ into itself, and move only finitely many elements
of that set.
If we choose an arbitrary element $(\alpha_i)\in\prod_i\<\Sigma_i,$
and let $D_i=\Sigma_0\times\ldots\times\Sigma_{i-1}$ for each $i,$
then we see easily that the conditions of Lemma~\ref{L.D_i} hold,
hence that $G\approx S.$
(The same argument works for the subgroup of the above $G$ consisting
of those elements $g$ for which there is a bound independent
of $i$ on the number of elements of $\Sigma_i$ moved by $g.)$
\section{Finite orbits of unbounded size.}\label{S.unbdd_orbs}
In this section, we again let $\Omega$ be a countably infinite set,
and will show that all closed subgroups $G\leq S=\Sym(\Omega)$ for which
\begin{xlist}\item\label{x.unbdd_orbs}
There exists a finite subset $\Gamma\subseteq\Omega$ such that all
orbits of $G_{(\Gamma)}$ are finite, but no such $\Gamma$ for
which the cardinalities of these orbits have a common finite bound,
\end{xlist}
are mutually $\approx\!\<$-equivalent.
The approach will parallel that of the preceding section, but
there are some complications.
First, there is not one natural subgroup that represents this
equivalence class, as $S$ represented the class considered
in the previous section.
Instead we will begin by defining a certain natural
family of closed subgroups of $S$ which we
will prove $\approx\!\<$-equivalent to one another.
Second, we do not have a result from the literature to serve in
the role of~(\ref{x.M&N.full}).
So we will prove such a result.
The fact that a finite symmetric group is not its own commutator
subgroup will complicate the latter task.
(Cf.\ the proof of~(\ref{x.M&N.full}) as
\cite[Lemma~3]{Sym_Omega:1}, which uses the fact, due to Ore~\cite{OO},
that every element of an infinite symmetric group is a commutator.)
So we will prepare for that proof by showing
that certain infinite products of finite symmetric groups within
$S$ are $\approx\!\<$-equivalent to
the corresponding products of alternating groups.
\vspace{2pt}
To define our set of representatives of the $\approx\!\<$-equivalence
class of subgroups of $S$ we are interested in, let
\begin{xlist}\item\label{x.def_P}
$\mathcal P\;=\;\{A: A$ is a partition of $\Omega$ into finite subsets,
and there is no common finite bound on the cardinalities of the
members of $A\,\}\<.$
\end{xlist}
For $A\in\mathcal P$ (and $S_{(A)}$ defined
by~(\ref{x.def_SA})), we see that
\begin{xlist}\item\label{x.SAiso}
$S_{(A)}\;\cong\;\prod_{\strut\Sigma\in A}\Sym(\Sigma).$
\end{xlist}
Note that if a partition $A_1$ is the image of a partition $A_2$ under
a permutation $f$ of $\Omega,$ then $S_{(A_2)}$ is the conjugate of
$S_{(A_1)}$ by $f;$ in particular, $S_{(A_1)}\approx S_{(A_2)}.$
Let us now show more; namely, that
\begin{xlist}\item\label{x.SAeqSB}
$S_{(A)}\;\approx\;S_{(B)}$~ for all $A,B\in\mathcal P.$
\end{xlist}
We claim first that given $A, B\in\mathcal P,$ we can find
two elements $f, g\in S$ such that
\begin{xlist}\item\label{x.Df_or_Dg}
For each $\Sigma\in B$ there exists $\Delta\in A$ such that
$\Sigma\subseteq\Delta f$ or $\Sigma\subseteq\Delta\_1}(ii)).
Combining these inequalities we have $G\approx S_{(A)}.$
This completes the main work of the proof of
\begin{theorem}\label{T.unbdd_orbs}
Let $\Omega$ be a countably infinite set,
and $\mathcal P$ the set of partitions of $\Omega$
defined in~{\rm(\ref{x.def_P})}.
Then the subgroups $S_{(A)}\leq S$ with $A\in\mathcal P$ {\rm(}which
are clearly all closed\/{\rm)} are mutually $\approx\!\<$-equivalent,
and a closed subgroup $G\leq S$ belongs to the equivalence class of
those subgroups if and only if it satisfies~{\rm(\ref{x.unbdd_orbs})}.
Moreover, the members of this $\approx\!\<$-equivalence class are
$\prec$ the members of the
equivalence class of Theorem~\ref{T.inf_orbs}.
\end{theorem}\begin{proof}
We have so far proved mutual equivalence of the $S_{(A)},$
and the sufficiency of~(\ref{x.unbdd_orbs}) for membership of a closed
subgroup $G$ in their common equivalence class.
To see necessity, consider any closed subgroup
$G$ which does not satisfy~(\ref{x.unbdd_orbs}).
Then either $G$ satisfies~(\ref{x.inf_orbs}),
or there exists a finite set $\Gamma$
such that $G_{(\Gamma)}$ has orbits of bounded finite cardinality.
In the former case, Theorem~\ref{T.inf_orbs} shows that
$G\approx S;$ but from the ``only if'' direction of that theorem
we see that for $A\in\mathcal P$ we have $S_{(A)}\not\approx S,$ and
hence $G\not\approx S_{(A)}.$
In the case where some
$G_{(\Gamma)}$ has all orbits of bounded finite cardinality,
let $A$ be the partition of $\Omega$ consisting of those orbits.
Then $G\approx G_{(\Gamma)}\preq S_{(A)},$ and by the
last sentence of Theorem~\ref{T.no_uncrwdd}, $S_{(A)}$ is not $\sucq$
the members of the equivalence class of this section,
hence $G$ is not in that equivalence class.
In the final sentence, the inequality $\preq$ holds because the
equivalence class of Theorem~\ref{T.inf_orbs} contains $S$ itself.
We have just seen that the two classes in question are distinct, so we
have strict inequality $\prec.$
\end{proof}
\section{Orbits of bounded size.}\label{S.bdd_orbs}
Moving on to still smaller subgroups,
we now consider $G\leq S$ satisfying
\begin{xlist}\item\label{x.bdd_orbs}
There exists a finite subset $\Gamma\subseteq\Omega$ and a positive
integer $n$ such that the cardinalities of all the orbits of
$G_{(\Gamma)}$ are bounded by $n,$ but there exists no
such $\Gamma$ with $G_{(\Gamma)}=\{1\}.$
\end{xlist}
Analogously to~(\ref{x.def_P}), we define
\begin{xlist}\item\label{x.def_Q}
$\mathcal Q\;=\;\{A: A$ is a partition of $\Omega$ for which there is
a common finite bound to the cardinalities of the members of $A,$ and
such that infinitely many members of $A$ have cardinality $>1\}.$
\end{xlist}
Unlike the $\mathcal P$ of the preceding section, $\mathcal Q$
has, up to isomorphism, a natural distinguished member, namely
a least isomorphism class with respect to refinement:
\begin{xlist}\item\label{x.A0}
We will denote by $A_0$ an element of $\mathcal Q,$ unique up
to isomorphism, which has infinitely many $1\!\<$-element
members, infinitely many $2\!\<$-element members, and no others.
\end{xlist}
Clearly any $A\in\mathcal Q$ can be refined to a partition $A'_0$
isomorphic to $A_0,$ hence $S_{(A)}\geq S_{(A'_0)}\approx S_{(A_0)},$
so $S_{(A)}\sucq S_{(A_0)}.$
We claim that the reverse inequality
$S_{(A)}\preq S_{(A_0)}$ also holds.
To show this, let us draw a graph with the elements
of $\Omega$ as vertices, and with edges making each member of our
given partition $A$ a chain (in an arbitrary way), and no other edges.
Now color the edges of each such chain alternately red and green,
subject to the condition that infinitely many chains have a
terminal red edge and infinitely many have a terminal green edge.
Clearly, the partition of $\Omega$ whose non-singleton members
are the pairs of points linked by red edges, all other
points forming singletons, is isomorphic to $A_0;$
hence the group of permutations whose general member acts by
transposing an arbitrary subset of the red-linked pairs of vertices
and fixing everything else can
be written $f^{-1}S_{(A_0)}f$ for some $f\in\Sym(\Omega).$
Similarly, the group of permutations which act by
transposing some pairs of green-linked vertices
and fixing everything else can be written $g^{-1}S_{(A_0)}\1$ be the largest integer such that for every finite
subset $\Gamma\subseteq\Omega,$ the group $G_{(\Gamma)}$ has
orbits of cardinality at least $M.$
Thus, there exists some finite $\Delta$ such that $G_{(\Delta)}$
has no orbits of cardinality $>M.$
Since $G_{(\Delta)}$
inherits from $G$ the property~(\ref{x.bdd_orbs}), we may replace $G$
by $G_{(\Delta)}$ and so assume without loss of generality that
\begin{xlist}\item\label{x.M=max}
For every finite subset $\Gamma\subseteq\Omega,$ the maximum
of the cardinalities of the orbits of $G_{(\Gamma)}$ is $M.$
\end{xlist}
%
A consequence is that for any such $\Gamma,$ every orbit
of $G_{(\Gamma)}$ of cardinality $M$ is also an orbit of $G$
(since the orbit of $G$ containing it cannot have larger cardinality).
Thus
\begin{xlist}\item\label{x.orbtimesG}
If $\Gamma$ is a finite subset of $\Omega,$ and $\alpha$ an
element of $\Omega$ such that $|\<\alpha\,G_{(\Gamma)}\<|=M,$
then for every $g\in G$ we have
$\alpha\,G_{(\Gamma)}\,g=\alpha\,G_{(\Gamma)}.$
\end{xlist}
We shall now construct recursively, for each $j\geq 0,$
elements $\alpha_j,\,\beta_j\in\Omega$ and a subset
$K_j\subseteq G,$ indexed as
\begin{xlist}\item\label{x.g()3}
$K_j\;=\;\{\_1},
for subgroups of symmetric groups $\Sym(\Omega),$
$\approx_{\strt\aleph_1}\!\!$-equivalence is the same as
$\approx_{\strt\aleph_0}\!\!$-equivalence, which is what we are
calling $\approx\!\<$-equivalence.
This gives the first assertion; the second is also immediate,
since the trivial subgroup is $\preq$ all subgroups,
and is $\not\approx$ the subgroups of Theorem~\ref{T.bdd_orbs} by
the ``only if'' assertion of that theorem.
To prove the equivalence of~(i)--(iii), we note first that~(ii)
and~(iii) are equivalent, since a neighborhood basis of the
identity in the function topology on $G$ is given by the subgroups
$G_{(\Gamma)}$ for finite $\Gamma,$ so the identity element (and hence
by translation, every element) is isolated in $G$ if and only if
some such subgroup is trivial.
To see that these equivalent conditions imply~(i), observe
that~(ii) implies that $G\approx G_{(\Gamma)}=\{1\},$ hence that
$G$ is countable, while~(iii) implies that $G$ is closed, by general
properties of topological groups.
(If $G$ is a discrete subgroup of a topological group $S,$
take a neighborhood $U$ of $1$ in $S$ containing no
nonidentity element of $G,$ and then a neighborhood $V$
of $1$ such that $VV^{-1}\subseteq U.$
One finds that for any $x\in S,$ $xV$ is a
neighborhood of $x$ containing at most one element of $G;$
so $G$ has no limit points in $S.)$
Conversely, we have seen that any countable $G$ is $\prec$ the members
of the equivalence class of Theorem~\ref{T.bdd_orbs}, hence does not
belong to the equivalence class of any of Theorems~\ref{T.inf_orbs},
\ref{T.unbdd_orbs} or~\ref{T.bdd_orbs}.
Hence if $G$ is also closed, those theorems exclude all
possible behaviors of its subgroups $G_{(\Gamma)}$ (for $\Gamma$ finite)
other than that there exist such a $\Gamma$ with $G_{(\Gamma)}=\{1\};$
so~(i) implies~(ii).
\end{proof}
For convenience in subsequent discussion, let us name the four
equivalence classes of subgroups of $S=\Sym(\Omega)$ which we have
shown to contain all closed subgroups:
\begin{xlist}\item\label{x.C1-C0}
$\!\mathcal C_S\;=$ the $\approx\!\<$-equivalence class of $S.$
\\[2pt]
$\mathcal{C_P}\;=$ the $\approx\!\<$-equivalence class to which
$S_{(A)}$ belongs for all $A\in\mathcal P.$
\\[2pt]
$\mathcal{C_Q}\;=$ the $\approx\!\<$-equivalence class to which
$S_{(A)}$ belongs for all $A\in\mathcal Q\<.$
\\[2pt]
$\mathcal C_1\;\;=$ the $\approx\!\<$-equivalence class
consisting of the countable subgroups\\
\indent of $S.$
\end{xlist}
\section{Notes and questions
on groups of bounded permutations.}\label{S.q.FN}
It would be of interest to investigate the equivalence relation
$\approx$ on classes of subgroups $G\leq\Sym(\Omega)$
other than the class of closed subgroups.
One such class is implicit in the techniques used above:
If $\Omega$ is any set and $d$ a generalized metric on $\Omega,$
let us define the subgroup
\begin{xlist}\item\label{x.defFN(O,d)}
$\mathrm{FN}(\Omega,d)\;\;=\;\;\{\n$ permutations of that set.
But by the last sentence of the preceding paragraph,
a group of order $>n$ can't be contained in any of the subgroups
$G_{(\{\gamma g\})},$ let alone in all but finitely many of them.
This contradiction completes the proof of the theorem.
\end{proof}
We remark that the analog of Lemma~\ref{L.move_fin} (and hence
of Theorem~\ref{C.hj}) fails for submonoids of $\Omega^\Omega\<.$
For instance, let $\Omega=\omega,$ and for
$i,j\in\omega$ define $f_i(j)=\max(i,j).$
Then $G=\{f_i\mid i\in\omega\}$ satisfies the hypotheses of
Lemma~\ref{L.move_fin} with ``submonoid
of $\omega^\omega$'' in place of ``subgroup of $\Sym(\Omega),$\!''
but does not satisfy the conclusion.
\section{Other preorderings, and
further directions for investigation.}\label{S.q.etc}
In the arguments of \S\S\ref{S.inf_orbs}-\ref{S.bdd_orbs},
when we obtained a relation $G\preq H,$
we often did this by showing that $G$ lay in the
subgroup of $S$ generated by finitely many conjugates of $H.$
This suggests
\begin{definition}\label{preqcj}
If $S$ is a group, $\kappa$ an infinite cardinal, and $G_1,$ $G_2$
subgroups of $S,$ let us write $G_1\preq_{\kappa,S}\cj G_2$
if there exists a subset $U\subseteq S$ of cardinality $<\kappa$
such that $G_1\leq \langle\,\bigcup_{f\in U}\ f^{-1}G_2 f\,\rangle.$
\end{definition}
As with $\preq_{\strt\kappa,S},$ we may omit the subscripts
$\kappa$ and $S$ from $\preq_{\kappa,S}\cj$
when their values are clear from context, and we will write
$\approx_{\kappa,S}\cj$ or
$\approx\cj$ for the induced equivalence relation.
For the remainder of this discussion, $\kappa$ will be $\aleph_0$ and
$S$ will be $\Sym(\Omega)$ for a countably infinite set $\Omega,$
and these subscripts will not be shown.
In general, $\preq\cj$ and $\approx\cj$ are finer relations
than $\preq$ and $\approx.$
Since not {\em all} the arguments in
\S\S\ref{S.inf_orbs}-\ref{S.bdd_orbs}
were based on combining conjugates of the given subgroup $G$
(in particular, some were based on conjugating a carefully constructed
element $s\in S$ {\em by} elements of $G),$ it is not obvious whether
those results can be strengthened to say that the classes of subgroups
that we proved $\approx$\!-equivalent are in fact
\mbox{$\approx\cj$\!-equivalent}.
Let us show that the answer is ``almost''.
Recall (cf.\ \cite[p.51, Theorem~6.3]{BhMMN})
that since $\Omega$ is countably infinite, the only
proper nontrivial normal subgroups of $S$ are the
group of permutations that move only finitely many points,
which we shall denote $S^{\mathrm{\prec}
$G\preq^{\mathrm{fix}}_{\strt\aleph_0}H\implies
G\preq_{\strt|\<\Omega\<|^+}H.$
\end{xlist}
The relations $\approx^{\mathrm{fix}}_\kappa$ and
$\preq^{\mathrm{fix}}_\kappa$ tend to be quite fine-grained.
For instance, given partitions $A_1$ and $A_2$ of $\Omega,$
it is not hard to see that
$S_{(A_1)}\approx^{\mathrm{fix}}_\kappa S_{(A_2)}$ if
and only if $A_1$ and $A_2$ ``disagree at $<\kappa$ elements'',
meaning that one can be obtained from the other by ``redistributing''
$<\kappa$ elements of $\Omega\<.$
\vspace{6pt}
In a different direction, one might define on abstract groups
(rather than subgroups of a fixed group) a preordering analogous
to $\preq_\kappa,$ by letting $G_1\preq^{\mathrm{emb}}_\kappa G_2$
mean that $G_1$ admits an embedding in a group $H$ which is generated
over $G_2$ by $<\kappa$ elements.
\vspace{6pt}
In our study of symmetric groups in this note,
we have considered only countable $\Omega,$ except when no additional
work or distraction was entailed by allowing greater generality.
It would be of interest to know what can be said about
$\approx_\kappa\!$-equivalence classes of closed subgroups
of $\Sym(\Omega)$ for general $\Omega$ and $\kappa\<;$ in particular,
whether there are simple criteria for a closed subgroup
$G\leq\Sym(\Omega)$ to be $\approx_{\aleph_0}\!$-equivalent
(equivalently, $\approx_{|\<\Omega\<|^+}\!$-equivalent)
to $\Sym(\Omega).$
A related topic which has been studied extensively
(e.g., \cite{SS+ST}, \cite{ST_surv})
is the {\it cofinality} of groups $\Sym(\Omega),$ defined as
the least cardinal $\kappa$ such that $\Sym(\Omega)$ can be written
as the union of a chain of $<\kappa$ proper subgroups.
If $S=\Sym(\Omega)$ is of cofinality $\geq\kappa,$
then our unary relation $\approx_{\aleph_0,\~~_1}(ii) above);
though the converse fails under some set-theoretic assumptions.
Still another direction for study would be to consider questions of
the sort studied here, but for structures other than groups.
Mesyan \cite{ZM} gets some results of this sort
for the ring of endomorphisms of the $\Omega\!$-fold direct sum
of copies of a module.
% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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\end{thebibliography}
\end{document}
~~