Re: Quotients of Sym(X)


On 14-Apr-2009, rusin@xxxxxxxxxxxx (Dave Rusin)
wrote in message <gs2oed$259g$1@xxxxxxxxxxxxxxxxxxxxxxxxx>:

In article <gs1niu$2fss$1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Dominic van der Zypen <dominic.zypen@xxxxxxxxx> wrote:

Which groups arise as quotients of symmetric groups?

A correct answer has been given for quotients of finite symmetric groups.
The OP clarified in email that his question concerns infinite groups,
e.g. whether there is ever a surjection from a Sym(X) onto the integers.

If C is any infinite cardinal <= card(X), then
Sym_C(X) = { f : X -> X | card( supp(f) ) ) <= C }
is easily seen to be a normal subgroup of Sym(X)
(where the supp(f) is the "support" of f : { x in X | f(x) \not = x } ).

Almost. You're certainly right that those are normal subgroups of
Sym(X) (assuming of course that f is a bijection), but you really
want the definition (I'll use Sym(X,C), per Dixon and Mortimer
below) to be |supp(f)| < C, not <= C, so that you have
Sym(X,aleph_0) = FSym(X), the group of permutations of X with
finite support, which is of course also normal in Sym(X). (With
this definition, Sym(X,C) = Sym(X) for C > |X|).

I believe it's true that these are the only normal subgroups of Sym(X)
for infinite X, apart from the "infinite alternating group", i.e.
the derived subgroup of Sym_C(X) when C = aleph_0.

Right. Note that here, you're implicitly using the "< C"
definition: Alt(X) is the group of even permutations in FSym(X).

(In particular,
there will be no surjections onto Z).

Well, |Sym(X,aleph_0)| = |X|, while |Sym(X,C)| = |X|^C for
aleph_0 < C <= |X| if |X| > aleph_0, and of course |Sym(X)| =
|X|^|X| = 2^|X|. It's not immediately obvious to me that this
precludes surjections to Z, since in particular |Sym(X)| =
|Sym(X,|X|)| if |X| > aleph_0. Does anyone know for sure?

I looked for references to
corroborate my memory on this and while I didn't quite find what I wanted,
I got some partial confirmation:

For a complete proof, see Theorem 8.1A of _Permutation Groups_ by
John D. Dixon and Brian Mortimer, 1996 Springer-Verlag.

--
Jim Heckman
.

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