From: mareg@csv.warwick.ac.uk (Dr D F Holt)
Newsgroups: sci.math
Subject: Re: alternating group analogue
Date: 14 Nov 1997 09:33:44 -0000
In article <346BCB7F.246@uwec.edu>,
smithaj@uwec.edu writes:
>Let S be the symmetric group on an infinite set. Take the
>set to be countable if you want to.
>
>Does S contain a subgroup of index 2?
>
No. Let the set be X. The infinite analogue of the alternating group
on X consists of all even permutations that move only finitely many
points of X. This is normal in Sym(X), but has infinite index if X
is infinite.
In general, for an infinite cardinal c, define Sym(X,c) to be all
permutations of X that move strictly fewer than c points. Then
there is a theorem due to Baer that the only normal subgroups of
X (for |X|>4) are 1, Alt(X), Sym(X) and the subgroups Sym(X,c).
There is a proof in Chapter 8 of "Permutation Groups" by J. Dixon
and B. Mortimer (Springer).
Derek Holt.