From: mareg@primrose.csv.warwick.ac.uk ()
Subject: Re: Large abelian subgroups of S_n
Date: Thu, 24 May 2001 09:09:23 +0000 (UTC)
Newsgroups: sci.math

In article <9eigbj$7tj$1@agate.berkeley.edu>,
	daw@mozart.cs.berkeley.edu (David Wagner) writes:
>About how large are the largest abelian subgroups of S_n, the symmetric
>group of permutations on n letters?  I'm particularly interested in a
>rough order-of-magnitude estimate of the order of large abelian subgroups
>of S_n for, say, the case where n=40.  (I apologize in advance if the
>answer should have been obvious; I don't immediately see how to approach
>the problem.)

This question came up only about a couple of months ago!!!

The answer is exact - let's see if I can get it right this time:
Let A be a largest abelian subgroup of S_n

If n=3k,   |A| = 3^k     
If n=3k+1, |A| = 4.3^(k-1)   (when k > 0)
If n=3k+2, |A| = 2.3^k

A is just a direct product of cyclic groups acting in the obvious way.
e.g., if n=10, then you could take A = <(1,2,3,4), (5,6,7), (8,9,10)>.
Or A = <(1,2), (3,4), (5,6,7), (8,9,10)>.
For n=3k or n=3k+2, A is unique up to conjugacy  in S_n.
For n=3k+1, there are two conjugacy classes.

Derek Holt.