From: Douglas Zare <zare@math.columbia.edu>
Subject: Re: Commutators in S_n
Date: Tue, 29 May 2001 14:05:59 -0400
Newsgroups: sci.math
Summary: Every element in Alt(n) is a commutator

Ken Silver wrote:

> The commutator subgroup of the symmetric group S_n is the alternating
> subgroup A_n but is it also true that every element in A_n is a
> commutator ?

Yes. In an article dated 5/13/01, Fred Galvin posted a short proof
(quoted below) that every even permutation is a product of two n-cycles,
which implies that every element of A_n is the commutator of an n-cycle
with something else: aba^-1b^-1 = a(ba^-1b^-1).

Douglas Zare

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THEOREM. Every even permutation in S_n is the product of two n-cycles.

PROOF. By induction on n. The cases n = 1 and n = 2 are trivial.
Suppose n > 2 and let f be an even permutation of {1,...,n}. Without
loss of generality we may assume that f(1) != n. Let x = f(1) and let
g = (x n)f(1 n). [Composition of permutations from right to left.]
Then g is an even permutation leaving n fixed. By the induction
hypothesis, g = jk where j and k are (n-1)-cycles leaving n fixed.
Then f = (x n)jk(1 n), where (x n)j and k(1 n) are n-cycles.
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