I am trying to prove that the commutator subgroup of $S_n$ ($S_n$ is a symmetric group on $[n]$), $[S_n,S_n]$ consists solely of commutators $s_1^{-1}s_2^{-1}s_1s_2$ for some $s_1,s_2\in S_n$. Any hints?
Thanks in advance!
2026-04-01 07:01:12.1775026872
Commutators of a symmetric group
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To prove that every element in $A_n$ is a commutator, you would have to divide it into several cases and prove that each k-cycle can be written as a commutator.
G.A. Miller's paper prove the statement: http://projecteuclid.org/euclid.bams/1183416038
as well as Noboru Ito's, which I can't find online.
Hope it is helpful.