From: Douglas Zare 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 ---- 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. ----