Commutators of n-cycles in the symmetric group

Abstract

We show that for $n\ge 6$ every even permutation on $n$ symbols is the commutator of two $n$-cycles. More precisely, let $S_n$ be the symmetric group and $A_n$ the alternating group. Let $C(n)\subset S_n$ denote the conjugacy class of $n$-cycles and $[\cdot ,\cdot ]$ be the commutator of two permutations. We prove: The map $C(n)\times C(n)\to A_n,(\tau ,\pi )\mapsto [\tau ,\pi ]$ is surjective for all $n\ge 6$.

The result presented in this article can be derived from [Aleš Vavpetič, Commutators of cycles in permutation groups, Ars Math. Contemp. 10 (2016)] and is therefore not intended for publication. We thank Goulnara Arzhantseva and Aleš Vavpetič for pointing out the reference as well as for the subsequent discussion and explanations