Commutators of cycles in permutation groups

Abstract

We prove that for n ≥ 5, every element of the alternating group A_n is a commutator of two cycles of A_n. Moreover we prove that for n ≥ 2, a (2n + 1)-cycle of the permutation group S_2n + 1 is a commutator of a p-cycle and a q-cycle of S_2n + 1 if and only if the following three conditions are satisfied (i) n + 1 ≤ p, q, (ii) 2n + 1 ≥ p, q, (iii) p + q ≥ 3n + 1.