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 (2 n + 1) -cycle of the permutation group S 2 n + 1 is a commutator of a p -cycle and a q -cycle of S 2 n + 1 if and only if the following three conditions are satisfied (i) n + 1 ≤ p , q , (ii) 2 n + 1 ≥ p , q , (iii) p + q ≥ 3 n + 1 .
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