Commutators of cycles in permutation groups
Keywords:Commutator, cycle, permutation, alternating group.
AbstractWe prove that for n ≥ 5, every element of the alternating group An is a commutator of two cycles of An. Moreover we prove that for n ≥ 2, a (2n + 1)-cycle of the permutation group S2n + 1 is a commutator of a p-cycle and a q-cycle of S2n + 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.
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