Even permutations not representable in the form of a product of two permutations of given order

V. G. Bardakov

Matematičeskie zametki, Tome 62 (1997) no. 2, pp. 169-177

Résumé

The paper gives a description the permutations from the alternating group $A_n$ that, for a given positive integer $k\ge4$, cannot be presented as a product of two permutations each of which contains only cycles of lengths 1 and 4 in the expansion into independent cycles. We construct a set $Q_k$ such that, for each $n$ from $Q_k$, the group $A_n$ contains a permutation not representable in the above form. We give answers to two questions of Brenner and Evans on the representability of even permutations in the form of a product of two permutations of a given order $k$.