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Even permutations as a product of two conjugate cycles

1972; Elsevier BV; Volume: 12; Issue: 3 Linguagem: Inglês

10.1016/0097-3165(72)90102-1

1096-0899

Edward A. Bertram,

Wireless Communication Networks Research

The central result of this paper is a generalization of the theorem that, for n ≥ 5, every even permutation defined on n symbols is a commutator a b a−1 b−1 of even permutations a and b. In particular, [3n4] ⩽ l ⩽ n is shown to be the necessary and sufficient condition on l, in order that every even permutation defined on n ⩾ 5 symbols can be expressed as a product of two cycles, each of length l. Various results follow, including the characterization of those l for which every odd permutation is a product of a cycle of length l and a cycle of length l + 1.