Covering the alternating groups by products of cycle classes

Abstract

Given integers k, l 2, where either l is odd or k is even, we denote by n = n(k, l) the largest integer such that each element of A n is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(A n , C l ), where C l is the set of all l-cycles in A n . We prove that if k 2 and l 9 is odd and divisible by 3, then 2 3 kl n(k, l) 2 3 kl + 1. This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99].

The p-width of the alternating groups

“…2.3, 2.4 below). This work was extended by Bertram and Herzog [2] to consider products of three and four p-cycles, and then by Herzog, Kaplan and Lev [4] to an arbitrary number of p-cycles (under some conditions). Products of conjugacy classes in A n have also been studied extensively by Dvir [3].…”