Covering the alternating groups by products of cycle classes

Given integers k, l >= 2, where either l is odd or k is even, we denote by n = n (k, 1) 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 2/3kl <= n (k, l) <= 2/3kl + 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]. (C) 2008 Elsevier Inc. All rights reserved.