Alternating groups as products of cycle classes - II

Given integers \(k,l\ge 2\), where either l is odd or k is even, let n(k, l) denote the largest integer n such that each element of \(A_n\) is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that \(\lfloor \frac{2kl}{3} \rfloor \le n(k,l)\le \lfloor \frac{2kl}{3}\rfloor +1\) [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when \(k=2,3,4\). Moreover, it is also true when \(3\mid l\). In this article, we determine the exact value of n(k, l) when \(3\not \mid l\) and \(k\ge 5\). As an immediate consequence, we get that \(n(k,l)<\lfloor \frac{2kl}{3}\rfloor \) when \(k\ge 5\) and \(3\not \mid l\), which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of n(k, l) and the conjectured value grows linearly in terms of k. Our results complete the determination of n(k, l) for all values of k and l.