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Given integers $k,l\geq 2$, 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 aproduct of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Levconjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor\frac{2kl}{3}\rfloor+1$. 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 theexact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediateconsequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$,which shows that the above conjecture is not true in general. In fact, thedifference between the exact value of $n(k,l)$ and the conjectured value growslinearly in terms of $k$. Our results also generalize the case of $k=2,3,4$.

Alternating groups as products of cycle classes - II