Covering the alternating groups by products of cycle classes. (Q958728)

Let \(A_n\) denote the group of even permutations on \(n\) letters. Given integers \(k,l\geq 2\) the integer \(n=n(k,l)\) is defined as the largest integer such that every permutation in \(A_n\) is a product of \(k\) cycles of length \(l\). The authors prove several interesting generalizations of earlier results due to \textit{E. Bertram} [J. Comb. Theory, Ser. A 12, 368-380 (1972; Zbl 0238.20004)] and \textit{E. Bertram} and \textit{M. Herzog} [J. Comb. Theory, Ser. A 94, No. 1, 87-99 (2001; Zbl 0983.20003)]. E.g. the following main result is proved. Theorem 3.4. Let \(k\) and \(l\) be such that \(k\geq 2\) and \(l\geq 9\) is odd and divisible by \(3\). Then \({2\over 3}kl\leq n(k,l)\leq {2\over 3}kl+1\). Furthermore, if \(k\) is odd then \(n(k,l)={2\over 3}kl\). Theorem 3.3 contains a result for arbitrary \(l>2\) which is more complicated in detail. Of course, the results are closely related to the covering numbers of conjugacy classes in \(A_n\).