Even permutations not representable in the form of a product of two permutations of given order (Q1277512)

Let \(k\) be a positive integer. A permutation is called \(k\)-representable if it is the product of two permutations of cycle types \(k^{\alpha_i}1^{\beta_i}\) where \(\alpha_i>0\). For each positive integer \(k\geq 4\) a non-empty set \(Q_k\) of natural numbers is defined; e.g., \(Q_4=\{4\ell+3\mid\ell=1,2,\ldots\}\) is infinite, \(Q_k\) is finite for \(k>4\). Then the following Theorem is proved: Let \(k\geq 4\) and \(n\in Q_k\). Then there exists a permutation in the alternating group \(A_n\) that is not \(k\)-representable. Results about two conjectures by \textit{J. L. Brenner} and \textit{R. J. Evans} [J. Comb. Theory, Ser. A 45, No. 2, 196-206 (1987; Zbl 0621.20017)] follow: 1) Let \(k\equiv 1\pmod 4\) be a prime number. Then no permutation of cycle type \(4^12^{(3k-5)/2}\) can be presented in the alternating group \(A_{3k-1}\) as the product of two permutations of order \(k\). 2) Let \(k\) be a prime number \(\geq 7\). If \(k\equiv 1\pmod 3\) then no permutation of cycle type \(3^12^{2k-2}\) can be presented in the alternating group \(A_{4k-1}\) as the product of two permutations of order \(k\). However, if \(k\equiv 2\pmod 3\) there exist such permutations.