On subgroups with non-zero Möbius numbers in the alternating and symmetric groups. (Q618258)

Let \(G\) be an alternating or symmetric group, \(L(G)\) be the subgroup lattice of \(G\) and, for every subgroup \(H\) of \(G\), let \(\mu(H,G)\) be the Möbius number of \(H\) in \(L(G)\). Denote also by \(b_m(G)\) the number of subgroups \(H\) of \(G\) of index \(m\) satisfying \(\mu(H,G)\neq 0\). This paper proves two interesting results, namely that there exist two absolute constants \(\alpha\) and \(\beta\) such that for any alternating or symmetric group \(G\), any subgroup \(H\) of \(G\) and any positive integer \(m\), we have \(b_m(G)\leq m^\alpha\) and \(|\mu(H,G)|\leq|G:H|^\beta\).