On subgroups with non-zero Möbius numbers in the alternating and symmetric groups

Abstract

For a subgroup H of an alternating or symmetric group G, we consider the Möbius number μ ( H , G ) of H in the subgroup lattice of G. Let b m ( G ) be the number of subgroups H of G of index m with μ ( H , G ) ≠ 0 . We prove that there exist two absolute constants α and β such that for any alternating or symmetric group G, any subgroup H of G and any positive integer m we have b m ( G ) ⩽ m α and | μ ( H , G ) | ⩽ | G : H | β .