Nilpotent covers of symmetric and alternating groups

Abstract

We prove that the symmetric group Sn has a unique minimal cover M by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of M. In addition, we prove that the order of M is equal to the order of a maximal non-nilpotent subset of Sn. This cover M has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of n into distinct positive integers. We show that these results contrast with those for the alternating group An. In particular, we prove that, for all but finitely many values of n, no minimal cover of An by maximal nilpotent subgroups is a normal cover and the order of a minimal cover of An by maximal nilpotent subgroups is strictly greater than the order of a maximal non-nilpotent subset of An.