Nilpotent covers of symmetric and alternating groups

Abstract

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of $\mathcal{M}$ is equal to the order of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ 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 $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the order of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the order of a maximal non-nilpotent subset of $A_n$.