Nilpotent covers of symmetric and alternating groups

TL;DR

This paper establishes the uniqueness and explicit structure of minimal nilpotent covers of symmetric groups, contrasting these properties with those of alternating groups, and explores their algebraic and combinatorial characteristics.

Contribution

It proves the uniqueness of the minimal nilpotent cover of symmetric groups, provides an explicit formula for its order, and compares these properties with those of alternating groups.

Findings

Unique minimal nilpotent cover for symmetric groups

Explicit formula for the order of the cover

Contrast with properties of 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 $A_n$.