Sylow subgroups of symmetric and alternating groups and the vertex of $S^{(kp-p,1^p)}$ in characteristic $p$

TL;DR

This paper characterizes Sylow p-subgroups of symmetric and alternating groups and applies this to determine the vertices of specific hook Specht modules in modular representation theory.

Contribution

It provides a new characterization of Sylow p-subgroups and computes vertices of certain Specht modules under specific modular conditions.

Findings

Sylow p-subgroups contain all elementary abelian p-subgroups up to conjugacy.

Vertices of hook Specht modules are explicitly computed for particular partitions.

Results depend on congruence conditions of k modulo p and p^2.

Abstract

We show that the Sylow $p$-subgroups of a symmetric group, respectively an alternating group, are characterized as the $p$-subgroups containing all elementary abelian $p$-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition $(kp-p,1^p)$ under the assumption that $k \equiv 1$ mod $p$ and $k \not\equiv 1$ mod $p^2$.