  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 $p^2$.

Representation Theory