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

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