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

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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,1ᵖ) under the assumption that k ≡ 1 mod p and k ≡ 1 mod p².