Every elementary abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic

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Suppose p is a prime number and V is an elementary Abelian p-group (equivalently, V is a vector space over the prime field \mathbb{F}_p. Then, there exists a finite p-group P such that P/\Phi(P) \cong V (where \Phi(P) denotes the Frattini subgroup of P) and such that every maximal subgroup of P is a characteristic subgroup.

Facts used

  1. Bryant-Kovacs theorem


For the case that V is one-dimensional, we can take P = V. For the case that \operatorname{dim}(V) > 1, apply fact (1) with the subgroup G as the trivial group.