Maximal elementary abelian subgroup of order four implies subgroup rank at most four

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Statement

Suppose P is a group whose order is a power of two, having an elementary abelian subgroup of order four (namely a Klein four-group) that is not contained in any bigger elementary abelian subgroup. In other words, P is a Group of prime power order having a maximal elementary abelian subgroup of prime-square order (?).

Then, the subgroup rank of P is at most four. In other words, every subgroup of P is generated by a subset of size at most four.

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