Maximal elementary abelian subgroup of prime-square order implies rank at most the prime for odd prime

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This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Statement

Let p be an odd prime.

Suppose P is a group of prime power order (i.e., a finite p-group for some prime p) and A is an elementary abelian subgroup of P of order p^2 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 rank of P is at most p. In other words, every elementary abelian subgroup of P is of order at most p^p.

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