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

Statement

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, $P$ has normal rank at most $p$. In other words, every elementary abelian normal subgroup of $P$ has order at most $p^{p}$.

Related facts

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