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

From Groupprops

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 be an odd prime.

Suppose is a group of prime power order (i.e., a finite -group for some prime ) and is an elementary abelian subgroup of of order that is not contained in any bigger elementary abelian subgroup. In other words, is a Group of prime power order having a maximal elementary abelian subgroup of prime-square order (?).

Then, the rank of is at most . In other words, every elementary abelian subgroup of is of order at most .