Group of prime power order having a maximal elementary abelian subgroup of prime-square order

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Let P be a group of prime power order, i.e., a finite p-group for some prime number p. We say that P has a maximal elementary abelian subgroup of prime-square order if it satisfies the following equivalent conditions:

  1. There exists an elementary abelian subgroup A of P of order p^2 that is not contained in any bigger elementary abelian subgroup of P.
  2. There exists an elementary abelian subgroup A of P of order p^2 such that \Omega_1(C_P(A)) = A.
  3. There exists an element t \in P such that \Omega_1(C_P(t)) is elementary abelian of order p^2.