Omega-1 of maximal among Abelian normal subgroups with maximum rank in odd-order p-group equals omega-1 of centralizer

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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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Suppose p is an odd prime, and P is an group of prime power order where the prime is p. Suppose A is Maximal among Abelian normal subgroups (?) of P, such that A is also an Abelian normal subgroup of maximum rank (?) in P. Then:

\Omega_1(C_P(\Omega_1(A))) = \Omega_1(A).

Here, \Omega_1 denotes the first omega subgroup and C_P denotes the centralizer inside P.

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