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

Statement
Suppose $$p$$ is an odd prime, and $$P$$ is an group of prime power order where the prime is $$p$$. Suppose $$A$$ is fact about::maximal among Abelian normal subgroups of $$P$$, such that $$A$$ is also an fact about::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 $$P$$.

Related facts

 * Abelian normal subgroup of maximum rank that is also maximal among Abelian normal subgroups exists

Journal references

 * , Page 796, Lemma 8.3, Section 8 (Miscellaneous Preliminary Lemmas), Chapter 2.