Abelian-to-normal replacement theorem for prime-fourth order

Statement in terms of normal replacement condition
Let $$p$$ be a prime number. The collection of abelian groups of order $$p^4$$ is a fact about::collection of groups satisfying a weak normal replacement condition.

Hands-on statement
Let $$p$$ be a prime number and $$P$$ be a finite $$p$$-group, i.e., a group of prime power order where the prime is $$p$$. Suppose $$P$$ has an abelian subgroup $$A$$ of order $$p^4$$. Then, $$P$$ has an abelian normal subgroup $$B$$ (hence, an fact about::abelian normal subgroup of group of prime power order) of order $$p^4$$.

Related facts

 * Congruence condition on number of abelian subgroups of prime-cube order
 * Congruence condition on number of abelian subgroups of prime-fourth order
 * Abelian-to-normal replacement theorem for prime-cube order
 * Group of prime-sixth or higher order contains abelian normal subgroup of prime-fourth order for prime equal to two
 * Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime

Facts used

 * 1) uses::Existence of abelian normal subgroups of small prime power order: This states that if $$n \ge 1 + k(k-1)/2$$, then any finite $$p$$-group of order $$p^n$$ has an abelian normal subgroup of order $$p^k$$.
 * 2) uses::Abelian-to-normal replacement theorem for prime-square index: This states that if a finite $$p$$-group has an abelian subgroup of index $$p^2$$, it has an abelian normal subgroup of index $$p^2$$.

Proof
Given: A finite $$p$$-group $$P$$ of order $$p^n$$ having a subgroup $$A$$ of order $$p^4$$.

To prove: $$P$$ has an abelian normal subgroup $$B$$ of order $$p^4$$.

Proof: Clearly, $$n \ge 4$$. We consider four cases:


 * 1) $$n = 4$$: This case is tautological.
 * 2) $$n = 5$$: In this case, any subgroup of order $$p^4$$ is of index $$p$$, hence normal, so any abelian subgroup is an abelian normal subgroup.
 * 3) $$n = 6$$: In this case, the result follows from fact (2).
 * 4) $$n \ge 7$$: In this case, fact (1) tells us that there is an abelian normal subgroup of order $$p^4$$.