Congruence condition on number of abelian subgroups of prime-fourth order

Statement in terms of universal congruence condition
Let $$p$$ be any prime number (including the case of odd $$p$$ and $$p = 2$$). Then ,the collection of abelian groups of order $$p^4$$ is a fact about::collection of groups satisfying a universal congruence condition for the prime $$p$$. Thus, it is also a fact about::collection of groups satisfying a strong normal replacement condition and a fact about::collection of groups satisfying a weak normal replacement condition.

Hands-on statement
Let $$p$$ be any prime number (including the case of odd $$p$$ and $$p = 2$$). Then, if $$P$$ is a finite $$p$$-group and $$A$$ is an abelian subgroup of $$P$$ of order $$p^4$$, the number of abelian subgroups of $$P$$ of order $$p^4$$ is congruent to 1 mod $$p$$.

Related facts
For a more complete list, refer collection of groups satisfying a universal congruence condition.


 * Congruence condition on number of abelian subgroups of prime-cube order
 * Abelian-to-normal replacement theorem for prime-cube order
 * Abelian-to-normal replacement theorem for prime-fourth order
 * Congruence condition on number of abelian subgroups of order eight and exponent dividing four
 * Congruence condition on number of abelian subgroups of order sixteen and exponent dividing eight
 * Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
 * Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
 * Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
 * Congruence condition on number of cyclic subgroups of small prime power order