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

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This article is about a congruence condition.
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Statement

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 Collection of groups satisfying a universal congruence condition (?) for the prime p. Thus, it is also a Collection of groups satisfying a strong normal replacement condition (?) and a 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#Examples/facts.