# 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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## Contents

## Statement

### Statement in terms of universal congruence condition

Let be any prime number (including the case of odd and ). Then ,the collection of abelian groups of order is a Collection of groups satisfying a universal congruence condition (?) for the prime . 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 be any prime number (including the case of odd and ). Then, if is a finite -group and is an abelian subgroup of of order , the number of abelian subgroups of of order is congruent to 1 mod .

## Related facts

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

- 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