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

From Groupprops

This article defines a replacement theorem

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## Contents

## Statement

### Statement in terms of normal replacement condition

Let be a prime number. The collection of abelian groups of order is a Collection of groups satisfying a weak normal replacement condition (?).

### Hands-on statement

Let be a prime number and be a finite -group, i.e., a group of prime power order where the prime is . Suppose has an abelian subgroup of order . Then, has an abelian normal subgroup (hence, an Abelian normal subgroup of group of prime power order (?)) of order .

## 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

- Existence of abelian normal subgroups of small prime power order: This states that if , then any finite -group of order has an abelian normal subgroup of order .
- Abelian-to-normal replacement theorem for prime-square index: This states that if a finite -group has an abelian subgroup of index , it has an abelian normal subgroup of index .

## Proof

**Given**: A finite -group of order having a subgroup of order .

**To prove**: has an abelian normal subgroup of order .

**Proof**: Clearly, . We consider four cases:

- : This case is tautological.
- : In this case, any subgroup of order is of index , hence normal, so any abelian subgroup is an abelian normal subgroup.
- : In this case, the result follows from fact (2).
- : In this case, fact (1) tells us that there is an abelian normal subgroup of order .