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

From Groupprops

This article defines a replacement theorem

View a complete list of replacement theorems| View a complete list of failures of replacement

## Contents

## Statement

### Statement in terms of weak normal replacement condition

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

### Hands-on statement

Suppose is a prime number and is a finite -group. If is an abelian subgroup of of order , there is an abelian normal subgroup of 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-fourth order
- Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime: For odd and , the number of abelian subgroups of order is either zero or mod .
- 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

## 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 .

## Proof

**Given**: A finite -group of order , containing an abelian subgroup of order .

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

**Proof**: If , then it is normal and we can set . Thus, we assume that is a proper subgroup of .

In this case, since , fact (1) tells us that has an abelian normal subgroup of order , and we are done.