# Abelian group of prime power order

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: abelian group and group of prime power order

View other group property conjunctions OR view all group properties

## Contents

## Definition

An **abelian group of prime power order** is a group of prime power order that is also an abelian group.

## Classification

As a particular case of the structure theorem for finitely generated abelian groups, we can say the following. For a prime and a nonnegative integer , the abelian groups of order correspond to unordered integer partitions of . Specifically, a partition corresponds to the group:

## Examples

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

cyclic group of prime power order | the corresponding partition has just one piece | |||

homocyclic group of prime power order | the corresponding partition has all pieces of equal size | |||

Finite elementary abelian group | the corresponding partition has all pieces of size 1 |