# Abelian group of prime power order

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

## 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 $p$ and a nonnegative integer $n$, the abelian groups of order $p^n$ correspond to unordered integer partitions of $n$. Specifically, a partition $n = k_1 + k_2 + \dots + k_r$ corresponds to the group: $\! \prod_{i=1}^r \mathbb{Z}/p_i^{k_i}\mathbb{Z}$

## Examples

Partition of $n$ Value of $n$ Case $p = 2$ Case $p = 3$ General case
empty 0 trivial group trivial group trivial group
1 1 cyclic group:Z2 cyclic group:Z3 group of prime order
2 2 cyclic group:Z4 cyclic group:Z9 cyclic group of prime-square order
1 + 1 2 Klein four-group elementary abelian group:E9 elementary abelian group of prime-square order
3 3 cyclic group:Z8 cyclic group:Z27 cyclic group of prime-cube order
2 + 1 3 direct product of Z4 and Z2 direct product of Z9 and Z3 direct product of cyclic group of prime-square order and cyclic group of prime order
1 + 1 + 1 3 elementary abelian group:E8 elementary abelian group:E27 elementary abelian group of prime-cube order
4 4 cyclic group:Z16 cyclic group:Z81 cyclic group of prime-fourth order
3 + 1 4 direct product of Z8 and Z2 direct product of Z27 and Z3 direct product of cyclic group of prime-cube order and cyclic group of prime order
2 + 2 4 direct product of Z4 and Z4 direct product of Z9 and Z9 direct product of cyclic group of prime-square order and cyclic group of prime-square order
2 + 1 + 1 4 direct product of Z4 and V4 direct product of Z9 and E9 direct product of cyclic group of prime-square order and elementary abelian group of prime-square order
1 + 1 + 1 + 1 4 elementary abelian group:E16 elementary abelian group:E81 elementary abelian group of prime-fourth order
5 5 cyclic group:Z32 cyclic group:Z243 cyclic group of prime-fifth order
4 + 1 5 direct product of Z16 and Z2 direct product of Z81 and Z3 direct product of cyclic group of prime-fourth order and cyclic group of prime order
3 + 2 5 direct product of Z8 and Z4 direct product of Z27 and Z9 direct product of cyclic group of prime-cube order and cyclic group of prime-square order
3 + 1 + 1 5 direct product of Z8 and V4 direct product of Z27 and E9 direct product of cyclic group of prime-cube order and elementary abelian group of prime-square order
2 + 2 + 1 5 direct product of Z4 and Z4 and Z2 direct product of Z9 and Z9 and Z3
2 + 1 + 1 + 1 5 direct product of E8 and Z4 direct product of E27 and Z9
1 + 1 + 1 + 1 + 1 5 elementary abelian group:E32 elementary abelian group:E243 elementary abelian group of prime-fifth order

## 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