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