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 |