# Periodic abelian group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: periodic group and abelian group
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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: locally finite group and abelian group
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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: group generated by periodic elements and abelian group
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## Definition

A periodic abelian group (also called torsion abelian group or abelian torsion group or locally finite abelian group) is a group satisfying the following equivalent conditions:

1. It is both a locally finite group and an abelian group.
2. it is both a periodic group (i.e., every element has finite order) and an abelian group.
3. It is both an abelian group and a group generated by periodic elements.
4. When it is endowed with the discrete topology, its Pontryagin dual is an abelian profinite group.

### Equivalence of definitions

Further information: equivalence of definitions of periodic abelian group

## Relation with other properties

### Stronger properties

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