# Periodic abelian group

From Groupprops

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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## Contents

## 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:

- It is both a locally finite group and an abelian group.
- it is both a periodic group (i.e., every element has finite order) and an abelian group.
- It is both an abelian group and a group generated by periodic elements.
- 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 |
---|---|---|---|---|

finite abelian group | |FULL LIST, MORE INFO |

### Weaker properties

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

periodic nilpotent group | |FULL LIST, MORE INFO | |||

periodic solvable group | |FULL LIST, MORE INFO | |||

locally finite group] | Periodic nilpotent group, Periodic solvable group|FULL LIST, MORE INFO | |||

periodic group | Locally finite group, Periodic nilpotent group, Periodic solvable group|FULL LIST, MORE INFO | |||

group generated by periodic elements | Locally finite group, Solvable group generated by periodic elements|FULL LIST, MORE INFO |