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:
- 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] | |FULL LIST, MORE INFO | |||
| periodic group | |FULL LIST, MORE INFO | |||
| group generated by periodic elements | |FULL LIST, MORE INFO |