Periodic 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: periodic group and abelian group
View other group property conjunctions OR view all group properties
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
View other group property conjunctions OR view all group properties
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
View other group property conjunctions OR view all group properties

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
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