Torsion-free 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: torsion-free group and abelian group
View other group property conjunctions OR view all group properties

Definition

A torsion-free abelian group or aperiodic abelian group or locally free abelian group is a group satisfying the following equivalent conditions:

  1. It is both an abelian group and a torsion-free group, i.e., no non-identity element of the group has finite order.
  2. Every finitely generated subgroup of it is a free abelian group.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Free abelian group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Torsion-free group |FULL LIST, MORE INFO
Locally reduced free group |FULL LIST, MORE INFO
Abelian group |FULL LIST, MORE INFO