Torsion-free abelian group
From Groupprops
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
Contents
Definition
A torsion-free abelian group or aperiodic abelian group or locally free abelian group is a group satisfying the following equivalent conditions:
- It is both an abelian group and a torsion-free group, i.e., no non-identity element of the group has finite order.
- 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 |
