# Pure subgroup of torsion-free abelian group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: pure subgroup with a group property imposed on theambient group: torsion-free abelian group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Definition

A subgroup of a group is termed a **pure subgroup of torsion-free abelian group** if the following equivalent conditions are satisfied:

- is a torsion-free abelian group and is a pure subgroup of .
- is a torsion-free abelian group and is a local powering-invariant subgroup of .
- is a torsion-free abelian group and the quotient group is also a torsion-free abelian group.