Pure subgroup of torsion-free abelian group

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


 * 1) $$G$$ is a torsion-free abelian group and $$H$$ is a pure subgroup of $$G$$.
 * 2) $$G$$ is a torsion-free abelian group and $$H$$ is a local powering-invariant subgroup of $$G$$.
 * 3) $$G$$ is a torsion-free abelian group and the quotient group $$G/H$$ is also a torsion-free abelian group.

Stronger properties

 * Direct factor of torsion-free abelian group: See pure subgroup of torsion-free abelian group not implies direct factor.