Local powering-invariant normal subgroup of nilpotent group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a local powering-invariant normal subgroup of nilpotent group if it satisfies the following equivalent conditions:


 * 1) $$G$$ is a nilpotent group and $$H$$ is a local powering-invariant normal subgroup of $$G$$, i.e., $$H$$ is both a local powering-invariant subgroup and a normal subgroup of $$G$$.
 * 2) $$G$$ is a nilpotent group and $$H$$ is a quotient-local powering-invariant subgroup of $$G$$.
 * 3) $$G$$ is a nilpotent group and $$H$$ is a quotient-torsion-freeness-closed subgroup of $$G$$.

Equivalence of definitions

 * Equivalence of (1) and (2) follows from local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group.