Local powering-invariance is quotient-transitive in nilpotent group

This article gives the statement, and possibly proof, of a subgroup property (i.e., local powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Suppose $G$ is a nilpotent group with subgroups $H\leq K$ satisfying the following:

• $H$ is a normal subgroup of $G$.
• $H$ is a local powering-invariant subgroup of $G$, i.e., for $h\in H$ and $n\in \mathbb {N}$ such that the equation $x^{n}=h$ has a unique solution for $x$ in $G$, we must have $x\in H$.
• $K/H$ is a local powering-invariant subgroup of the quotient group $G/H$.

Then, $K$ is a local powering-invariant subgroup of $G$.

Related facts

Applications

• Upper central series members are local powering-invariant in nilpotent group

Opposite facts

• Second center not is local powering-invariant in solvable group

Facts used

1. Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
2. Local powering-invariant over quotient-local powering-invariant implies local powering-invariant

Proof

The proof follows directly by combining Facts (1) and (2).