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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about local powering-invariant subgroup |Get facts that use property satisfaction of local powering-invariant subgroup | Get facts that use property satisfaction of local powering-invariant subgroup|Get more facts about quotient-transitive subgroup property

Statement

Suppose $G$ is a nilpotent group with subgroups $H\le K$ 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).