Lower central series members are quotient-powering-invariant in nilpotent group

Statement

Suppose $G$ is a nilpotent group. Then, all members of the lower central series of $G$ are quotient-powering-invariant subgroups of $G$.

Related facts

• Upper central series members are quotient-powering-invariant
• Lower central series members are divisibility-invariant in nilpotent group

Facts used

1. Lower central series members are divisibility-invariant in nilpotent group
2. Divisibility-invariant implies powering-invariant
3. Normal subgroup contained in the hypercenter that is powering-invariant is quotient-powering-invariant

Proof

The proof follows by combining Facts (1), (2), and (3), with the observation that in a nilpotent group, the hypercenter is the whole group, so every normal subgroup is contained in the hypercenter, and moreover, the lower central series members are normal (in fact, they are verbal subgroups, hence fully invariant subgroups and also characteristic subgroups).