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

Statement
Suppose $$G$$ is a nilpotent group. Then, all members of the upper central series of $$G$$ are quotient-local powering-invariant subgroups and hence also local powering-invariant subgroups in $$G$$.

Similar facts

 * Center is local powering-invariant
 * Center is quotient-local powering-invariant in nilpotent group
 * Upper central series members are local powering-invariant in Lie ring

Opposite facts

 * Second center not is local powering-invariant in solvable group
 * Center not is quotient-local powering-invariant in solvable group

Style (A)

 * 1) uses::Center is quotient-local powering-invariant in nilpotent group
 * 2) uses::Nilpotency is quotient-closed
 * 3) uses::Quotient-local powering-invariance is quotient-transitive
 * 4) uses::Quotient-local powering-invariant implies local powering-invariant

Style (B)

 * 1) uses::Center is local powering-invariant
 * 2) uses::Local powering-invariance is quotient-transitive in nilpotent group
 * 3) uses::Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group

Using Style (A)
The proof of quotient-local powering-invariance follows directly from Facts (1)-(3), using the principle of mathematical induction. The proof of local powering-invariance follows by combining with Fact (4).

Using Style (B)
This proof is also fairly straightforward from the given facts.