Center is quotient-local powering-invariant in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$Z(G)$$ is the center of $$G$$. Then, $$Z(G)$$ is a quotient-local powering-invariant subgroup of $$G$$: if any element of $$G$$ has a unique $$p^{th}$$ root in $$G$$ for some prime number $$p$$, then its image in $$G/Z(G)$$ has a unique $$p^{th}$$ root in $$G/Z(G)$$.

Opposite facts

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

Facts used

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

Proof
The proof follows directly by combining Facts (1), (2), and (3).