Center is quotient-local powering-invariant in nilpotent group

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., quotient-local powering-invariant subgroup)}
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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 $G/Z(G)$.

Related facts

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. Center is local powering-invariant
2. Center is normal
3. 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).