Center is local powering-invariant

Statement
The center of a group is a local powering-invariant subgroup. Explicitly, suppose $$G$$ is a group and $$Z$$ is the center. Suppose $$z \in Z$$ and $$n$$ is a natural number such that there is a unique $$x \in G$$ satisfying $$x^n = z$$. Then, $$x \in Z$$.

Generalizations

 * Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant

Similar facts

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

Analogues in other algebraic structures

 * Center is local powering-invariant in Lie ring

Opposite facts

 * Center not is quotient-local powering-invariant
 * Derived subgroup not is local powering-invariant
 * Second center not is local powering-invariant in solvable group
 * Characteristic not implies powering-invariant

Facts used

 * 1) uses::Group acts as automorphisms by conjugation

Proof
Given: Group $$G$$ with center $$Z$$. Element $$z \in Z$$ and natural number $$n$$ such that there exists a unique $$x \in G$$ satisfying $$x^n = z$$.

To prove: $$x \in Z$$. In other words, $$yxy^{-1} = x$$ for all $$y \in G$$.

Proof: We have by Fact (1) that:

$$\! (yxy^{-1})^n = yx^ny^{-1}$$

Simplifying further, we get that:

$$\! (yxy^{-1})^n = yx^ny^{-1} = yzy^{-1} = z$$

where we use that $$x^n = z \in Z$$. Since $$x$$ is the unique element of $$G$$ whose n^{th} power is $$z$$, the above forces that $$yxy^{-1} = x$$.