# Center is local powering-invariant

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., local powering-invariant subgroup)}
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## 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, $x \in Z$.

## Facts used

1. 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 <mah>n^{th}[/itex] power is $z$, the above forces that $yxy^{-1} = x$.