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, $x\in Z$.

Related facts

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. 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:

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

Simplifying further, we get that:

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

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