Center is local powering-invariant

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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

Similar facts

Analogues in other algebraic structures

Opposite facts

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}</math> power is z, the above forces that yxy^{-1} = x.