# Center is local powering-invariant

From Groupprops

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)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

The center of a group is a local powering-invariant subgroup. Explicitly, suppose is a group and is the center. Suppose and is a natural number such that there is a unique satisfying . Then, .

## Related facts

### Generalizations

### Similar facts

### Analogues in other algebraic structures

### 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

## Proof

**Given**: Group with center . Element and natural number such that there exists a unique satisfying .

**To prove**: . In other words, for all .

**Proof**: We have by Fact (1) that:

Simplifying further, we get that:

where we use that . Since is the unique element of whose <mah>n^{th}</math> power is , the above forces that .