# Center is quotient-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., quotient-powering-invariant subgroup)}

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

## Statement

Suppose is a group and denotes the center of . Then, is a quotient-powering-invariant subgroup of . Explicitly, if is a prime number such that is powered over , the quotient group (which can be identified with the inner automorphism group of ) is also powered over .

## Facts used

We essentially use that the center is a fixed-point subgroup of a subgroup of the automorphism group (for Fact (1)) and that it is a central subgroup (for Fact (2)) to get the result.

- Center is local powering-invariant (and hence, the center is powering-invariant). This is a special case of the fact that fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
- Powering-invariant and central implies quotient-powering-invariant

## Proof

The proof follows directly by combining Facts (1) and (2).