# Power automorphism not implies universal power automorphism

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., power automorphism) neednotsatisfy the second automorphism property (i.e., universal power automorphism)

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## Contents

## Statement

A power automorphism of a group (i.e., an automorphism that sends every element to a power of itself) need not be a universal power automorphism.

## Definitions used

### Power automorphism

`Further information: Power automorphism, power map`

Let be a group. An automorphism of is termed a power automorphism of if, for every , there exists a such that .

### Universal power automorphism

`Further information: Universal power automorphism, universal power map`

Let be a group. An automorphism of is termed a universal power automorphism (or a uniform power automorphism) if there exists a such that for all .

## Proof

### Example of the quaternion group

In the quaternion group:

conjugation by is a power automorphism that is *not* a universal power automorphism.

- It is a power automorphism: It fixes , and sends the other elements to their inverses.
- It is not a universal power automorphism: It sends the elements to their inverses. Since each of these elements have order four, any for which this automorphism is the power map, must be congruent to mod . On the other hand, since it fixes , which also has order four, must be modulo , a contradiction.