# Universal power not implies class-preserving

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., universal power automorphism) neednotsatisfy the second automorphism property (i.e., class-preserving)

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## Contents

## Statement

A universal power automorphism of a group (i.e., an automorphism obtained by taking the power for some integer ) need not be a class-preserving automorphism: it need not preserve conjugacy classes of elements.

## Related facts

### Stronger facts

### Corollaries

## Proof

### Example of an Abelian group

Consider the cyclic group of order , where is an odd prime. The inverse map is a universal power automorphism of this group, and is *not* the identity map. Since the group is Abelian, the conjugacy classes are singleton, so we have a universal power automorphism that is not class-preserving.

The example generalizes to any Abelian group whose exponent is bigger than two.