# Class-inverting automorphism

From Groupprops

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)

View other automorphism properties OR View other function properties

## Contents

## Definition

### Symbol-free definition

An automorphism of a group is termed a **class-inverting automorphism** if it sends every element to an element that is in the conjugacy class of its inverse.

### Definition with symbols

An automorphism of a group is termed a **class-inverting automorphism** if, for any , there exists such that .

Note that the set of class-inverting automorphisms, if non-empty, is a group only if the group is an ambivalent group, i.e., every element is conjugate to its inverse. In this case, it is the group of class-preserving automorphisms.

Otherwise, it is a single coset of the group of class-preserving automorphisms, and the class-preserving and class-inverting automorphisms together form a normal subgroup of the automorphism group.

## Relation with other properties

### Weaker properties

- Extended class-preserving automorphism
- Normal automorphism:
`For full proof, refer: Class-inverting implies normal`

### Related group properties

## Facts

### Alternating and linear groups

- Classification of alternating groups having a class-inverting automorphism
- Transpose-inverse map is class-inverting automorphism for general linear group
- Transpose-inverse map induces class-inverting automorphism on projective general linear group
- Special linear group of degree two has a class-inverting automorphism
- Projective special linear group of degree two has a class-inverting automorphism