# Class-inverting automorphism

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)
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## 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 $\sigma$ of a group $G$ is termed a class-inverting automorphism if, for any $g \in G$, there exists $x \in G$ such that $\sigma(g) = xg^{-1}x^{-1}$.

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.