Class-inverting automorphism

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.

Weaker properties

 * Stronger than::Extended class-preserving automorphism
 * Stronger than::Normal automorphism:

Related group properties

 * Group having a class-inverting automorphism

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