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.

Relation with other properties

Weaker properties

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

Related group properties

• Group having a class-inverting automorphism

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