Group having a class-inverting automorphism

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group having a class-inverting automorphism is a group for which there exists a class-inverting automorphism: an automorphism that sends every element to an element in the conjugacy class of its inverse.

Relation with other properties

Stronger properties

• Ambivalent group
• Abelian group

Weaker properties

• Group in which every element is automorphic to its inverse

Group families

Group family	What groups in the family have a class-inverting automorphism?	Proof
alternating group $A_{n}$	$n=1,2,3,4,5,6,7,8,10,12,14$	Classification of alternating groups having a class-inverting automorphism
symmetric group $S_{n}$	all $n$	Symmetric groups are rational, rational implies ambivalent (every element is conjugate to its inverse), ambivalent implies there is a class-inverting automorphism (the identity automorphism works)
general linear group $GL(n,F)$ for a natural number $n$ and field $F$	all $n,F$	transpose-inverse map is class-inverting automorphism for general linear group
projective general linear group $PGL(n,F)$ for a natural number $n$ and field $F$	all $n,F$	transpose-inverse map induces class-inverting automorphism on projective general linear group
special linear group of degree two $SL(2,F)$ for a field $F$	all $F$	special linear group of degree two has a class-inverting automorphism
projective special linear group of degree two $PSL(2,F)$ for a field $F$	all $F$	projective special linear group of degree two has a class-inverting automorphism