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

Weaker properties

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