# Group having a class-inverting automorphism

## Contents

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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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.

## 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