# 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

View a complete list of group propertiesVIEW 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.

## 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 | Classification of alternating groups having a class-inverting automorphism | |

symmetric group | all | 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 for a natural number and field | all | transpose-inverse map is class-inverting automorphism for general linear group |

projective general linear group for a natural number and field | all | transpose-inverse map induces class-inverting automorphism on projective general linear group |

special linear group of degree two for a field | all | special linear group of degree two has a class-inverting automorphism |

projective special linear group of degree two for a field | all | projective special linear group of degree two has a class-inverting automorphism |