Group having a class-inverting automorphism: Difference between revisions
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| [[general linear group]] <math>GL(n,F)</math> for a natural number <math>n</math> and field <math>F</math> || all <math>n,F</math> || [[transpose-inverse map is class-inverting automorphism for general linear group]] | | [[general linear group]] <math>GL(n,F)</math> for a natural number <math>n</math> and field <math>F</math> || all <math>n,F</math> || [[transpose-inverse map is class-inverting automorphism for general linear group]] | ||
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| [[projective general linear group]] <math>PGL(n,F)</math> for a natural number <math>n</math> and field <math>F</math> || all <math>n,F</math> || [[transpose-inverse map | | [[projective general linear group]] <math>PGL(n,F)</math> for a natural number <math>n</math> and field <math>F</math> || all <math>n,F</math> || [[transpose-inverse map induces class-inverting automorphism on projective general linear group]] | ||
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| [[special linear group of degree two]] <math>SL(2,F)</math> for a field <math>F</math> || all <math>F</math> || [[special linear group of degree two has a class-inverting automorphism]] | | [[special linear group of degree two]] <math>SL(2,F)</math> for a field <math>F</math> || all <math>F</math> || [[special linear group of degree two has a class-inverting automorphism]] | ||
Latest revision as of 03:55, 12 January 2014
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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 properties
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.
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 |
