Class-inverting automorphism: Difference between revisions
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An [[automorphism]] <math>\sigma</math> of a [[group]] <math>G</math> is termed a '''class-inverting automorphism''' if, for any <math>g \in G</math>, there exists <math>x \in G</math> such that <math>\sigma(g) = xg^{-1}x^{-1}</math>. | An [[automorphism]] <math>\sigma</math> of a [[group]] <math>G</math> is termed a '''class-inverting automorphism''' if, for any <math>g \in G</math>, there exists <math>x \in G</math> such that <math>\sigma(g) = xg^{-1}x^{-1}</math>. | ||
Note that the set of class-inverting automorphisms, if non-empty, is | Note that the set of class-inverting automorphisms, if non-empty, is a group only if the group is an [[ambivalent group]], i.e., every element is conjugate to its inverse. In this case, it is the group of [[class-preserving automorphism]]s. | ||
Otherwise, it is a single coset of the group of [[class-preserving automorphism]]s, and the class-preserving and class-inverting automorphisms together form a [[normal subgroup]] of the automorphism group. | |||
==Relation with other properties== | ==Relation with other properties== | ||
===Weaker properties=== | |||
* [[Stronger than::Extended class-preserving automorphism]] | |||
* [[Stronger than::Normal automorphism]]: {{proofat|[[Class-inverting implies normal]]}} | |||
===Related group properties=== | ===Related group properties=== | ||
* [[Group having a class-inverting automorphism]] | * [[Group having a class-inverting automorphism]] | ||
==Facts== | |||
===Alternating and linear groups=== | |||
* [[Classification of alternating groups having a class-inverting automorphism]] | |||
* [[Transpose-inverse map is class-inverting automorphism for general linear group]] | |||
* [[Transpose-inverse map induces class-inverting automorphism on projective general linear group]] | |||
* [[Special linear group of degree two has a class-inverting automorphism]] | |||
* [[Projective special linear group of degree two has a class-inverting automorphism]] | |||
Latest revision as of 15:00, 3 September 2009
This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties
Definition
Symbol-free definition
An automorphism of a group is termed a class-inverting automorphism if it sends every element to an element that is in the conjugacy class of its inverse.
Definition with symbols
An automorphism of a group is termed a class-inverting automorphism if, for any , there exists such that .
Note that the set of class-inverting automorphisms, if non-empty, is a group only if the group is an ambivalent group, i.e., every element is conjugate to its inverse. In this case, it is the group of class-preserving automorphisms.
Otherwise, it is a single coset of the group of class-preserving automorphisms, and the class-preserving and class-inverting automorphisms together form a normal subgroup of the automorphism group.
Relation with other properties
Weaker properties
- Extended class-preserving automorphism
- Normal automorphism: For full proof, refer: Class-inverting implies normal
Related group properties
Facts
Alternating and linear groups
- Classification of alternating groups having a class-inverting automorphism
- Transpose-inverse map is class-inverting automorphism for general linear group
- Transpose-inverse map induces class-inverting automorphism on projective general linear group
- Special linear group of degree two has a class-inverting automorphism
- Projective special linear group of degree two has a class-inverting automorphism