About: Class automorphism

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In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts: For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a for the action that is locally a coboundary but is not a global coboundary.

Attributes	Values
rdfs:label	Class automorphism (en)
rdfs:comment	In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts: For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a for the action that is locally a coboundary but is not a global coboundary. (en)
dcterms:subject	Group theory Group automorphisms
Wikipage page ID	5632720 (xsd:integer)
Wikipage revision ID	596522057 (xsd:integer)
Link from a Wikipage to another Wikipage	Group theory Mathematics Normal subgroup Quotientable automorphism Conjugacy class Automorphism Abelianization Group (mathematics) Group automorphisms Group theory Inner automorphism Stability group IA automorphism Coboundary Outer automorphism dbr:Cocycle_(algebraic_topology) dbr:Center-fixing_automorphism dbr:Family_automorphism
sameAs	Class automorphism Class automorphism Class automorphism
dbp:wikiPageUsesTemplate	dbt:Unreferenced
has abstract	In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts: * Every inner automorphism is a class automorphism. * Every class automorphism is a and a quotientable automorphism. * Under a quotient map, class automorphisms go to class automorphisms. * Every class automorphism is an IA automorphism, that is, it acts as identity on the Abelianization. * Every class automorphism is a , that is, it fixes all points in the center. * Normal subgroups are characterized as subgroups invariant under class automorphisms. For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation. This conjugation defines an outer automorphism on the group of finitary permutations. However, for any specific finitary permutation, we can find a finitary permutation whose conjugation has the same effect as this infinitary permutation. This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support. For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a for the action that is locally a coboundary but is not a global coboundary. (en)
gold:hypernym	Automorphism
prov:wasDerivedFrom	wikipedia-en:Class_automorphism?oldid=596522057&ns=0
page length (characters) of wiki page	1919 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Class_automorphism
is Link from a Wikipage to another Wikipage of	Quotientable automorphism Conjugacy-closed subgroup
is foaf:primaryTopic of	wikipedia-en:Class_automorphism

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