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- Class_automorphism 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 family automorphism 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 center-fixing automorphism, 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 cocycle for the action that is locally a coboundary but is not a global coboundary.".
- Class_automorphism wikiPageID "5632720".
- Class_automorphism wikiPageLength "1919".
- Class_automorphism wikiPageOutDegree "18".
- Class_automorphism wikiPageRevisionID "596522057".
- Class_automorphism wikiPageWikiLink Abelianization.
- Class_automorphism wikiPageWikiLink Automorphism.
- Class_automorphism wikiPageWikiLink Category:Group_automorphisms.
- Class_automorphism wikiPageWikiLink Category:Group_theory.
- Class_automorphism wikiPageWikiLink Center-fixing_automorphism.
- Class_automorphism wikiPageWikiLink Chain_complex.
- Class_automorphism wikiPageWikiLink Coboundary.
- Class_automorphism wikiPageWikiLink Cocycle_(algebraic_topology).
- Class_automorphism wikiPageWikiLink Commutator_subgroup.
- Class_automorphism wikiPageWikiLink Conjugacy_class.
- Class_automorphism wikiPageWikiLink Family_automorphism.
- Class_automorphism wikiPageWikiLink Group_(mathematics).
- Class_automorphism wikiPageWikiLink Group_theory.
- Class_automorphism wikiPageWikiLink IA_automorphism.
- Class_automorphism wikiPageWikiLink Inner_automorphism.
- Class_automorphism wikiPageWikiLink Mathematics.
- Class_automorphism wikiPageWikiLink Normal_subgroup.
- Class_automorphism wikiPageWikiLink Outer_automorphism.
- Class_automorphism wikiPageWikiLink Outer_automorphism_group.
- Class_automorphism wikiPageWikiLink Quotientable_automorphism.
- Class_automorphism wikiPageWikiLink Stability_group.
- Class_automorphism wikiPageWikiLinkText "class automorphism".
- Class_automorphism hasPhotoCollection Class_automorphism.
- Class_automorphism wikiPageUsesTemplate Template:Unreferenced.
- Class_automorphism subject Category:Group_automorphisms.
- Class_automorphism subject Category:Group_theory.
- Class_automorphism hypernym Automorphism.
- Class_automorphism type Article.
- Class_automorphism type Article.
- Class_automorphism type Function.
- Class_automorphism 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: Every inner automorphism is a class automorphism. Every class automorphism is a family automorphism and a quotientable automorphism. Under a quotient map, class automorphisms go to class automorphisms.".
- Class_automorphism label "Class automorphism".
- Class_automorphism sameAs m.0dxbbf.
- Class_automorphism sameAs Q5128014.
- Class_automorphism sameAs Q5128014.
- Class_automorphism wasDerivedFrom Class_automorphism?oldid=596522057.
- Class_automorphism isPrimaryTopicOf Class_automorphism.