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Statements

Subject Item
dbr:Quotientable_automorphism
dbo:wikiPageWikiLink
dbr:Class_automorphism
Subject Item
dbr:Conjugacy-closed_subgroup
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dbr:Class_automorphism
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dbr:Class_automorphism
rdfs:label
Class automorphism
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.
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dbc:Group_theory dbc:Group_automorphisms
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596522057
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dbr:Inner_automorphism dbr:Quotientable_automorphism dbr:Automorphism dbr:Cocycle_(algebraic_topology) dbr:IA_automorphism dbc:Group_automorphisms dbr:Family_automorphism dbr:Group_theory dbr:Outer_automorphism dbr:Normal_subgroup dbr:Conjugacy_class dbr:Center-fixing_automorphism dbc:Group_theory dbr:Group_(mathematics) dbr:Abelianization dbr:Coboundary dbr:Mathematics dbr:Stability_group
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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: * 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.
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dbr:Automorphism
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