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In [[mathematics]], in the realm of [[group theory]], a '''class automorphism''' is an [[automorphism]] of a [[Group (mathematics)|group]] that sends each element to within its [[conjugacy class]]. The class automorphisms form a subgroup of the automorphism group. Some facts:<br />
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* Every [[inner automorphism]] is a class automorphism.<br />
* Every class automorphism is a [[family automorphism]] and a [[quotientable automorphism]].<br />
* Under a quotient map, class automorphisms go to class automorphisms.<br />
* Every class automorphism is an [[IA automorphism]], that is, it acts as identity on the [[abelianization]].<br />
* Every class automorphism is a [[center-fixing automorphism]], that is, it fixes all points in the center.<br />
* [[Normal subgroup]]s are characterized as subgroups invariant under class automorphisms.<br />
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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.<br />
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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 (algebraic topology)|cocycle]] for the action that is locally a [[coboundary]] but is not a global coboundary.<br />
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==References==<br />
{{reflist}}<br />
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{{DEFAULTSORT:Class Automorphism}}<br />
[[Category:Group theory]]<br />
[[Category:Group automorphisms]]<br />
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{{group-theory-stub}}</div>imported>Boleyn