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Center (group theory)
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== Conjugation == Consider the map {{math|''f'' : ''G'' β Aut(''G'')}}, from {{math|''G''}} to the [[automorphism group]] of {{math|''G''}} defined by {{math|1=''f''(''g'') = ''Ο''{{sub|''g''}}}}, where {{math|''Ο''{{sub|''g''}}}} is the automorphism of {{math|''G''}} defined by :{{math|1=''f''(''g'')(''h'') = ''Ο''{{sub|''g''}}(''h'') = ''ghg''{{sup|β1}}}}. The function, {{math|''f''}} is a [[group homomorphism]], and its [[kernel (algebra)|kernel]] is precisely the center of {{math|''G''}}, and its image is called the [[inner automorphism group]] of {{math|''G''}}, denoted {{math|Inn(''G'')}}. By the [[first isomorphism theorem]] we get, :{{math|''G''/Z(''G'') β Inn(''G'')}}. The [[cokernel]] of this map is the group {{math|Out(''G'')}} of [[outer automorphism]]s, and these form the [[exact sequence]] :{{math|1 βΆ Z(''G'') βΆ ''G'' βΆ Aut(''G'') βΆ Out(''G'') βΆ 1}}.
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