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In [[mathematics]], a [[group (mathematics)|group]] {{math|''G''}} is said to be '''complete''' if every [[automorphism]] of {{math|''G''}} is [[inner automorphism|inner]], and it is centerless; that is, it has a trivial [[outer automorphism group]] and trivial [[center (group theory)|center]]. Equivalently, a group is complete if the [[conjugation (group theory)|conjugation]] map, {{math|''G'' β Aut(''G'')}} (sending an element {{math|''g''}} to conjugation by {{math|''g''}}), is an [[group isomorphism|isomorphism]]: [[injective|injectivity]] implies that only conjugation by the [[identity element]] is the identity automorphism, meaning the group is centerless, while [[surjective|surjectivity]] implies it has no outer automorphisms. == Examples == As an example, all the [[symmetric group]]s, {{math|S{{sub|''n''}}}}, are complete except when {{math|''n'' β {2, 6}}}. For the case {{math|''n'' {{=}} 2}}, the group has a non-trivial center, while for the case {{math|''n'' {{=}} 6}}, there is an [[automorphisms of the symmetric and alternating groups#exceptional outer automorphism|outer automorphism]]. The automorphism group of a [[simple group]] is an [[almost simple group]]; for a non-[[abelian group|abelian]] simple group {{math|''G''}}, the automorphism group of {{math|''G''}} is complete. == Properties == A complete group is always [[group isomorphism|isomorphic]] to its [[automorphism group]] (via sending an element to conjugation by that element), although the converse need not hold: for example, the [[dihedral group]] of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see {{harv|Robinson|1996|loc=section 13.5}}. == Extensions of complete groups == Assume that a group {{math|''G''}} is a group extension given as a [[short exact sequence]] of groups : {{math|1 βΆ ''N'' βΆ ''G'' βΆ ''G''β² βΆ 1}} with [[kernel (group theory)|kernel]], {{math|''N''}}, and quotient, {{math|''G''β²}}. If the kernel, {{math|''N''}}, is a complete group then the extension splits: {{math|''G''}} is isomorphic to the [[direct product of groups|direct product]], {{math|''N'' Γ ''G''β²}}. A proof using [[homomorphism]]s and exact sequences can be given in a natural way: The action of {{math|''G''}} (by [[conjugation (group theory)|conjugation]]) on the [[normal subgroup]], {{math|''N''}}, gives rise to a group homomorphism, {{math|''Ο'' : ''G'' β Aut(''N'') β ''N''}}. Since {{math|Out(''N'') {{=}} 1}} and {{math|''N''}} has trivial center the homomorphism {{math|''Ο''}} is surjective and has an obvious section given by the inclusion of {{math|''N''}} in {{math|''G''}}. The kernel of {{math|''Ο''}} is the [[centralizer]] {{math|C{{sub|''G''}}(''N'')}} of {{math|''N''}} in {{math|''G''}}, and so {{math|''G''}} is at least a [[semidirect product]], {{math|C<sub>''G''</sub>(''N'') β ''N''}}, but the action of {{math|''N''}} on {{math|C<sub>''G''</sub>(''N'')}} is trivial, and so the product is direct. This can be restated in terms of elements and internal conditions: If {{math|''N''}} is a normal, complete [[subgroup]] of a group {{math|''G''}}, then {{math|''G'' {{=}} C{{sub|''G''}}(''N'') Γ ''N''}} is a direct product. The proof follows directly from the definition: {{math|''N''}} is centerless giving {{math|C<sub>''G''</sub>(''N'') ∩ ''N''}} is trivial. If {{math|''g''}} is an element of {{math|''G''}} then it induces an automorphism of {{math|''N''}} by conjugation, but {{math|''N'' {{=}} Aut(''N'')}} and this conjugation must be equal to conjugation by some element {{math|''n''}} of {{math|''N''}}. Then conjugation by {{math|''gn''{{sup|β1}}}} is the identity on {{math|''N''}} and so {{math|''gn''<sup>β1</sup>}} is in {{math|C<sub>''G''</sub>(''N'')}} and every element, {{math|''g''}}, of {{math|''G''}} is a product {{math|(''gn''<sup>β1</sup>)''n''}} in {{math|C<sub>''G''</sub>(''N'')''N''}}. == References == * {{Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}} * {{Citation | last1=Rotman | first1=Joseph J. | title=An introduction to the theory of groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94285-8 | year=1994}} (chapter 7, in particular theorems 7.15 and 7.17). == External links == * [[Joel David Hamkins]]: [https://arxiv.org/abs/math/9808094v1 How tall is the automorphism tower of a group?] [[Category:Properties of groups]]
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