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In [[mathematics]], a [[group (mathematics)|group]] ''G'' is said to be '''complete''' if every [[automorphism]] of ''G'' is [[inner automorphism|inner]], and the group is a centerless group; that is, it has a trivial [[outer automorphism group]] and trivial [[Center (group theory)|center]]. | |||
Equivalently, a group is complete if the conjugation map <math>G \to \mbox{Aut}(G)</math> (sending an element ''g'' to conjugation by ''g'') is an isomorphism: one-to-one implies centerless, as no inner automorphisms are the identity, while onto corresponds to no outer automorphisms. | |||
== Examples == | |||
As an example, all the [[symmetric group]]s ''S''<sub>''n''</sub> are complete except when ''n'' = 2 or 6. For the case ''n'' = 2 the group has a nontrivial center, while for the case ''n'' = 6 there is an [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|outer automorphism]]. | |||
The automorphism group of a simple group ''G'' is an [[almost simple group]]; | |||
for a nonabelian [[simple group]] ''G'', the automorphism group of ''G'' is complete. | |||
== Properties == | |||
A complete group is always [[isomorphism|isomorphic]] to its [[automorphism group]] (via sending an element to conjugation by that element), although the reverse need not hold: for example, the [[dihedral group]] of eight 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 ''G'' is a group extension given as a [[short exact sequence]] of groups | |||
:<math> 1 \rightarrow N \rightarrow G \rightarrow G' \rightarrow 1 </math> | |||
with [[Kernel (group theory)|kernel]] ''N'' and quotient '' G' ''. If the kernel ''N'' is a complete group then the extension splits: G is [[isomorphic]] to the [[direct product of groups|direct product]] N × G'. A proof using homomorphisms and exact sequences can be given in a natural way: The action of ''G'' (by [[Conjugation (group theory)|conjugation]]) on the normal subgroup ''N'' gives rise to a [[group homomorphism]] <math> \phi: G \rightarrow \mathrm{Aut}(N) \cong N </math>. Since Out(''N'') = 1 and ''N'' has trivial center the homomorphism φ is [[surjective]] and has an obvious section given by the inclusion of ''N'' in ''G''. The kernel of φ is the [[centralizer]] C<sub>''G''</sub>(''N'') of ''N'' in ''G'', and so ''G'' is at least a [[semidirect product]] C<sub>''G''</sub>(''N'') ⋊ ''N'', but the action of ''N'' on C<sub>''G''</sub>(''N'') is trivial, and so the product is direct. This proof is somewhat interesting since the original exact sequence is reversed during the proof. | |||
This can be restated in terms of elements and internal conditions: If ''N'' is a normal, complete subgroup of a group ''G'', then ''G'' = C<sub>''G''</sub>(''N'') × ''N'' is a direct product. The proof follows directly from the definition: ''N'' is centerless giving C<sub>''G''</sub>(''N'') ∩ ''N'' is trivial. If ''g'' is an element of ''G'' then it induces an automorphism of ''N'' by conjugation, but ''N'' = Aut(''N'') and this conjugation must be equal to conjugation by some element ''n'' of ''N''. Then conjugation by ''gn''<sup>−1</sup> is the identity on ''N'' and so ''gn''<sup>−1</sup> is in C<sub>''G''</sub>(''N'') and every element ''g'' of ''G'' is a product (''gn''<sup>−1</sup>)''n'' in 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]]: [http://arxiv.org/abs/math/9808094v1 How tall is the automorphism tower of a group?] | |||
[[Category:Properties of groups]] | |||
Revision as of 06:18, 27 January 2014
In mathematics, a group G is said to be complete if every automorphism of G is inner, and the group is a centerless group; that is, it has a trivial outer automorphism group and trivial center. Equivalently, a group is complete if the conjugation map (sending an element g to conjugation by g) is an isomorphism: one-to-one implies centerless, as no inner automorphisms are the identity, while onto corresponds to no outer automorphisms.
Examples
As an example, all the symmetric groups Sn are complete except when n = 2 or 6. For the case n = 2 the group has a nontrivial center, while for the case n = 6 there is an outer automorphism.
The automorphism group of a simple group G is an almost simple group; for a nonabelian simple group G, the automorphism group of G is complete.
Properties
A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the reverse need not hold: for example, the dihedral group of eight elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see Template:Harv.
Extensions of complete groups
Assume that a group G is a group extension given as a short exact sequence of groups
with kernel N and quotient G' . If the kernel N is a complete group then the extension splits: G is isomorphic to the direct product N × G'. A proof using homomorphisms and exact sequences can be given in a natural way: The action of G (by conjugation) on the normal subgroup N gives rise to a group homomorphism . Since Out(N) = 1 and N has trivial center the homomorphism φ is surjective and has an obvious section given by the inclusion of N in G. The kernel of φ is the centralizer CG(N) of N in G, and so G is at least a semidirect product CG(N) ⋊ N, but the action of N on CG(N) is trivial, and so the product is direct. This proof is somewhat interesting since the original exact sequence is reversed during the proof.
This can be restated in terms of elements and internal conditions: If N is a normal, complete subgroup of a group G, then G = CG(N) × N is a direct product. The proof follows directly from the definition: N is centerless giving CG(N) ∩ N is trivial. If g is an element of G then it induces an automorphism of N by conjugation, but N = Aut(N) and this conjugation must be equal to conjugation by some element n of N. Then conjugation by gn−1 is the identity on N and so gn−1 is in CG(N) and every element g of G is a product (gn−1)n in CG(N)N.
References
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