Complete group: Difference between revisions
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* It is [[centerless group|centerless]] and every [[automorphism]] of it is [[inner automorphism|inner]] | * It is [[centerless group|centerless]] and every [[automorphism]] of it is [[inner automorphism|inner]] | ||
* The natural homomorphism to the automorphism group that sends each element to the conjugation via that element, is an isomorphism | * The natural homomorphism to the automorphism group that sends each element to the conjugation via that element, is an isomorphism | ||
* Whenever it is embedded as a [[normal subgroup]] inside a bigger group, it is actually a [[direct factor]] inside that bigger group | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
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* <math>Z(G)</math> (viz the [[center]] of <math>G</math>) is trivial and <math>Inn(G) = Aut(G)</math> (viz every automorphism of <math>G</math> is inner) | * <math>Z(G)</math> (viz the [[center]] of <math>G</math>) is trivial and <math>Inn(G) = Aut(G)</math> (viz every automorphism of <math>G</math> is inner) | ||
* The natural homomorphism <math>G \to Aut(G)</math> given by <math>g \mapsto c_g</math> (where <math>c_g = x \mapsto gxg^{-1}</math>) is an isomorphism | * The natural homomorphism <math>G \to Aut(G)</math> given by <math>g \mapsto c_g</math> (where <math>c_g = x \mapsto gxg^{-1}</math>) is an isomorphism | ||
* For any embedding of <math>G</math> as a [[normal subgroup]] of some group <math>K</math>, <math>G</math> is a [[direct factor]] of <math>K</math> | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 20:04, 24 December 2007
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition
Symbol-free definition
A group is said to be complete if it satisfies the following equivalent conditions:
- It is centerless and every automorphism of it is inner
- The natural homomorphism to the automorphism group that sends each element to the conjugation via that element, is an isomorphism
- Whenever it is embedded as a normal subgroup inside a bigger group, it is actually a direct factor inside that bigger group
Definition with symbols
A group is said to be complete if it satisfies the following equivalent conditions:
- (viz the center of ) is trivial and (viz every automorphism of is inner)
- The natural homomorphism given by (where ) is an isomorphism
- For any embedding of as a normal subgroup of some group , is a direct factor of
Relation with other properties
Stronger properties
- Symmetric group whose order is not 2 or 6
- Automorphism group of a non-Abelian characteristically simple group
Weaker properties
- Quasi-complete group is a group where every automorphism is inner, but where the center may be nontrivial
- Centerless group is a group where the center is trivial
- EAC-true group is a group where every extensible automorphism is inner