Complete group: Difference between revisions

Latest revision as of 05:07, 9 October 2015

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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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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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 $G$ is said to be complete if it satisfies the following equivalent conditions:

$Z(G)$ (the center of $G$ ) is trivial and $\operatorname {Inn} (G)=\operatorname {Aut} (G)$ (i.e., every automorphism of $G$ is inner)
The natural homomorphism $G\to \operatorname {Aut} (G)$ given by $g\mapsto c_{g}$ (where $c_{g}=x\mapsto gxg^{-1}$ ) is an isomorphism
For any embedding of $G$ as a normal subgroup of some group $K$ , $G$ is a direct factor of $K$

Examples

Extreme examples

The trivial group is complete.

Groups satisfying the property

Here are some basic/important groups satisfying the property:

	GAP ID
Symmetric group:S3	6 (1)

Here are some relatively less basic/important groups satisfying the property:

Here are some even more complicated/less basic groups satisfying the property:

Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

	GAP ID
Alternating group:A6	360 (118)

Here are some even more complicated/less basic groups that do not satisfy the property:

Formalisms

In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (direct factor), and vice versa.
View other group properties obtained in this way

A group $G$ is complete if and only if whenever $G$ is embedded as a normal subgroup in some group $K$ , $G$ is a direct factor of $K$ .

Relation with other properties

Stronger properties

Symmetric group on a set of size other than $2$ or $6$ : For full proof, refer: Symmetric groups on finite sets are complete, Symmetric groups on infinite sets are complete
Automorphism group of a non-Abelian characteristically simple group: For full proof, refer: Characteristically simple and non-Abelian implies automorphism group is complete

Weaker properties

Testing

GAP code

One can write code to test this group property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this group property at: IsCompleteGroup
View other GAP-codable group properties | View group properties with in-built commands

While there is no built-in command to test completeness, this can be done with a short snippet of code available at GAP:IsCompleteGroup. The function is invoked as follows:

IsCompleteGroup(group);

@@ Line 1: / Line 1: @@
+{{semibasicdef}}
 {{group property}}
-{{semibasicdef}}
 ==Definition==
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 A [[group]] is said to be '''complete''' if it satisfies the following equivalent conditions:
-* It is [[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===
@@ Line 16: / Line 16: @@
 A [[group]] <math>G</math> is said to be '''complete''' if it satisfies the following equivalent conditions:
-* <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> (the [[center]] of <math>G</math>) is trivial and <math>\operatorname{Inn}(G) = \operatorname{Aut}(G)</math> (i.e., 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 \operatorname{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>
+==Examples==
+===Extreme examples===
+* The [[trivial group]] is complete.
+===Groups satisfying the property===
+{{groups satisfying property sorted by importance rank}}
+===Groups dissatisfying the property===
+{{groups dissatisfying property sorted by importance rank}}
+==Formalisms==
+{{supergroup property collapse|normal subgroup|direct factor}}
+A group <math>G</math> is complete if and only if whenever <math>G</math> is embedded as a [[normal subgroup]] in some group <math>K</math>, <math>G</math> is a direct factor of <math>K</math>.
 ==Relation with other properties==
 ===Stronger properties===
-* [[Symmetric group]] whose order is not 2 or 6
+* [[Symmetric group]] on a set of size other than <math>2</math> or <math>6</math>: {{proofat|[[Symmetric groups on finite sets are complete]], [[Symmetric groups on infinite sets are complete]]}}
-* Automorphism group of a non-Abelian [[characteristically simple group]]
+* Automorphism group of a non-Abelian [[characteristically simple group]]: {{proofat|[[Characteristically simple and non-Abelian implies automorphism group is complete]]}}
 ===Weaker properties===
-* [[Quasi-complete group]] is a group where every automorphism is inner, but where the center may be nontrivial
+* [[Stronger than::Group in which every automorphism is inner]]
-* [[Centerless group]] is a group where the center is trivial
+* [[Stronger than::Group in which every normal subgroup is characteristic]]
-* [[EAC-true group]] is a group where every [[extensible automorphism]] is inner
+* [[Stronger than::Centerless group]]
+* [[Stronger than::Group isomorphic to its automorphism group]]
+==Testing==
+{{GAP code for group property|
+test = IsCompleteGroup}}
+While there is no built-in command to test completeness, this can be done with a short snippet of code available at [[GAP:IsCompleteGroup]]. The function is invoked as follows:
+<pre>IsCompleteGroup(group);</pre>