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This article defines a group property: a property that can be evaluated to true/false for any given group
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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Formalisms 2.1 In terms of the supergroup property collapse operator 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 4 Testing 4.1 GAP code

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
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of semi-basic definitions on this wiki

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) (viz the center of G) is trivial and Inn(G) = Aut(G) (viz every automorphism of G is inner)
• The natural homomorphism given by (where ) is an isomorphism
• For any embedding of G as a normal subgroup of some group K, G is a direct factor of K

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 whose order is not 2 or 6
• Automorphism group of a non-Abelian characteristically simple group

Weaker properties

• Group in which every automorphism is inner
• Group in which every normal subgroup is characteristic
• Centerless group
• EAC-true group
• Group isomorphic to its automorphism group

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);