Complete group

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:

	GAP ID
Symmetric group:S4	24 (12)
Symmetric group:S5	120 (34)

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:A5	60 (5)
Alternating group:A6	360 (118)

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

	GAP ID
Alternating group:A7
M16	16 (6)

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

• Group in which every automorphism is inner
• Group in which every normal subgroup is characteristic
• Centerless 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);