Complete group

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 $G$ 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 $G\to Aut(G)$ 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 $K$

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