Complete group

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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 $K$