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 $K$

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

• Quasi-complete group is a group where every automorphism is inner, but where the center may be nontrivial
• Centerless group is a group where the center is trivial
• EAC-true group is a group where every extensible automorphism is inner
• Group that is isomorphic to its automorphism group (the property of being complete is stronger, because we require a particular map to give the isomorphism. An example of a group that is not complete, but is isomorphic to its automorphism group, is the dihedral group of order eight)