Complete group

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 $$K$$

Extreme examples

 * The trivial group is complete.

Formalisms
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 $$K$$.

Stronger properties

 * Symmetric group on a set of size other than $$2$$ or $$6$$:
 * Automorphism group of a non-Abelian characteristically simple group:

Weaker properties

 * Stronger than::Group in which every automorphism is inner
 * Stronger than::Group in which every normal subgroup is characteristic
 * Stronger than::Centerless group
 * Stronger than::Group isomorphic to its automorphism group

Testing
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);