Complete group (reviewed)

A group $G$ is complete if one of the following equivalent conditions holds:

• it has trivial center and every automorphism is inner,
• the natural map from $G$ to its inner automorphism group is an isomorphism,
• whenever it is embedded as a normal subgroup of a larger group, it is a direct factor.

For $n \ne 2, 6$, the symmetric group $S_n$ is complete.