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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Roe on 2021-10-06 02:24:32

**Referred to by:**

**History:**(expand/hide all)

- 2021-10-06 02:24:32 by David Roe (Reviewed)