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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.

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