An outer group automorphism is not conjugation by any element of the group. Its nonidentity class belongs to . A permutation-group example is the exceptional automorphism of that swaps its natural and transitive copies of ; it sends a transposition to a product of three disjoint transpositions and hence cannot be inner.
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In group theory, the outer automorphism group is a concept that quantifies the symmetries of a group that are not inherent to the group itself but arise from the way it can be related to other groups. To understand this concept, we should first cover some related definitions: 1. **Automorphism**: An automorphism of a group \( G \) is an isomorphism from the group \( G \) to itself.