Outer automorphism group ($\mathrm{Out}(G)$)

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.