Outer automorphism group

Definition
Let $$G$$ be a group. The outer automorphism group of $$G$$, denoted $$\operatorname{Out}(G)$$, is defined as the quotient group of the defining ingredient::automorphism group $$\operatorname{Aut}(G)$$ by the defining ingredient::inner automorphism group $$\operatorname{Inn}(G)$$.

Elements of the outer automorphism group are sometimes termed outer automorphism classes. Each outer automorphism class is a coset of the inner automorphism group -- it is not a single automorphism. Thus, the outer automorphism group is not the set of outer automorphisms.

Facts

 * Homomorphism from automorphism group of connected groupoid to outer automorphism group at a point