Outer automorphism group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Let $G$ be a group. The outer automorphism group of $G$, denoted $\mathrm{O}\mathrm{u}\mathrm{t}(G)$, is defined as the quotient group of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ by the inner automorphism group $\mathrm{I}\mathrm{n}\mathrm{n}(G)$.

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