Outer automorphism group

From Groupprops
Jump to: navigation, search
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Outer automorphism group, all facts related to Outer automorphism group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]

Definition

Let G be a group. The outer automorphism group of G, denoted \operatorname{Out}(G), is defined as the quotient group of the automorphism group \operatorname{Aut}(G) by the 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