Automorphism group of a group

From Groupprops
Revision as of 02:54, 28 May 2013 by Vipul (talk | contribs)
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 Automorphism group of a group, all facts related to Automorphism group of a 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

Symbol-free definition

The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.

Definition with symbols

The automorphism group of a group G, denoted Aut(G), is a set whose elements are automorphisms \sigma:G \to G, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of \operatorname{Sym}(G), the group of all permutations on G.

Subgroups

Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Examples are the property of being an inner automorphism, class automorphism, extensible automorphism.