outer automorphism group

The outer automorphism groupMathworldPlanetmath of a group is the quotientPlanetmathPlanetmath (http://planetmath.org/QuotientGroup) of its automorphism groupMathworldPlanetmath by its inner automorphism group:


There is some variance in terminology about “an outer automorphism.” Some authors define an outer automorphism as any automorphismPlanetmathPlanetmathPlanetmathPlanetmath ϕ:GG which is not an inner automorphism. In this way an outer automorphism still permutes the group G. However, an equally common definition is to declare an outer automorphism as an element of Out(G) and consequently the elements are cosets of Inn(G)ϕ, and not a map ϕ:GG. In this definition it is not generally possible to treat the element as a permutationMathworldPlanetmath of G. In particular, the outer automorphism group of a general group G does not act on the group G in a natural way. An exception is when G is abelianMathworldPlanetmath so that Inn(G)=1; thus, the elements of Out(G) are canonically identified with those of Aut(G) so we can speak of the action by outer automorphisms.

Title outer automorphism group
Canonical name OuterAutomorphismGroup
Date of creation 2013-03-22 14:01:26
Last modified on 2013-03-22 14:01:26
Owner Thomas Heye (1234)
Last modified by Thomas Heye (1234)
Numerical id 13
Author Thomas Heye (1234)
Entry type Definition
Classification msc 20F28
Related topic InnerAutomorphism
Defines outer automorphism group