inner automorphism

Let G be a group. For every xG, we define a mapping


called conjugationMathworldPlanetmath by x. It is easy to show the conjugation map is in fact, a group automorphismMathworldPlanetmath.

An automorphismPlanetmathPlanetmathPlanetmathPlanetmath of G that corresponds to conjugation by some xG is called inner. An automorphism that isn’t inner is called an outer automorphism.

The composition operation gives the set of all automorphisms of G the structureMathworldPlanetmath of a group, Aut(G). The inner automorphisms also form a group, Inn(G), which is a normal subgroupMathworldPlanetmath of Aut(G). Indeed, if ϕx,xG is an inner automorphism and π:GG an arbitrary automorphism, then


Let us also note that the mapping


is a surjectivePlanetmathPlanetmath group homomorphism with kernel Z(G), the centre subgroupMathworldPlanetmathPlanetmath. Consequently, Inn(G) is naturally isomorphic to the quotient of G/Z(G).

Note: the above definitions and assertions hold, mutatis mutandi, if we define the conjugation action of xG on B to be the right action


rather than the left action given above.

Title inner automorphism
Canonical name InnerAutomorphism
Date of creation 2013-03-22 12:49:53
Last modified on 2013-03-22 12:49:53
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 12
Author rmilson (146)
Entry type Definition
Classification msc 20A05
Synonym inner
Defines conjugation
Defines outer
Defines outer automorphism
Defines automorphism group