opposite group

Let G be a group under the operationMathworldPlanetmath *. The opposite group of G, denoted Gop, has the same underlying set as G, and its group operationMathworldPlanetmath is * defined by g1*g2=g2*g1.

If G is abelianMathworldPlanetmath, then it is equal to its opposite group. Also, every group G (not necessarily abelian) is isomorphicPlanetmathPlanetmathPlanetmath to its opposite group: The isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupIsomorphism) φ:GGop is given by φ(x)=x-1. More generally, any anti-automorphism ψ:GG gives rise to a corresponding isomorphism ψ:GGop via ψ(g)=ψ(g), since ψ(g*h)=ψ(g*h)=ψ(h)*ψ(g)=ψ(g)*ψ(h)=ψ(g)*ψ(h).

Opposite groups are useful for converting a right action to a left action and vice versa. For example, if G is a group that acts on X on the , then a left action of Gop on X can be defined by gopx=xg.

constructions occur in opposite ring and opposite categoryMathworldPlanetmath.

Title opposite group
Canonical name OppositeGroup
Date of creation 2013-03-22 17:09:56
Last modified on 2013-03-22 17:09:56
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Definition
Classification msc 08A99
Classification msc 20-00