homogeneous group

A homogeneous group is a set G together with a map ():G×G×GG satisfying:




for all a,b,c,d,eG.

A map f:GH of homogeneous groups is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if it f(a,b,c)=(fa,fb,fc), for all a,b,cG.

A non-empty homogeneous group is essentially a group, as given any xG, we may define the following productPlanetmathPlanetmathPlanetmath on G:


This gives G the of a group with identityPlanetmathPlanetmathPlanetmath x. The choice of x does not affect the isomorphism class of the group obtained.

One may recover a homogeneous group from a group obtained this way, by setting


Also, every group may be obtained from a homogeneous group.

Homogeneous groups are homogeneousPlanetmathPlanetmathPlanetmath: Given a,bG we have a homomorphism f taking a to b, given by fx=(x,a,b).

In this way homogeneous groups differ from groups, as whilst often used to describe symmetryPlanetmathPlanetmath, groups themselves have a distinct element: the identity.

Also the definition of homogeneous groups is given purely in of identities, and does not exclude the empty setMathworldPlanetmath.

Title homogeneous group
Canonical name HomogeneousGroup
Date of creation 2013-03-22 16:12:12
Last modified on 2013-03-22 16:12:12
Owner whm22 (2009)
Last modified by whm22 (2009)
Numerical id 15
Author whm22 (2009)
Entry type Definition
Classification msc 20A05
Defines homomorphism of homogeneous groups