G-module


Let V a vector space over some field K (usually K= or K=). Let G be a group which acts on V. This means that there is an operationMathworldPlanetmath ψ:G×VV such that

  1. 1.

    gvV.

  2. 2.

    g(hv)=(gh)v

  3. 3.

    ev=v

where gv stands for ψ(g,v) and e is the identity elementMathworldPlanetmath of G.

If in addition,

g(cv+dw)=c(gv)+d(gw)

for any gG, v,wV, c,dK, we say that V is a G-module. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with the existence of a group representationMathworldPlanetmathPlanetmath from G to GL(V).

Title G-module
Canonical name Gmodule
Date of creation 2013-03-22 14:57:53
Last modified on 2013-03-22 14:57:53
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 20C99
Related topic GroupRepresentation
Related topic Group