G-Module

A G-module is formed by taking a G-set which is also an abelian group, and imposing some level of compatibility between the operation of the abelian group and the group action.

Definition

Given a group G, a (left) G-module is an abelian group (M,+) and a map .:G×M→M such that

  1. id.a=a∀a∈M
  2. g.(g′.a)=(gg′).a∀g,g′∈G∀a∈M
  3. g.(a+b)=g.a+g.b∀g∈G∀a,b∈M

The first two axioms are those required for the map to be a group action, and thus on top of a group action, the only additionally imposed axiom is the third,

g.(a+b)=g.a+g.b.

This definition can be generalised further with rings acting on abelian groups to the notion of a module.