\(G\)-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 \(. : G \times M \to M\) such that

  1. \(\mathrm{id}.a = a \quad \forall a \in M\)
  2. \(g.(g'.a) = (g g').a \quad \forall g, g' \in G \quad \forall a \in M\)
  3. \(g.(a + b) = g.a + g.b \quad \forall g \in G \quad \forall a, b \in M\)

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.