Definition:G-Module

Definition
Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$.

Let $G$ be a group.

Let $\phi : G \times V \to V$ be an linear group action of $G$ on $V$.

Then $(V,\phi)$ is called a $G$-module.