\begin{definition}[Definition:G-Submodule]
Let $\struct {G, \cdot}$ be a finite group.
Let $\struct {V, \phi}$ be a $G$-module.
Let $W$ be a vector subspace of $V$.
Let $\phi$ be a linear group action when restricted to $G \times W \subseteq G \times V$.
Let $\phi_W$ be the restriction of $\phi$ to $G \times W$.
Then $\struct {W, \phi_W}$ is called a '''$G$-submodule of $\struct {V, \phi}$'''.
\end{definition}