Definition:G-Submodule

Definition
Let $\left({G, \cdot}\right)$ be a finite group.

Let $\left({V, \phi}\right)$ be a $G$-module.

Let $W$ be a vector subspace of $V$.

If $\phi$ is a linear group action when restricted to $G \times W \subseteq G \times V$.

Then $W$ is called a $G$-submodule of $V$.

Remark
$\forall g\in G,\ \forall w\in W,\ \phi(g,w)\in V$; so $W$ is a $G$-submodule if $\phi(G,W)\subseteq W$.