Definition:G-Submodule

Definition
Let $(G,\cdot)$ be a finite group and let $(V,\phi)$ be a $G$-module.

We say that $W$ a vector subspace of $V$ is a $G$-submodule if $\phi$ is still a linear group action when restricted to $G\times W\subseteq G\times V$.

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