Definition:G-Submodule

Definition
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 $\struct {V, \phi}$.

Also see

 * G-Submodule Test