G-Submodule Test

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

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

Then $\left({W, \phi_W}\right)$, where $\phi_W: G \times W \to W$ is the restriction of $\phi$ to $G \times W$, is a $G$-submodule of $V$ iff $\phi \left({G, W}\right) \subseteq W$.

Necessary Condition
Assume that $W$ is a $G$-submodule of $V$.

Hence by definition $\phi_W: G \times W \to W$ is a linear action on $W$.

Also by definition, $\phi_W \left({G, W}\right) = \phi \left({G, W}\right) \subseteq W$.

Sufficient Condition
Assume now that $\phi \left({G, W}\right) = \phi_W \left({G, W}\right) \subseteq W$.

Then it is correct to define $\phi_W: G \times W \to W$; it is a well-defined mapping.

We need to check if $\phi_W$ is a linear action on $W$:

Assume $a,b \in W$ and $g \in G$; in particular, then, $a,b \in V$ and:

Further, assume $\lambda \in k$ and $g\in G$, then:

Thus $W$ is a $G$-submodule of $V$.