G-Submodule Test

Theorem
Let $\struct {V, \phi}$ be a $G$-module over a field $k$.

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

Let $\phi_W: G \times W \to W$ denote the restriction of $\phi$ to $G \times W$.

Then:
 * $\struct {W, \phi_W}$ is a $G$-submodule of $V$


 * $\map \phi {G, W} \subseteq W$
 * $\map \phi {G, W} \subseteq W$

Necessary Condition
Let $W$ be a $G$-submodule of $V$.

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

Also by definition:
 * $\map {\phi_W} {G, W} = \map \phi {G, W} \subseteq W$

Sufficient Condition
Let:
 * $\map \phi {G, W} = \map {\phi_W} {G, W} \subseteq W$

We have that $\phi_W: G \times W \to W$ 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:
 * $a, b \in V$

and so:

Further, let $\lambda \in k$ and $g \in G$.

Then:

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