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 $W$ is a $G$-submodule of $V$ iff $\phi(G,W)\subseteq W$

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

Hence by definition $\phi:G\times W\to W$ is a linear action on $W$ and $\phi(G,W)\subseteq W$

Sufficient Condition
Assume now that $\phi(G,W)\subseteq W$.

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

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


 * Take $a,b\in W$ and $g\in G$, then $a,b\in V$. Since $\phi$ is a linear action on $V$ we get by definition that $\phi(g,a+b)=\phi(g,a)+\phi(g,b)$
 * Take $b\in W$, $\lambda\in k$ and $g\in G$, then $b\in V$. Since $\phi$ is a linear action on $V$ we get by definition that $\phi(g,\lambda b)=\lambda\phi(g,b)$

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