Let $f: V\to W$ be a homomorphism of $G$ modules.
Show $\ker f $ is a $G$ submodule of $V$, and $\mbox{im}f$ is $G$ submodule of $W$ .
Proof: Since $v$ is linear, then we only need to check for closure under the action of $G$. Let $a \in\ker f$ and $g\in G$. Then $f(ga) = gf(a) = g(0) = 0.$ So $ga \in \ker f$.
So the $\ker f$ is a $G$ submodule of $V$.
In addition, similarly suppose $a \in V$ and for $g\in G$ we have $gf(a) = f(ga) \in \mbox{im}f$.
So $\mbox{im}f$ is a submodule of $W$
Can someone please verify this and provide feedback? Thank you