Kernel is G-Module

Theorem
Let $\struct {G, \cdot}$ be a group.

Let $f: \struct {V, \phi} \to \struct {V', \mu}$ be a homomorphism of $G$-modules.

Then its kernel $\map \ker f$ is a $G$-submodule of $V$.

Proof
From G-Submodule Test it suffices to prove that $\phi \sqbrk {\struct {G, \map \ker f} } \subseteq \map \ker f$.

That is, it is to be shown that, if $g \in G$ and $v \in \map \ker f$, then $\map \phi {g, v} \in \map \ker f$.

Assume that $g \in G$ and $v \in \map \ker f$.

Thus $\map \phi {g, v} \in \map \ker f$.

Hence $\map \ker f$ is a $G$-submodule of $V$.