Schur's Lemma (Representation Theory)

Theorem
Let $(G,\cdot)$ be a finite group.

Let $V$ and $V^\prime$ be two irreducible $G$-modules.

Let $f: V \to V^\prime$ be a homomorphism of $G$-modules.

Then $f (v)=0$ for all $v\in V$ or $f$ is an isomorphism.

Proof
From Kernel is G-module, $\ker(f)$ is a $G$-submodule of $V$ and from Image is G-module, $\operatorname{Im}(f)$ is a $G$-submodule of $V^\prime$.

By the definition of irreducible, $\ker(f)=\{0\}$ or $\ker(f)=V$ from which follows that $f$ is inyective or $f=0$.

It also follow that $\operatorname{Im}(f)=\{0\}$ or $Im(f)=V^\prime$, thus $f$ is surjective or $f=0$.

In conclusion, $f=0$ or $f$ is inyective and surjective.