Schur's Lemma (Representation Theory)

Theorem
Let $(G,\cdot)$ be a finite group and let $V$ and $V^\prime$ be two irreducible $G$-modules.

Consider $f:V\to V^\prime$ an homomorphism of $G$-modules, then $f\equiv 0$ or $f$ is an isomorphism.

Proof
Since $f$ is a linear mapping, $\ker(f)$ is a $G$-submodule of $V$ and $\operatorname{Im}(f)$ 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.