Schur's Lemma (Representation Theory)

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

Let $V$ and $V'$ be two irreducible $G$-modules.

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

Then either:
 * $\map f v = 0$ for all $v \in V$

or:
 * $f$ is an isomorphism.

Proof
From Kernel is G-Module, $\map \ker f$ is a $G$-submodule of $V$.

From Image is G-Module, $\Img f$ is a $G$-submodule of $V'$.

By the definition of irreducible:
 * $\map \ker f = \set 0$

or:
 * $\map \ker f = V$

If $\map \ker f = V$ then by definition:
 * $\map f v = 0$ for all $v \in V$

Let $\map \ker f = \set 0$.

Then from Linear Transformation is Injective iff Kernel Contains Only Zero:
 * $f$ is injective.

It also follows that:
 * $\Img f = V'$

Thus $f$ is surjective and injective.

Thus by definition $f$ is a bijection and thence an isomorphism.