Commutative Linear Transformation is G-Module Homomorphism

Theorem
Let $\rho: G \to \GL V$ be a representation.

Let $f: V \to V$ be a linear mapping.

Let:
 * $\forall g \in G: \map \rho g \circ f = f \circ \map \rho g$

Then $f: V \to V$ is a $G$-module homomorphism.

Proof
Let:
 * $\forall g \in G: \map \rho g \circ f = f \circ \map \rho g$

Let $v$ be a vector $v \in V$.

Then:
 * $\map {\map \rho g} {\map f v} = \map f {\map {\map \rho g} v}$

Using the properties from Correspondence between Linear Group Actions and Linear Representations:
 * there exists a $G$-module $\struct {V, \phi}$ associated with $\rho$ such that:
 * $\map \phi {g, v} = \map {\map \rho g} v$

Applying the last formula:


 * $\map {\map \rho g} {\map f v} = \map \phi {g, \map f v}$

and:


 * $\map f {\map \phi {g, v} } = \map f {\map {\map \rho g} v}$

Thus our assumption is equivalent to:
 * $\map f {\map \phi {g, v} } = \map \phi {g, \map f v}$

Hence, by definition of $G$-module homomorphism, $f: V \to V$ is a $G$-module homomorphism.