Commutative Linear Transformation is G-Module Homomorphism

Theorem
Let $\rho:G\to\operatorname{GL}\left(V\right)$ be a representation, and $h\in G$. Then:

$\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $g\in G\ \Rightarrow\ \rho(h):V\to V$ is a $G$-module homomorphism.

Proof
Assume that $\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $g\in G$.

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

Then $\rho(g)\rho(h)(v)=\rho(h)\rho(g)(v)$.

Using the properties from Equivalence of Representation Definitions there exists $\left(V,\phi\right)$, a $G$-module associated with $\rho$, such that $\phi(g,v)=\rho(g)(v)$.

Applying the last formula:

$\rho(g)\rho(h)(v)=\phi(g,\rho(h)(v))$

and

$\rho(h)(\phi(g,v))=\rho(h)\rho(g)(v)$.

Thus $\rho(h)(\phi(g,v))=\phi(g,\rho(h)(v))$.

Hence by definition of $G$-module homomorphism $\rho(h):V\to V$ is a $G$-module homomorphism.