If $G$ is a group and $\mathbb K$ is a field let $\mathbb KG$ be the usual group ring. We know a representation $\rho:G\longrightarrow GL(V)$, where $V$ is a $\mathbb K$-vetor space, is the same as a structure of $\mathbb KG$-module on $V$.
Suppose $G$ is a finite abelian group and let $\mathbb K=\mathbb C$. How can I show that if $V$ is a $\mathbb CG$-module then we can choose a basis for $V$ (as $\mathbb K$-vector space) such that $\rho(g)$ is a diagonal matrix, where $\rho$ is the afforded representation?
Obs: I'm always supposing the $\mathbb KG$-modules are finite dimensional $\mathbb K$ vector space.