Projective normality of finite group quotients and EGZ theorem

In this note, we prove that for any finite dimensional vector space $V$ over $\mathbb{C}$, and for a finite cyclic group $G$, the projective variety $\mathbb{P}(V)\mathrm{/}G$ is projectively normal with respect to the descent of $\mathcal{O}(1{)}^{\otimes \mathrm{\mid }G\mathrm{\mid }}$ by a method using toric variety, and deduce the EGZ theorem as a consequence.