# Necessary condition for orbits to be separated by invariant polynomials.

Suppose we have an algebraic subgroup $$G \subset GL(V)$$, where $$V$$ is a finite dimensional vector space over the field of complex numbers $$\mathbb{C}$$. I'm trying to prove the following:

If $$G$$-orbits in $$V$$ can be separated by $$G$$-invariant polynomials, then $$G$$ is finite.

I've found out that it's important that the field is $$\mathbb{C}$$ (I think because $$\mathbb{C}$$ is closed). Over $$\mathbb{R}$$ we have a counterexample $$SO_1 \subset GL(\mathbb{R}^3)$$. But the requirement for subgroup to be algebraic prevented me to construct something of this kind for $$\mathbb{C}$$. So this condition is also crucial.

I have a feeling it's an easy question, but I didn't succeed in making all those conditions work together. Any hint would be appreciated.