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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.

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