Let $G\subseteq GL_n(\mathbb{C})$ a Zariski-closed linear subgroup and $X\subseteq G$ closed with $X*X\subseteq X$ and $e \in X$. Then $X$ is a subroup. I am not sure how to start here. I know that $X^{-1}$ is also closed and that $e\in X\cap X^{-1}$. Is that going to help me?
Thank you.