Well known that any finite number of points can be seen as intersection of two algebraic curves. Is it true that any finite group $G$ can be seen as intersection of two (connected) one dimensional algebraic subgroups $G_1, G_2$ of two dimensional algebraic group $G^*$? Also is it true that if $G$ is abelian?
In this questions groups $G_1, G_2, G^*$ are over field $\mathbb{C}$. I found this question after study of complex representation theory of finite groups, and i want to understand what is geometric nature of characters $G\rightarrow \mathbb{C}$.