Suppose there is a faithful representation $\rho:G\to SL_2(\mathbb{R})$. Prove that $G$ is cyclic.
I know there has to be something special about its representation being special (no pun intended) because e.g. the Klein 4 group has a non-special representation. Also it has to be important that it's in two dimensions, because $SO(3)$ contains non-cyclic groups.
Apart from that, I haven't really made any progress. Any hints?