Non-conjugate subgroups

I want to show that $\left\{\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} : \theta \in \mathbb{R} \right\}$ and $\left\{\begin{pmatrix} t & 0 \\ 0 & \frac{1}{t} \end{pmatrix} : t >0 \right\}$ are not conjugate in $SL(2,\mathbb{R})$. To do so, can I just say that the former has complex eigenvalues while the latter has real eigenvalues and conjugate matrices must have the same eigenvalues??

• Even stronger: they're not isomorphic as abstract groups (since only the first has torsion) nor homeomorphic (since only the first is compact). – anon Sep 12 '17 at 1:56

• @okaskadasd conjugation by an element is a homeomorphism from $\text{SL}_2(\Bbb R)$ to itself. – Lord Shark the Unknown Sep 12 '17 at 3:01