Maximal compact subgroup of SL(2,R)

Question

I want to classify up conjugacy the compact subgroups of SL(2,R) (I need a proof)... Please help me.

Answer 1

Write $SL(2,R)=T SO(2,R)$ with $T$ the set of upper triangular matrices with determinant 1 and positive diagonal elements. Now for $A \in T$, we have $(A^n)_{1,1}=(A_{1,1})^n$ so if $(A_{1,1})\neq 1$ the sequence $(A^n)_n$ will have hard time having a subsequence converging in $SL(2,R)$. If $A_{1,1}=1$ then it follows that $A_{2,2}=1$ and in this case $(A^n)_{1,2}=nA_{1,2}$ so the above sequence has a converging subsequence iff $A_{1,2}=0$ that is $A=I$.