Automorphism of the upper half plane on $\mathbb C$

Question

I know that the automorphisms of the upper half plane $\mathbb C$ have the form $F_A=\frac{az+b}{cz+d}$, where $\det A=ad-bc\neq0$. I want to show that if it preserves $\mathbb R\cup\{\infty\}$, then $a,b,c,d\in \mathbb R$ and $\det A>0$. I can prove that the determinant will be positive if $a,b,c,d$ are reals, but I can not show the first.

I know that $F_A$ is 2-transitive, this meaning that if I choose $p,q,r$ distinct points and $p',q',r' $ also distinct points, then there is a choice of $A$ such that $F_A(p)=p',F_A(q)=q',F_A(r)=r'$, my professor said that with a good choice of the image could give me that the coeficients are real.

Thanks in advance

Answer 1

If an automorphism preserves $\Bbb R$, then there must be three real numbers that are sent to real numbers. This gives three homogenous linear equations with real coefficients when solving for $a,b,c,d$, so $a,b,c,d$ may be taken to be real.