I want to find all linear fractional transformations that fix the points 1 and -1. In particular i'd like to give this set a group structure and see if it is some familiar group or not. I wrote conditions for such $f$:
$$f(z)=\frac{az+b}{cz+d}$$
and i found
$$a+b-c-d=0$$ $$a-b+c-d=0$$ which give $a=d$ and $b=c$, thus the determinant of associated matrix is $a^2-b^2$. And now, how can i go on?
The function $f$ does not change if you scale the coefficients by a constant, so you can just as well assume that $a^2-b^2=1$.
But this gives (after scaling by $-1$ if necessary) the hyperbolic rotations $$\begin{pmatrix} \cosh x & \sinh x \\ \sinh x & \cosh x \end{pmatrix}.$$
Now, you can just have a look at the change of $x$ when you multiply two matrices to understand the group structure.