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 $f$:
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?