The task given in my textbook was to find which algebraic structure is $(X, *)$, where $X$ is set of Möbius transformations $x\rightarrow y=\frac{ax+b}{cx+d}$ in $\mathbb R$ and $*$ is composition. I showed that $*$ is a closed binary operation, then that it's associative and commutative and then that there is the neutral element. However, I don't know how to show that for any element exists an inverse element. I know that it is$$y^{-1}=\frac{dx-b}{-cx+a},$$but I have no idea how to find it(but I know how to check that it's true). So my queations are:
- How to find it?
- Is it possible to do without using matrices(they can be and are usually associated with matrices, but matrices are covered later in the book, so i think that they shouldn't be necessary)?
$$y=\frac{ax+b}{cx+d}$$
$$y(cx+d)=ax+b$$
$$cxy-ax=b-dy$$
$$(cy-a)x=b-dy$$
$$x = \frac{-dy+b}{cy-a}$$
Is that what you were hoping for?