How could I show that the 3 groups
$$x\to x+a$$ $$x\to \exp(b) x$$ $$x\to {x\over 1-cx}$$ where $a,b,c\in \mathbb R$ and also $x\in \mathbb R$,
generate a group $$x\to {\alpha x+\beta\over \gamma x+\delta}$$ where $\alpha\delta-\beta\gamma=1$?
Thank you.
I can see how the 3 smaller groups are subgroups of the bigger group, but I don't know how to show that they generate the larger group.