Is the Möbius group $PSL_2(\mathbb C)$ an infinite group?

Question

Is the Möbius group, $PSL(2, \mathbb{C}) $ an infinite group? Is it not cyclic and not abelian?

I beleive it is an infinite group because it contains elements of infinite order. If a group has an element of infinite order, it Is neccessarily infinite.

My guess Is that it Is not cyclic and not abelian. Not abelian cause the composition of functions doesn't commute.

Thanks for the help with this stupid question

Answer 1

It is not cyclic because it is infinite and has nontrivial elements of finite order, such as $z \mapsto 1/z$.

It is not abelian because $z \mapsto z+1$ and $z \mapsto 1/z$ do not commute. This also proves that it is not cyclic.

Answer 2

Yes. You are right. Consider the element $f(z)=z+1$. It has infinite order.