Let $G$ be a finite subgroup of $\mathrm{Aut}(\mathbb P^{1}(k))=\mathrm{PGL}(2,k)$, where $k$ is an algebrically closed field of characteristic $0$. Suppose that there is a common fixed point for all the elements of $G$. Is $G$ cyclic?
In other words, is it true that $ \exists [p] \in \mathbb P^{1}(k): \, \phi([p])=[p] \, \quad \forall \phi \in G \Rightarrow G \quad \text{is cyclic?} $
I am really puzzled and I do not know what to say. What do you think?
I started considering $k = \mathbb C$ but I couldn't conclude anything. Can you help me, please?
Thanks.