I'm having a hard time trying to prove the following statement :
Prove that for finite sets $\{ P_1, ... , P_m \}$ and $\{ Q_1, ... , Q_n \}$ of $\mathbb{P}^1$, there exists $ A \in GL(2,\mathbb{C}) $ such that $p_{A}(\{P_1,...,P_m\}) \cap \{Q_1, ... , Q_n\} = \emptyset$ . (Here $p_A$ is the automorphism on $\mathbb{P}^1$ determined by $A \in GL(2,\mathbb{C})$).
I've tried using mathematical induction on $n$ and $m$, which didn't turn out to be any helpful. Any helps? Thanks!