What is wrong with this argument?

The problem is the umption you make in your inductive step that p2 occurs in both groups. It seems like it should obviously be true, especially when you list out p1, p2, ... pk like that. However, in the case that k = 1, it is false. In your "proof", the group of k+1 persons would be {p1, p2}. In this case, you could split the set into two subsets, {p1} and {p2}, as per your suggestion. However, p2 clearly does not appear in both subsets, which means your inductive step fails at the very first case.