Will anyone give me a hand on this problem? I am trying to show that if and contains , a Sylow p-subgroup in . Then . I am not sure that this is true, but I see that since and is contained in , then must contain all the conjugates of . By Sylow's theorem, we know all Sylow p-subgroups of are conjugates. So, . Do I need to construct a 1-1 correspondence here? I think that and are just the number of Sylow p-subgroups, how can I really construct a 1-1 correspondence if they are not sets? I don't claim that and have the same set of Sylow p-subgroups.