Here is the question:

Using the hint it was very easy to show a contradiction. Briefly the result is

and

I have two questions. The first: Is there a specific reason that

uses <e> rather than {e}?

The second is by way of a counter-example. I note that

contains 3 subgroups of order 2. Now this doesn't directly contradict the theorem since p = q = 2 and the problem statement said p > q. However the proof by contradiction never used this fact and I can find nothing in the proof to indicate that p = 2 and |G| = 4 should be a special case?

Thanks.

-Dan