Prove that no group of order pq,where p and q are both prime, is simple.

So this is what I have been asked to prove, and I have included my proof.

http://img14.imageshack.us/img14/665...10301at819.png

I have now been asked to prove this:

http://img824.imageshack.us/img824/6...10301at819.png

I feel like the proof for this is exactly the same as the previous proof. Am I missing something? It seems too easy. I think I can use the exact same argument to show that $\displaystyle S_q=1$ and that therefore the Sylow q-subgroup is normal.