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

1. ## 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.

I have now been asked to prove this:

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 $S_q=1$ and that therefore the Sylow q-subgroup is normal.

2. Originally Posted by CropDuster
So this is what I have been asked to prove, and I have included my proof.

I have now been asked to prove this:

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 $S_q=1$ and that therefore the Sylow q-subgroup is normal.

No, it's not exactly the same: this time it may be $p^2=1+qn\Longrightarrow q\mid (p^2-1)$ , which
it's easily possible.

Tonio