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

• Mar 1st 2011, 05:44 PM
CropDuster
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.
• Mar 1st 2011, 06:39 PM
tonio
Quote:

Originally Posted by CropDuster
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.

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

Tonio