What is the significance of Op(G) in finite groups with a prime order?

[TheForumLord]

Homework Statement

Let G be a finite group of order n and let p be a prime number that divides n.
Let's mark as Op(G) as the intersection of all p-sylow groups of G.

1. Prove that Op(G) is a normal p-subgroup in G. -I've managed to prove it..
2. Prove that every normal p-subgroup of G is contained in Op(G) -I've managed to prove this also...
3. Let's mark: G] = G/Op(G). Prove that Op(G] ) ={1} -I have no idea about this...
Help is needed :(

Tnx

Homework Equations

The Attempt at a Solution

 

[ystael]

Use the correspondence between subgroups of $$G/O_p(G)$$ and subgroups of $$G$$ containing $$O_p(G)$$.

 

[TheForumLord]

I've managed to prove it on my own...TNX anyway