Finite group with two prime factors

[moont14263]

Homework Statement

I am trying to prove the following:

Let G be a finite group and let \{p,q\} be the set of primes dividing the order of G. Show that PQ=QP for any P Sylow p-subgroup of G and Q Sylow q-subgroup of G. Deduce that G=PQ.

Homework Equations

The set PQ=\{xy: x \in P \text{ and } y \in Q\}

The Attempt at a Solution

I know that PQ=QP means that I must prove PQ is a subgroup of G. Let r, s be elements of PQ. Then r=x_{1}y_{1} and s=x_{2}y_{2}.

Now rs^{-1}=x_{1}y_{1}(x_{2}y_{2})^{-1}=x_{1}y_{1}y_{2}^{-1}x_{2}^{-1} but I am not sure whether this is an element of PQ or not.

The second part by the following:
|G|=|P||Q|=\frac{|P||Q|}{|P \cap Q|}=|PQ| as P,Q are Sylow for distinct primes of G and P \cap Q=\{1\}. This imply that G=PQ as PQ is a subgroup of G with the same order of G.

Thanks in advance.

 

[jbunniii]

The second part by the following:
|G|=|P||Q|=\frac{|P||Q|}{|P \cap Q|}=|PQ| as P,Q are Sylow for distinct primes of G and P \cap Q=\{1\}. This imply that G=PQ as PQ is a subgroup of G with the same order of G.

What happens if you reverse the roles of P and Q?

 

[moont14263]

In case PQ is a subgroup of G, then PQ=QP.