Groups of order p^2*q^2 are solvable?



Greetings,

This is not a homework problem, but it has bothered me for quite a
while: I want to show that a group G of order p^2*q^2 are solvable,
where p, q are distinct odd prime with p<q, but got stocked. And
here's what I had:

Let P be a p-sylow subgroup and Q a q-sylow subgroup. If either of
them are normal, it would be easy problem. If not, let N be the
normalizer of Q, then by Sylow's theorem, we must have [G:N] = p^2.
Now let G act on conjugates of Q, then the kernel K must be of order 1
or q, since G/K is a subgroup of S_p^2. If |K| = q, I can handle the
rest, but if |K| = 1, this is not very useful.

Also if the normalizer of P has index q, it would also be enough to
solve the problem, but I don't know how to get there neither.

Thanks in advance for any hints.

.



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