ord(G) = ( p^2)*(q^2)
( p , q = prime number)
G not has normal subgroup order p^2
I want to prove :
ord(G)=36
If you don't know Sylow's Third Theorem, then this problem is considerably harder.
However, trying to solve this problem, I can't help but feel that there might be some information missing from the problem.
Can you explain to me what you have covered in terms of group theory. What concepts/theorems have you learned thus far?
I think the OP meant: let G be a group of order $\displaystyle p^2q^2\,,\,\,p>q$ primes, and such that G has no normal subgroup of order $\displaystyle p^2$ ; then, it must be that $\displaystyle |G| = 36$.
We can argue as follows: any sbgp. of order $\displaystyle p^2$ in G is a Sylow p-sbgp. of G, so it is non-normal iff there's more than one such sbgps. Now, we know that the number $\displaystyle n_p$ of such sbgps. is congruent to $\displaystyle 1\!\!\pmod p$ and that it divides $\displaystyle q^2$ , so we get that:
$\displaystyle n_p=q^2\Longrightarrow q^2=1\!\!\pmod p\Longrightarrow q=\pm 1\!\!\pmod p\Longrightarrow q=-1\!\!\pmod p $ (the other option is impossible: why?) , and as $\displaystyle p>q$ we must have that $\displaystyle p=q+1\Longrightarrow q=2\,,\,p=3$ and we're done.
Tonio