ord(G) = ( p^2)*(q^2)
( p , q = prime number)
G not has normal subgroup order p^2
I want to prove :
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 primes, and such that G has no normal subgroup of order ; then, it must be that .
We can argue as follows: any sbgp. of order 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 of such sbgps. is congruent to and that it divides , so we get that:
(the other option is impossible: why?) , and as we must have that and we're done.