Do you know sylow's third theorem?
Also, do you know the theorem that states that every group of order is Abelian?
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.
Tonio