Let p, q be primes with p < q . Prove that a nonabelian group G of order pq

has a nonnormal subgroup of index q , so there exists an injective

homomorphism into Sq. Deduce that G is isomorphic to a subgroup of the normalizer in S(q) of the cyclic group

generated by the q-cycle.

I think I need to construct the group and see it's nonabelian.I thought of using

conjugacy and group actions , but I could't get anywhere.Can someone help?