Let p,q be distinct primes with q < p and let G be a finite group with |G| = pq.

(i) Use sylow's theorem to show that G has a normal subgroup K with $\displaystyle K \cong G $

(ii) Use the Recogition Criterion to show $\displaystyle G \cong C_p \rtimes_h C_q $ for some homomorphism $\displaystyle h:C_q \rightarrow Aut(C_p) $

(iii) Describle explicitly all homomorphisms $\displaystyle h:C_5 \rightarrow Aut(C_7) $. Hence describe all groups of order 35. How many such subgroups are there?

(iv) Describe explicitly all homomorphisms $\displaystyle h:C_3 \rightarrow Aut(C_{13}) $. Hence describe all groups of order 39. How many such groups are there, up to isomorphism?

any help is highly appreciated as usual. i will attempt the rest myself once i have good idea. thnx a lot.