Want exact and accurate solution of these hw problems. Detail work required

Let p and q be distinct prime numbers. Let G be a group of order p3 . q such that its commutator subgroup K is of order q. Let H be a p-Sylow subgroup of G. (a) Show that H is abelian and G = HK. (b) Show that there are elements h E H and k E K such that he # kh. (c) From (b) show that p divides q - 1.(a) Let p, q be primes, p > q > 2. Let G be a group of order pq. Show G has a subgroup of order pq. (b) What can you say if q = 2 (and p > q)?Let A be the abelian group on generators e, f, and g, subject to the relations ge + 3f + 6g = 0, 3e + 3f = 0, 3e - 3f + 6g = 0. Give a decomposition of A as a direct sum of cyclic groups of prime order or infinite order