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5. Let P be a Sylow p-subgroup of G and suppose a, b are in the centre of P. Suppose further that a = xbx" for some x e G. Prove that 3 y e N(P) s.t. a = yby 1 (Hint: P S N(a), P S N(b) as a, b e Z(P) x P x 1 5 xN(b)x ' s N(xbx]) = N(a)). 6. If o(G) = p q where p, q are primes, show that either Sylow p-subgroup or Sylow q-subgroup is normal in G. 7. If G is a group of order 385, show that both Sylow 7-subgroup and Sylow 11-subgroup are normal in G and Sylow 7-subgroup is in the centre of G. (Hint: Let H, K, L be the subgroups of order 5, 7, 11, then K, L G and HK S G where o(HK) = 35 and so HK is cyclic. G = HKL = KLH If x e K, g E G then xg = xklh KEKIEL, heH = kxlh as K is abelian = klxh as Ko L = {e}, K, L normal = klhx as HK is abelian = gx or that x E Z(G)) 8. Let G be group. Prove that o G Z(G) cannot be 77. 9. Let G be a group of order 255. Show that (i) Sylow 17-subgroup is normal in G. (ii) 3 a normal subgroup K of order 85. (iii) K c Z(G). (iv) G is cyclic. 10. If G is a non-abelian group of order 231, show that Z(G) is Sylow 11 subgroup of G 11. Let o(G) = 108. Show that either Slow 3-subgroup is normal or there exits a normal subgroup of order 9. 12. If H is a normal subgroup of order p* of a finite group G, show that H is contained in every Sylow p-subgroup of G. 13. If o(G) = p"q with p > q primes, show that G contains a unique normal subgroup of index q. 14. Show that groups of order 12, 28, 56 and 200 must contain a normal Sylow subgroup. 15. Show that groups of order 36, 42, 99, 112 and 120 are not simple. 16. Let o(G) = pq where p and q are distinct primes. Prove that G has a proper normal subgroup. Prove further that if neither prime is congruent to 1 modulo the other, then G is abelian