MTH 411-1 Abstract Algebra II S17 Homework 8 due on 3/31/17 1. Let G be a group and let G G G, (a, b) aba1 be the action of G on G by conjugation. Define Z(G) = {a G ab = ba for all b G}. (a) Show that Z(G) = FixG (G). (b) Suppose G is a p-group with G > 1. Show that Z(G) > 1. (Hint: Use (a) and the Fixed Point Formula) 2. Let G be a finite group of order n. Suppose that n = pk l, where p is a prime and k and l are positive integer with p > l. Show that G has a unique Sylow p-subgroup. For exercises 3 and 4, recall that a group G is called simple if {e} and G are the only normal subgroups of G. 3. Let G be a simple group of order 660. Show that G has 12 Sylow 11-subgroups and that G is isomorphic to a subgroup of Sym(12). 4. Let G be a group of order 6,10,15,18,20,21,22,26,28,33,34,35,38,39,40,42,44,45,46,50,51,52,54,55,57,or 58. Show that G is not simple. (Hint: In each case find a Sylow p-subgroup which is normal.) 5. Find all Sylow 2-subgroups of Sym(4). 6. Let G = Sym(5) and I = Syl5 (G). (a) List all elements of I. (b) Let G Sym(I) be the homomorphism associated to the action of G on I. Determine the cycle type of ((1, 2)). 1