33. Let H and K be subgroups of a finite group G with H CKC G. 34. Suppose that a group contains elements of orders 1 through 10. 35. Give an example of the dihedral group of smallest order that con- 5 Groups 158 IG:KI IK:HI. Prove that IG:H What is the minimum possible order of the group? tains a subgroup isomorphic to Z12 and a subgroup isomorphic to Zp No need to prove anything, but explain your reasoning. 36. Show that in any group of order 100, either every element has order that is a power of a prime or there is an element of order 10 37. Suppose that a finite Abelian group G has at least three elements of order 3. Prove that 9 divides IGI. 55 5 38. Prove that if G is a finite group, the index of Z(G) cannot be prime 39. Find an example of a subgroup H of a group G and elements a and b in G such that aH Hb and aHH .(Compare with prop- erty 5 of cosets.) 55 40. Prove that a group of order 63 must have an element of order 3. 41. Let G be a group of order 100 that has a subgroup H of order 25. Prove that every element of G of order 5 is in H. 42. Let G be a group of order n and k be any integer relatively prime to n. Show that the mapping from G to G given by g - one. If G is also Abelian, show that the mapping given by g g is an automorphism of G. 43. Let G be a group of permutations of a set S. Prove that the orbits of the members of S constitute a partition of S. (This exercise is re- ferred to in this chapter and in Chapter 29.) 44. Prove that every subgroup of D of odd order is cyclic. 45. Let G = {(1), (12) (34), (1234) (56), (13)(24), (1432)(56), (56)(13), (14)(23), (24)(56)}. a. Find the stabilizer of 1 and the orbit of 1. b. Find the stabilizer of 3 and the orbit of 3. c. Find the stabilizer of 5 and the orbit of 5. gk is one-to- 55 9 46. Prove that a group of order 12 must have an element of order 2. 47. Show that in a group G of odd order, the equation x2 : unique solution for all a in G. 48. Let G be a group of order pqr, where p, q, and r are distinct primes. a has a If H and K are subgroups of G with IH that IHn Kl q. Pq and IKI = qr, prove 49. Prove that a group that has more than one subgroup of order 5 must have order at least 25. 50. Prove that A, has a subgroup of order 12.