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51. 52. 53. 54. 55. 56. 57. 58. 59. 1 - .- - _J_'___'_'__J is a subgroup of Z Suppose that G is a nite group with the property that every non- identity element has prime order (for example, .03 and Ds)' If Z(G) is not trivial, prove that every nonidentity element of G has the same order. Prove that an innite group must have an innite number of subgroups. Let p be a prime. If a group has more than p 1 elements of order p, why can't the group be cyclic? Suppose that G is a cyclic group and that 6 divides IGI. How many elements of order 6 does G have? If 8 divides IGI , how many ele- ments of order 8 does G have? If a is one element of order 8, list the other elements of order 8. List all the elements of 240 that have order 10. Let lxl : 40. List all the elements of (x) that have order 10. Reformulate the corollary of Theorem 4.4 to include the case when the group has innite order. Determine the orders of the elements of D33 and how many there are of each. If G is a cyclic group and 15 divides the order of G, determine the number of solutions in G of the equation x15 = a. If 20 divides the order of G, determine the number of solutions of x\" : e. Generalize. If G is an Abelian group and contains cyclic subgroups of orders 4 and 5, what other sizes of cyclic subgroups must G contain? npnprgl i 79