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67 69. 70. 71. 72. 73. 74. 75. 76 Suppose that G is a group of order 16 and that, by direct computa- tion, you know that G has at least nine elements it: such that x8 2 2. Can you conclude that G is not cyclic? What if G has at least ve elements A: such that x4 = e? Generalize. . Prove that Zn has an even number of generators if n 3 2. What does this tell you about (i301)? If lajl : 12, what are the possibilities for lal'? If |a4| = 12, what are the possibilities for Ial? Suppose that le = :1. Find a necessary and sufcient condition on r and s such that {x'} Q (x3). Suppose a is a group element such thatlazsl = 10 and lal = 20. Determine |a|. Let a be a group element such that Ial = 48. For each part, find a divisor k of 48 such that a- (121) = (at); IL (1214} I (talk); c. ((113) : (ck). Let p be a prime. Show that in a cyclic group of order p\" 1, every element is a pth power (that is, every element can be written in the form a!\" for some a). 1 n Prove that H = {[ ] 0 1 0142, R). Leta and b belong to a group. If lal : 12, Ibl : 22, and (a) F} (b) i {c}, prove that a6 = b\". (2008 GRE Practice Exam) If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal, determine |x13|. n E Z} is a cyclic subgroup of