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36. 37. 38. 39. 40. 41. 42. 43. 45. 46. 47. 48. 49. 50. rationals under multiplication is not cyclic? Consider the set {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20 by constructing its Cayley table. What is the identity element? Is the group cyclic? If so, find all of its generators. Give an example of a group that has exactly 6 subgroups (including the trivial subgroup and the group itself). Generalize to exactly :1 subgroups for any positive integer :1. Let m and n be elements of the group Z. Find a generator for the grow (\"1) n (n)- Suppose that a and b are group elements that commute. If lal is nite and lbl innite, prove that labl has innite order. Suppose that a and b belong to a group G, a and b commute, and Ial and lbl are nite. What are the possibilities for labl? Leta belong to a group and Ial = 100. Find legal and laml. Let F and F' be distinct reflections in D21. What are the possibilities for IFF'I? Suppose that H is a subgroup of a group G and [HI = 10. If a belongs to G and .55 belongs to H, what are the possibilities for Ial'? . Which of the following numbers could be the exact number of elements of order 21 in a group: 21600, 21602, 21604? If G is an innite group, what can you say about the number of elements of order 8 in the group? Generalize. If G is a cyclic group of order n, prove that for every element a in G, a" = e. For each positive integer n, prove that C*, the group of nonzero complex numbers under multiplication, has exactly (Mn) elements of order n. Prove or disprove that H = {n E Z | n is divisible by both 8 and 10} is a subgroup of Z. What happens if \"divisible by both 8 and 10" is changed to \"divisible by 8 or 10?\" Suppose that G is a finite group with the property that every non- identity element has prime order {for example, l.)3 and D5). 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