#35 please 4:53Aa <» Q4| Cyclic Groups8931. Let G be a finite group. Show that there exists a fixed positive integern such that a" = e for all a in G. (Note that n is independent of a.)32. Determine the subgroup lattice for Z2:33. Determine the subgroup lattice for Ze where p and q are distinctprimes.34. Determine the subgroup lattice for Z,.35. Determine the subgroup lattice for Z, where p is a prime and n issome positive integer.36. Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic.37. Show that the group of positive rational numbers under multiplica-tion is not cyclic.38. Consider the set {4, 8, 12, 16}. Show that this set is a group undermultiplication modulo 20 by constructing its Cayley table. Whatis the identity element? Is the group cyclic? If so, find all of itsgenerators.39. Give an example of a group that has exactly 6 subgroups (includingtrivial subgroup and the group itself). Generalize to exactly nthesubgroups for any positive integer n.40. Let m andn be elements of the group Z. Find a generator for thegroup (m) n (n).41. Suppose that a and b are group elements that commute and haveorders m and n. If (a) n (b) = {e}, prove that the group contains anelement whose order is the least common multiple of m and n.Show that this need not be true if a and b do not commute.Supposelal and Ibl are finite. What are the possibilities for labl?43. Suppose that a and b belong to a group G, a and b commute, and laland Ibl are finite. Prove that G has an element of order Icm(lal, Ibl).et F and F' be distinct reflections in Dj. What are the possibili-ties for IFF'1?42.e that a and b belong to a group G, a and b commute, and45. Suppose that H is a subgroup of a group G and lH| = 10. If abelongs to G and aº belongs to H, what are the possibilities for lal?Whichelements of order 21 in a group: 21600, 21602, 21604?46.ofthe following numbers could be the exact number of47. If G is an infinite group, what can you say about the number ofelements of order 8 in the group? Generalize.48. Suppose that K is a proper subgroup of D3s and K contains at leasttwo reflections. What are the possible orders of K? Explain yourreasoning.C CmLing Rene e td ile D niig<>89日ReaderContentsNotebookBookmarksFlashcards

Name: #35 please 4:53Aa <» Q4| Cyclic Groups8931. Let G be a finite group. S
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Description: #35 please 4:53Aa <» Q4| Cyclic Groups8931. Let G be a finite group. Show that there exists a fixed positive integern such that a" = e for all a in G. (Note that n is independent of a.)32. Determine the subgroup lattice for Z2:33. Determine the subgr