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32. 33. 37. 39. 41. Determine the subgroup lattice for Zu- Determine the subgroup lattice for sz q, where p and q are distinct primes. . Determine the subgroup lattice for 23- 35. Determine the subgroup lattice for Zn", where p is a prime and n is some positive integer. . Prove that a nite group is the union of proper subgroups if and only if the group is not cyclic. Show that the group of positive rational numbers under multiplica- tion 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 I: subgroups for any positive integer n. . Let m and n be elements of the group Z. Find a generator for the group (m) H (:1). Suppose that a and b are group elements that commute and have orders at and at. If (a) I\": (b) = {a}, prove that the group contains an element whose order is the least common multiple of m. and n