Abstract Algebra/Group Theory/Cyclic groups

Cyclic groups.
In the previous section about subgroups we saw that if formula_1 is a group with formula_2, then the set of powers of formula_3, formula_4 constituted a subgroup of formula_1, called the "cyclic subgroup generated by formula_3". In this section, we will generalize this concept, and in the process, obtain an important family of groups which is very rich in structure.
Definition 1: Let formula_1 be a group with an element formula_2 such that formula_9. Then formula_1 is called a "cyclic" group, and formula_3 is called a "generator" of formula_1. Alternatively, formula_3 is said to "generate" formula_1. If there exists an integer formula_15 such that formula_16, and formula_15 is the smallest positive such integer, formula_1 is denoted formula_19, the cyclic group of order formula_15. If no such integer exists, formula_1 is denoted formula_22, the infinite cyclic group.
The infinite cyclic group can also be denoted formula_23, the "free group" with one generator. This is foreshadowing for a future section and can be ignored for now.
Theorem 2: Any cyclic group is abelian.
"Proof": Let formula_1 be a cyclic group with generator formula_3. Then if formula_26, then formula_27 and formula_28 for some formula_29. To show commutativity, observe that formula_30 and we are done. 
Theorem 3: Any subgroup of a cyclic group is cyclic.
"Proof": Let formula_1 be a cyclic group with generator formula_3, and let formula_33. Since formula_34, in particular every element of formula_35 equals formula_36 for some formula_37. We claim that if formula_38 the lowest positive integer such that formula_39, then formula_40. To see this, let formula_41. Then formula_42 and formula_43 for unique formula_44. Since formula_35 is a subgroup and formula_39, we must have formula_47. Now, assume that formula_48. Then formula_49 contradicts our assumption that formula_38 is the least positive integer such that formula_39. Therefore, formula_52. Consequently, formula_41 only if formula_54, and formula_40 and is cyclic, as was to be shown. 
As the alert reader will have noticed, the preceding proof invoked the notion of "division with remainder" which should be familiar from number theory. Our treatment of cyclic groups will have close ties with notions from number theory. This is no coincidence, as the next few statements will show. Indeed, an alternative title for this section could have been "Modular arithmetic and integer ideals". The notion of an "ideal" may not yet be familiar to the reader, who is asked to wait patiently until the chapter about rings.
Theorem 4: Let formula_56 with addition defined modulo formula_15. That is formula_58, where formula_59. We denote this operation by formula_60. Then formula_61 is a cyclic group.
"Proof": We must first show that formula_61 is a group, then find a generator. We verify the group axioms. Associativity is inherited from the integers. The element formula_63 is an identity element with respect to formula_64. An inverse of formula_65 is an element formula_66 such that formula_67. Thus formula_68. Then, formula_69, and so formula_70, and formula_61 is a group. Now, since formula_72, formula_73 generates formula_74 and so formula_61 is cyclic. 
Unless we explicitly state otherwise, by formula_74 we will always refer to the cyclic group formula_61. Since the argument for the generator of formula_74 can be made valid for any integer formula_38, this shows that also formula_80 is cyclic with the generator formula_73.
Theorem 5: An element formula_82 is a generator if and only if formula_83.
"Proof": We will need the following theorem from number theory: If formula_84 are integers, then there exists integers formula_85 such that formula_86, if and only if formula_87. We will not prove this here. A proof can be found in the number theory section.
For the right implication, assume that formula_88. Then for all formula_89, formula_90 for some integer formula_91. In particular, there exists an integer formula_92 such that formula_93. This implies that there exists another integer formula_94 such that formula_95. By the above-mentioned theorem from number theory, we then have formula_83. For the left implication, assume formula_83. Then there exists integers formula_98 such that formula_99, implying that formula_100 in formula_74. Since formula_73 generates formula_74, it must be true that formula_38 is also a generator, proving the theorem. 
We can generalize Theorem 5 a bit by looking at the orders of the elements in cyclic groups.
Theorem 6: Let formula_105. Then, formula_106.
"Proof": Recall that the order of formula_107 is defined as the lowest positive integer formula_94 such that formula_109 in formula_74. Since formula_111 is cyclic, there exists an integer formula_92 such that formula_113 is minimal and positive. This is the definition of the least common multiple; formula_114. Recall from number theory that formula_115. Thus, formula_116, as was to be proven. 
Theorem 7: Every subgroup of formula_80 is of the form formula_118.
"Proof": The fact that any subgroup of formula_80 is cyclic follows from Theorem 3. Therefore, let formula_15 generate formula_121. Then we see immediately that formula_122. 
Theorem 8: Let formula_123 be fixed, and let formula_124. Then formula_35 is a subgroup of formula_80 generated by formula_127.
"Proof": We msut first show that formula_35 is a subgroup. This is immediate since formula_129. From the proof of Theorem 3, we see that any subgroup of formula_80 is generated by its lowest positive element. It is a theorem of number theory that the lowest positive integer formula_131 such that formula_132 for fixed integers formula_133 and formula_134 equals the greatest common divisor of formula_38 and formula_66 or formula_127. Thus formula_127 generates formula_35. 
Theorem 9: Let formula_140 and formula_141 be subgroups of formula_80. Then formula_143 is the subgroup generated by formula_144.
"Proof": The fact that formula_143 is a subgroup is obvious since formula_140 and formula_141 are subgroups. To find a generator of formula_143, we must find its lowest positive element. That is, the lowest positive integer formula_91 such that formula_91 is both a multiple of formula_15 and of formula_152. This is the definition of the least common multiple of formula_15 and formula_152, or formula_144, and the result follows. 
It should be obvious by now that formula_19 and formula_74, and formula_22 and formula_80 are the "same groups". This will be made precise in a later section but can be visualized by denoting any generator of formula_19 or formula_22 by formula_73.
We will have more to say about cyclic groups later, when we have more tools at our disposal.