30 Chapter 1. Groups I A group G is called cyclic if there exists a ∈ G with G = a , in which case a is called a generator of G. The Laws of Exponents, Proposition 1.23, show that a is, in fact, a subgroup: 1 = a0 ∈ a anam = an+m ∈ a a−1 ∈ a . Example 1.37. (i) The multiplicative group μn of all nth roots of unity [see Example 1.17(iv)] is a cyclic group a generator is the primitive nth root unity ζ = e2πi/n [by De Moivre’s Theorem, e2πik/n = cos ( 2πk n ) + i sin ( 2πk n )of = ζk]. (ii) The (additive) group Z is an infinite cyclic group with generator 1. (iii) We recall the definition of the integers modulo m. Given m ≥ 0 and a ∈ Z, the congruence class of a mod m, denoted by [a], is [a] = {b ∈ Z : b ≡ a mod m} = {a + km : k ∈ Z}. Definition. The integers mod m, denoted by Im,18 is the family of all congruence classes mod m with binary operation [a] + [b] = [a + b]. It is easy to see that Im is a group it is a cyclic group, for [1] is a generator. Note that if m ≥ 1, then Im has exactly m elements, namely, [0], [1],..., [m − 1]. Even though the definition of Im makes sense for all m ≥ 0, one usually assumes that m ≥ 2 because the cases m = 0 and m = 1 are not very interesting. If m = 0, then Im = I0 = Z, for a ≡ b mod 0 means 0 | (a − b) that is, a = b. If m = 1, then Im = I1 = {[0]}, for a ≡ b mod 1 means 1 | (a − b) that is, a and b are always congruent. The next proposition relates the two usages of the word order in Group Theory. Proposition 1.38. Let G be a group. If a ∈ G, then the order of a is equal to | a |, the order of the cyclic subgroup generated by a. Proof. The result is obviously true when a has infinite order, and so we may assume that a has finite order n. We claim that A = {1,a,a2,...,an−1} has exactly n elements that is, the displayed elements are distinct. If ai = aj for 0 ≤ i j ≤ n − 1, then aj−i = 1 as 0 j − i n, this contradicts n being the smallest positive integer with an = 1. It suﬃces to show that A = a . Clearly, A ⊆ a . For the reverse inclusion, take ak ∈ a . By the Division Algorithm, k = qn + r, where 0 ≤ r n hence, ak = aqn+r = aqnar = (an)qar = ar. Thus, ak = ar ∈ A, and a = A. • A cyclic group can have several different generators for example, a = a−1 . 18We introduce this new notation because there is no commonly agreed one the most popular contenders are Z/mZ and Zm. The former notation is too complicated to use many times in a proof the latter is ambiguous because, when p is prime, Zp often denotes the ring of p-adic integers and not the integers mod p.

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