《Discrete Mathematics II教学-华南理工》Lecture 6 Cyclic Groups.pdfVIP

I.6 Cyclic Groups 1 Section I.6. Cyclic Groups Note. We’ll see that cyclic groups are fundamental examples of groups. In some sense, all finite abelian groups are “made up of” cyclic groups. Recall that the order of a finite group is the number of elements in the group. Definition. Let G be a group and a ∈ G. If G is cyclic and G = a, then (1) if G is finite of order n, then element a is of order n, and (2) if G is infinite then element a is of infinite order. Theorem 6.1. Every cyclic group is abelian. Note. We now state a result from number theory. The text offers “an intuitive digrammatic explanation.” A truly rigorous proof would require a clear definition of N in terms of sets. Theorem 6.3. The Division Algorithm for Z If m is a positive integer (i.e., m ∈ N) and n is any integer (i.e., n ∈ Z), then there exist unique integers q and r such that n = mq + r and 0 ≤ r m. q is called the quotient and r the remainder when n is divided by m. I.6 Cyclic Groups 2 Example. For n = 25 and m = 3, we have 25 = 3(8) + (1), so q = 8 and r = 1. For n = −15 and m = 6 we have −15 = 6(−3) + (3), so q = −3 and r = 3 (notice that r ≥ 0). In terms of the least integer function x, we have for general m and n that q = n and r = n − m n = n − mq . m m Note. The division algorithm is necessary when studying subgroups of cyclic groups. Theorem 6.6. A subgroup of a cyclic group is cyclic. Recall. From the previous section we know that for all n ∈ Z, n = −n = nZ. Since Z itself is cyclic (Z = 1), then by Theorem 6.6 every subgroup of Z must be cyclic. We therefore have the following. Corollary 6.7. The subgroup of Z, + are precisely the groups nZ = n (under +) for