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Chapter 15 GROUP THEORY and APPLICATIONS

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Chapter 15 GROUP THEORY and APPLICATIONS Chapter 15 GROUP THEORY AND APPLICATIONS GOALS In Chapter 11, Algebraic Systems, groups were introduced as a typical algebraic system. The associated concepts of subgroup, group isomor- phism, and direct products of groups were also introduced. Groups were chosen for that chapter because they are among the simplest types of algebraic systems. Despite this simplicity, group theory abounds with interesting applications, many of which are of interest to the computer scientist. In this chapter we intend to present the remaining important concepts in elementary group theory and some of their applications. 15.1 Cyclic Groups Groups are classified according to their size and structure. A group's structure is revealed by a study of its subgroups and other properties (e.g., whether it is abelian) that might give an overview of it. Cyclic groups have the simplest structure of all groups. Definitions: Cyclic Group, Generator. Group G is cyclic if there exists a œ G such that the cyclic subgroup generated by a, HaL, equals all of G. That is, G = 8n a n œ Z<, in which case a is called a generator of G. The reader should note that additive notation is used for G. Example 15.1.1. Z12 = @Z12 , +12 D, where +12 is addition modulo 12, is a cyclic group. To verify this statement, all we need to do is demonstrate that some element of Z12 is a generator. One such element is 5; that is, (5) = Z12 . One more obvious generator is 1. In fact, 1 is a generator of every @Zn; +nD. The reader is asked to prove that if an element is a generator, then its inverse is also a generator. Thus, -5 = 7 and -1 = 11 are the other generators of Z12. 0 0 11 1 11 1 10 2 10 2 9 3 9 3 8 4 8 4 7 5 7 5 6 6 HaL HbL Figure 15.1.1 Examples of "string art" Figure 15.1.1(a) is an example of "string art" that illustrates how 5 generates Z12. Twelve tacks are placed along a circle and numbered. A string is tied to tack 0, and is then looped around every fifth tack. As a result, the numbers of the tacks that are reached are exactly the ordered li l f 5 dl 12 5 10 3 7 0 N h if h k d h k ld b bi d If hi d Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States Chapter 15 - Group Theory and Applications multiples of 5 modulo 12: 5, 10, 3, ... , 7, 0. Note that if every seventh tack were used, the same artwork would be obtained. If every third tack were connected, as in Figure 15.1.1(b), the resulting loop would only use four tacks; thus 3 does not generate Z12 . Example 15.1.2. The group of additive integers, [Z; +], is cyclic: Z = H1L = 8nÿ1 n œ Z<. This observation does not mean that every integer is the product of an integer times 1. It means that n terms n terms Z = 80< ‹ :1 + 1 + º⋯ + 1 n œ P> ‹ :H-1L + H-1L + º⋯ + H-1L n œ P> Theorem 15.1.1. If @G *D is cyclic, then it is abelian. Proof: Let a be any generator of G and let b, c œ G. By the definition of the generator of a group, there exists integers m and n such that b = m a and c = n a. Thus b*c = Hm aL*Hn aL = Hm + nL a by Theorem 11.3.7(ii) = Hn + mL a = Hn aL*Hn bL = c*b ‡ One of the first steps in proving a property of cyclic groups is to use the fact that there exists a generator. Then every element of the group can be expressed as some multiple of the generator. Take special note of how this is used in theorems of this section. Up to now we have used only additive notation to discuss cyclic groups. Theorem 15.1.1 actually justifies this practice since it is customary to use additive notation when discussing abelian groups. Of course, some concrete groups for which we employ multiplicative notation are cyclic. If one of its elements, a, is a generator, n HaL = 8a n œ Z< * Example 15.1.3. The group of positive integers modulo 11 with modulo 11 multiplication, @Z11 ;µ11 D, is cyclic. One of its generators is 6: 61 = 6, 62 = 3, 63 = 7,… , 69 = 2, and 610 = 1, the identity of the group. Example 15.1.4. The real numbers with addition, @R; +D is a noncyclic group. The proof of this statement requires a bit more generality since we are saying that for all r œ R, HrL is a proper subset of R. If r is nonzero, the multiples of r are distributed over the real line, as in Figure 15.1.2. It is clear then that there are many real numbers, like rê2, that are not in HrL. -3r -2r -1r 0r 1r 2r 3r Figure 15.1.2 Elements of HrL, r > 0 The following theorem shows that a cyclic group can never be very complicated. Theorem 15.1.2. If G is a cyclic group, then G is either finite or countably infinite. If G is finite and †G§ = n, it is isomorphic to @Zn, +nD. If G is infinite, it is isomorphic to @Z, +D. Proof: Case 1: †G§ < ¶. If a is a generator of G and †G§ = n, define f : Zn Ø G by fHkL = k a for all k œ Zn Since HaL is finite, we can use the fact that the elements of HaL are the first n nonnegative multiples of a. From this observation, we see that f is a surjection. A surjection between finite sets of the same cardinality must be a bijection. Finally, if p, q œ Zn, fHpL + fHqL = p a + q a = Hp + qL a = Hp +n qL a see exercise 10 = fHp +n qL Therefore f is an isomorphism. Case 2; †G§ = ¶. We will leave this case as an exercise. ‡ The proof of the next theorem makes use of the division property for integers, which was introduced in Section 11.4: If m, n are integers, m > 0, there exist unique integers q (quotient) and r (remainder) such that n = qm + r and 0 § r < m. Theorem 15.1.3. Every subgroup of a cyclic group is cyclic. Proof: Let G be cyclic with generator a and let H § G. If H = 8e<, H has e as a generator. We may now assume that †H§ ¥ 2 and a ¹≠ e. Let m be the least positive integer such that m a belongs to H. (This is the key step. It lets us get our hands on a generator of H.) We will now show that c = m a generates H. Suppose that HcL ¹≠ H. Then there exists b œ H such that b – HcL. Now, since b is in G, there exists n œ Z such that b = n a. We now apply the division property and divide n by m. b = n a = Hq m + rL a = Hq mL a + r a, where 0 § r < m. We note that r cannot be zero for otherwise we would have b = n a = qHm aL = q c œ HcL. Therefore, Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States Chapter 15 - Group Theory and Applications r a = n a - Hq mL a œ H This contradicts our choice of m because 0 < r < m. ‡ Example 15.1.5. The only proper subgroups of Z10 are H1 = 80, 5< and H2 = 80, 2, 4, 6, 8<. They are both cyclic: H1 = H5L, while H2 = H2L = H4L = H6L = H8L. The generators of Z10 are 1, 3, 7, and 9. Example 15.1.6. With the exception of 80<, all subgroups of Z are isomorphic to Z. lf H § Z, then H is the cyclic subgroup generated by the least positive element of H. It is infinite and so by theorem 15.1.2 it is isomorphic to Z. We now cite a useful theorem for computing the order of cyclic subgroups of a cyclic group: Theorem 15.1.4. If G is a cyclic group of order n and a is a generator of G, the order of k a is nêd, where d is the greatest common divisor of n and k. The proof of this theorem is left to the reader. Example 15.1.7. To compute the order of H18L in Z30, we first observe that 1 is a generator of Z30 and 18 = 18(1). The greatest common divisor of 18 and 30 is 6. Hence, the order of (18) is 30/6, or 5. APPLICATION: FAST ADDERS At this point, we will introduce the idea of a fast adder, a relatively modern application (Winograd, 1965) to an ancient theorem, the Chinese Remainder Theorem. We will present only an overview of the theory and rely primarily on examples. The interested reader can refer to Dornhoff and Hohn for details. Out of necessity, integer addition with a computer is addition modulo n, for n some larger number. Consider the case where n is small, like 64. Then addition involves the addition of six-digit binary numbers. Consider the process of adding 31 and 1. Assume the computer’s adder takes as input two bit strings a = 8a0, a1, a2, a3, a4, a5< and b = 8b0, b1, b2, b3, b4, b5< and outputs s = 8s0, s1, s2, s3, s4, s5<, the sum of a and b.
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