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CHAPTER 2 Groups Definition (Binary Operation). Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. Note. This condition of assigning an element of G to each ordered pair of G is called the closure of the set G under the given binary operation. Example. Addition, subtraction, and multiplication in Z are binary operations; division in Z is not (8 ÷ 3 62 Z). Example. Let Zn = {0, 1, 2, . . . , n 1}, the integers modulo n. Addition modulo n and multiplication modulo n are binary operations. Definition (Group). Let G be a nonempty set together with a binary operation on G (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element of G denoted by ab. G is a group under this operation if (1) The operation is associative: (ab)c = a(bc) 8 a, b, c 2 G. (2) There is an identity element e in G such that ae = ea = a 8 a 2 G. (3) For each a 2 G, there is an inverse element b 2 G such that ab = ba = e. Note. b is often denoted as a 1. 27 28 2. GROUPS A group G is Abelian ( or commutative) if ab = ba 8 a, b 2 G. It is non-Abelian if there exist a, b 2 G such that ab 6= ba. Example. (1) Z, Q, and R are groups under + with identity 0 and inverse a for a. None of these are groups under ⇥ since 1 is the multiplicative identity and so 0 has no inverse in each case. However, Q⇤ = Q\{0} and R⇤ = R\{0} are groups under ⇥, but Z⇤ = Z\{0} is not (e.g., 3 has no inverse). (2) {1, 1, i, i} is a group under complex multiplication. 1 and own inverses, and i and i are inverses of each other. 1 are their (3) The set S of positive irrational numbers along with 1 satisfy properties (1), (2), and (3) of the definition of group under ⇥. But p Spis not a group since ⇥ is not a binary operation on S. Closure fails (e.g., 2 2 = 2 62 S). ⇢ a b (4) Let M = a, b, c, d 2 R , the set of 2 ⇥ 2 matrices with c d a1 b1 a b a + a2 b1 + b2 + 2 2 = 1 . c1 d1 c2 d2 c1 + c2 d1 + d2 0 0 M is a group with this operation. is the identity and the inverse of 0 0 a b a b is . c d c d (5) Zn = {0, 1, 2, . . . , n 1} is a group under addition modulo n where, for j > 0, n j is the inverse of j. This is the group of integers modulo n. 2. GROUPS a b (6) The determinant of the 2 ⇥ 2 matrix A = is det A = ad c d Consider ⇢ a b GL(2, R) = a, b, c, d 2 R, ad bc 6= 0 c d with multiplication a1 b1 a2 b2 a a + b1c2 a1b2 + b1d2 = 1 2 . c1 d1 c2 d2 c1a2 + d1c2 c1b2 + d1d2 29 bc. Multiplication is closed in GL(2, R) since det(AB) = (det A)(det B). Associa1 0 a b tivity is true, but messy; the identity is ; the inverse of is 0 1 c d d b 1 d b ad bc ad bc = . c a c a ad bc ad bc ad bc This is the general linear group of 2 ⇥ 2 matricies over R. Since 1 2 2 3 10 13 2 3 1 2 11 16 = and = , 3 4 4 5 22 29 4 5 3 4 19 28 GL(2, R) is non-Abelian. The set of all 2 ⇥ 2 matrices over R with matrix multiplication is not a group since matrices with 0 determinant do not have inverses. (7) Consider Zn with multiplication modulo n. Are there multiplicative inverses? If so, we have a group. Suppose a 2 Zn and ax mod n = 1 has a solution (i.e., a has an inverse). Then ax = qn + 1 for some q 2 Z =) ax + n( q) = 1 =) a and n are relatively prime by Theorem 0.2. Now suppose a is relatively prime to n. Then, again by Theorem 0.2, 9 s, t 2 Z 3 as+nt = 1 =) as = ( t)n+1 =) as mod n = 1 =) s = a 1. Thus we have proven Page 24 # 11: 30 2. GROUPS Lemma (Restatement of Page 24 # 11). a 2 Z has a multiplicative inverse modulo n () a and n are relatively prime. Definition (U (n)). For each n > 1, define U (n) = {x 2 Zn|x and n are relatively prime}. U (n) will be a group under multiplication if multiplication modulo n is closed. Lemma. If a, b 2 U (n), then ab 2 U (n). Proof. a, b 2 U (n) =) 9 s1, t1, s2, t2 2 Z 3 as1 + nt1 = 1 and bs2 + nt2 = 1 =) as1 = 1 nt1 and bs2 = 1 nt2 =)(ab)(s1s2) = 1 nt1 nt2 + n2t1t2 =) (ab)(s1s2) + n(t1 + t2 + n2t1t2) = 1. Let s = s1s2 and t = t1 + t2 + n2t1t2. Then (ab)s + nt = 1 =) ab and n are relatively prime =) ab 2 U (n). ⇤ So multiplication modulo n is closed in U (n), and U (n) is an Abelian group. Example. Consider U (14) = {1, 3, 5, 9, 11, 13}. mod14 1 3 5 9 11 13 1 1 3 5 9 11 13 3 3 9 1 13 5 11 5 5 1 11 3 13 9 9 9 13 3 11 1 5 11 11 5 13 1 9 3 13 13 11 9 5 3 1 Corollary. Z⇤n (the nonzero integers modulo n) is a group under multiplication modulo n () n is prime. 2. GROUPS 31 (8) Rn = {(a1, a2, . . . , an)|a1, a2, . . . , an 2 R} is an Abelian group under vector addition. (9) For (a, b, c) 2 R3, define Ta,b,c : R3 ! R3 by Ta,b,c(x, y, z) = (x + a, y + b, z + c). Then T = {Ta,b,c|a, b, c 2 R} is a group under function composition. Ta,b,cTd,e,f = Ta+d,b+e,c+f . T0,0,0 is the identity, and the inverse of Ta,b,c is T is Abelian. a, b, c . This translation group (10) Let p be prime and F 2 {Q, R, C, Zp}. The special linear group SL(2, F ) is ⇢ a b SL(2, F ) = a, b, c, d 2 F, ad c d bc = 1 , where the operation is matrix multiplication (modulo p in Zp). It is a non a b d b Abelian group. The inverse of is . c d c a 6 2 In SL(2, Z7), consider A = . det A = 6 · 5 2 · 4 = 22 = 1 mod 7. 4 5 5 2 5 5 A 1= = mod 7 since 4 6 3 6 6 2 5 5 36 42 1 0 AA 1 = = = mod 7 and 4 5 3 6 35 50 0 1 5 5 6 2 50 35 1 0 A 1A = = = mod 7. 3 6 4 5 42 36 0 1 32 2. GROUPS GL(2, F ) is also a group under matrix multiplication (modulo p for Zp). Also, in the case of Zp, interpret division by ad bc as multiplication by the inverse of ad bc modulo p. GL(2, F ) is non-Abelian. 9 6 In GL(2, Z11), consider A = . det A = 63 48 = 15 = 4 mod 11. The 8 7 inverse of 4 mod 11 is 3 mod 11 since 4 · 3 = 12 = 1 mod 11. Then 7·3 6·3 21 18 10 4 A 1= = = mod 11 since 8·3 9·3 24 27 9 5 9 6 10 4 144 66 1 0 AA 1 = = = mod 11 and 8 7 9 5 143 67 0 1 10 4 9 6 122 88 1 0 A 1A = = = mod 11. 9 5 8 7 121 89 0 1 Theorem (2.1 — Uniqueness of the Identity). The identity of a group G is unique. Proof. Suppose e and e0 are identity elements of G. Then e = ee0 = e0 from the definition of identity element, so e = e0 and the identity is unique. ⇤ Theorem (2.2 — Cancellation). In a group G, the right and left cancellation laws hold; that is, ba = ca =) b = c and ab = ac =) b = c. Proof. Suppose ba = ca and let a0 be an inverse of a. Then (ba)a0 = (ca)a0 =) b(aa0) = c(aa0) by associativity =) be = ce =) b = c. The proof for left cancellation is similar. ⇤ 2. GROUPS 33 Theorem (2.3 — Uniqueness of Inverses). For each element a in a group G, there exists a unique b 2 G such that ab = ba = e. Proof. Suppose b and c are inverses of a. Then ab = e and ac = e =) ab = ac =) b = c by cancellation. ⇤ Note. This allows us to ambiguously denote the inverse of g 2 G as g 1. Notation. g 0 = e, gn = ggg · · · g} | {z (unambiguous by associativity). n factors, n positive For n < 0, g n = (g 1)|n|, e.g., g For every m, n 2 Z snd g 2 G, 4 = (g 1)4. g mg n = g m+n and (g m)n = g mn. However, in general, (ab)n 6= anbn. Translations to use if the group operation is “+” instead of “·.” multiplicative ab or a · b e or 1 a 1 an ab 1 additive a+b 0 a na a b 34 2. GROUPS Theorem (2.4 — Socks-Shoes Property). In a group G, (ab) Proof. By definition and Theorem 2.3, (ab) 1 (ab)(ab) But 1 = b 1a 1. is the unique element in G such that = (ab) 1(ab) = e. (ab)(b 1a 1) = a(bb 1)a and 1 1 = aea 1 = aa 1 =e (b 1a 1)(ab) = b 1(a 1a)b = b 1eb = b 1b = e, so (ab) 1 = b 1a 1. (In other words, b 1a 1 is the inverse of ab since it acts like an inverse, and the inverse is unique.) ⇤ Problem (Page 56 # 25). 1 A group G is Abelian () (ab) Proof. = a 1b 1 8 a, b 2 G. G is Abelian () ab = ba 8 a, b 2 G () aba aba 1 = b () aba 1b aba 1b 1 = e () (ab) 1 1 = bb 1 () = a 1b 1. 1 = baa 1 () ⇤ 2. GROUPS 35 Theorem (2). If G is a group and a, b 2 G, there exist unique c, d 2 G 3 ac = b and da = b (i.e., the equations ax = b and xa = b have unique solutions in G). Proof. Let c = a 1b. Then ac = a(a 1b) = (aa 1)b = eb = b, so c is a solution of ax = b. Suppose also ac0 = b. Then c = ec = (a 1a)c = a 1(ac) = a 1b = a 1(ac0) = (a 1a)c0 = ec0 = c0. Thus the solution of ax = b is unique. The proof of the second half is similar. ⇤