Example. Addition, subtraction, and multiplication in Z are binary operations; division in Z is not ( Z).

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1 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 ( Z). Example. Let Z n {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 inary 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 Note. b is often denoted as a 1. ab ba e. 27

2 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 1 are their own inverses, and i and i are inverses of each other. (3) The set S of positive irrational numbers along with 1 satisfy properties (1), (2), and (3) of the definition of group under. But S is not a group since is not inary operation on S. Closure fails (e.g., p 2 p S). (4) Let M c d a, b, c, d 2 R, the set of 2 2 matrices with a1 b 1 c 1 d 1 + a2 b 2 c 2 d 2 M is a group with this operation. c d is c d a1 + a 2 b 1 + b 2 c 1 + c 2 d 1 + d 2. is the identity and the inverse of (5) Z n {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.

3 (6) The determinant of the 2 2 matrix A Consider GL(2, R) c d with multiplication a1 b 1 a2 b 2 c 1 d 1 c 2 d 2 c d 2. GROUPS 29 a, b, c, d 2 R, ad bc 6 0 a1 a 2 + b 1 c 2 a 1 b 2 + b 1 d 2 c 1 a 2 + d 1 c 2 c 1 b 2 + d 1 d 2. is det A ad bc. Multiplication is closed in GL(2, R) since det(ab) (det A)(det B). Associativity is true, but messy; the identity is ; the inverse of c d is d b 1 ad bc ad bc d b c. ad bc c a ad bc a ad bc This is the general linear group of 2 2 matricies over R. Since and , 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 Z n with multiplication modulo n. Are there multiplicative inverses? If so, we have a group. Suppose a 2 Z n 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:

4 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 Z n 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). a, b 2 U(n) ) 9 s 1, t 1, s 2, t 2 2 Z 3 as 1 + nt 1 1 and bs 2 + nt 2 1 ) as 1 1 nt 1 and bs 2 1 nt 2 )(ab)(s 1 s 2 ) 1 nt 1 nt 2 + n 2 t 1 t 2 ) (ab)(s 1 s 2 ) + n(t 1 + t 2 + n 2 t 1 t 2 ) 1. Let s s 1 s 2 and t t 1 + t 2 + n 2 t 1 t 2. 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}. mod Corollary. Z n (the nonzero integers modulo n) is a group under multiplication modulo n () n is prime.

5 2. GROUPS 31 (8) R n {(a 1, a 2,..., a n ) a 1, a 2,..., a n 2 R} is an Abelian group under vector addition. (9) For (a, b, c) 2 R 3, define T a,b,c : R 3! R 3 by T a,b,c (x, y, z) (x + a, y + b, z + c). Then T {T a,b,c a, b, c 2 R} is a group under function composition. T a,b,c T d,e,f T a+d,b+e,c+f. T 0,0,0 is the identity, and the inverse of T a,b,c is T a, b, c. This translation group is Abelian. (10) Let p be prime and F 2 {Q, R, C, Z p }. The special linear group SL(2, F ) is SL(2, F ) a, b, c, d 2 F, ad bc 1, c d where the operation is matrixmultiplication (modulo p in Z p ). It is a nona b d b Abelian group. The inverse of c d is. c a 6 2 In SL(2, Z 7 ), consider A. det A mod A mod 7 since AA mod 7 and 0 1 A A mod

6 32 2. GROUPS GL(2, F ) is also a group under matrix multiplication (modulo p for Z p ). Also, in the case of Z p, 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, Z 11 ), consider A. det A mod 11. The 8 7 inverse of 4 mod 11 is 3 mod 11 since mod 11. Then A mod 11 since AA mod 11 and 0 1 A A mod Theorem (2.1 Uniqueness of the Identity). The identity of a group G is unique. Suppose e and e 0 are identity elements of G. Then e ee 0 e 0 from the definition of identity element, so e e 0 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. Suppose ba ca and let a 0 be an inverse of a. Then (ba)a 0 (ca)a 0 ) b(aa 0 ) c(aa 0 ) by associativity ) be ce ) b c. The proof for left cancellation is similar.

7 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. 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, g n ggg {z g } (unambiguous by associativity). n factors, n positive For n < 0, g n (g 1 ) n, e.g., g 4 (g 1 ) 4. For every m, n 2 Z snd g 2 G, However, in general, g m g n g m+n and (g m ) n g mn. (ab) n 6 a n b n. Translations to use if the group operation is + instead of. multiplicative additive ab or a b a + b e or 1 0 a 1 a a n na ab 1

8 34 2. GROUPS Theorem (2.4 Socks-Shoes Property). In a group G, (ab) 1 b 1 a 1. By definition and Theorem 2.3, (ab) 1 is the unique element in G such that (ab)(ab) 1 (ab) 1 (ab) e. But (ab)(b 1 a 1 ) a(bb 1 )a 1 aea 1 aa 1 e and (b 1 a 1 )(ab) b 1 (a 1 a)b b 1 eb b 1 b e, so (ab) 1 b 1 a 1. (In other words, b 1 a 1 is the inverse of ab since it acts like an inverse, and the inverse is unique.) Problem (Page 56 # 25). A group G is Abelian () (ab) 1 a 1 b 1 8 a, b 2 G. G is Abelian () ab ba 8 a, b 2 G () aba 1 baa 1 () aba 1 b () aba 1 b 1 bb 1 () aba 1 b 1 e () (ab) 1 a 1 b 1.

9 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). Let c a 1 b. Then ac a(a 1 b) (aa 1 )b eb b, so c is a solution of ax b. Suppose also ac 0 b. Then c ec (a 1 a)c a 1 (ac) a 1 b a 1 (ac 0 ) (a 1 a)c 0 ec 0 c 0. Thus the solution of ax b is unique. The proof of the second half is similar.