Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 1 Groups and Subgroups 1.1 Introduction and Examples 1.1.1 Revisit some examples in brief Ex 1.1. (R, +). Ex 1.2. (Z, +). Ex 1.3. (R+ , ·). Ex 1.4. (V, +) where V is a vector space. Ex 1.5. Rotations on a plane, with composition. Ex 1.6. The set of bijections from R to R, with function composition. Question: What are in common? A set together with an operation. The operation acts “closely” and “associatively” in the set. The set has an identity. Every element in the set has an “inverse”. These are the examples of an algebraic structure called group. 1.1.2 Complex numbers • Real numbers are numbers on a line. • Complex numbers are numbers on a plane (with x-axis and yi-axis). The √ 2 number i satisfies that i = −1. We may write i = −1. The set of complex number is C = {a + bi | a, b ∈ R}. 7 8 CHAPTER 1. GROUPS AND SUBGROUPS A complex number can be expressed in two ways: (by graphs) 1. a + bi, for a, b ∈ R. 2. Euler’s formula states that eiθ = cos θ + i sin θ. So we may write z = |z|(cos θ + i sin θ) = |z|eiθ , |z| ≥ 0 and θ ∈ R. p |a + bi| = a2 + b2 . Multiplication of complex numbers: (by graphs) 1. (a + bi)(c + di) = ac + adi + bci + (bi)(di) = (ac − bd) + (ad + bc)i. 2. aeiα · beiβ = abei(α+β) . Below are some examples of isomorphisms: 1.1.3 Algebra on Circles • Addition modulo 2π on R2π . • Multiplication on unit circle U . • Let φ : U → R2π be the bijection given by eiθ ↔ θ. We see that: if eiθ1 ↔ θ1 and eiθ2 ↔ θ2 , then eiθ1 · eiθ2 ↔ (θ1 +2π θ2 ). This is called an isomorphism. 1.1.4 Roots of Unity The elements of Un = {z ∈ C | z n = 1} is called the n-th roots of unity. Using Euler’s formula, we see that 2π 2π Un = cos m + i sin m | m = 0, 1, 2, · · · , n − 1 n n 2π 2π 0 1 2 n−1 = {1 = ξ , ξ , ξ , · · · , ξ }, ξ = cos + i sin . n n Let Zn = {0, 1, 2, · · · , n − 1} be a subset of Rn . There is an isomorphism Un ∼ Zn in the way that if ξ i ↔ i and ξ j ↔ j, 1.1.5 I-1, p19 Homework 8, 17, 38 then ξ i · ξ j ↔ (i +n j). 1.2. BINARY OPERATIONS 1.2 1.2.1 9 Binary Operations Definitions and Examples Def 1.7. A binary operation ∗ on a set S is a function mapping S × S to S. We write a ∗ b instead of ∗((a, b)). In other words, for every a, b ∈ S, there is exactly one element c ∈ S with c = a ∗ b. Ex 1.8 (Ex 2.2, p21). The usual addition “+” is a binary operation on R, Z = the set of all integers, C. The usual multiplication “·” is a binary operation on R, C, Z, R+ , respectively. Ex 1.9 (Ex 2.5, p21). Let R∗ = R−{0} = the set of nonzero real numbers. Then “·” is closed on R∗ and hence “·” is a binary operation on R∗ . However, “+” is not closed on R∗ (e.g. −2 + 2 = 0 ∈ / R∗ ). So “+” is NOT a ∗ binary operation on R . Ex 1.10 (Ex 2.8, p22, projection). Given a set S, define a ∗0 b = a for (a, b) ∈ S × S. Then ∗0 is a binary operation. Let ∗ be a binary operation ∗ on a set S. Def 1.11. The ∗ is commutative iff a ∗ b = b ∗ a for all a, b ∈ S. Revisit the previous three examples. Def 1.12. The ∗ is associative iff (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S. Thm 1.13. Let S be a set. Let f, g, h be functions mapping S into S. Then (f ◦ g) ◦ h = f ◦ (g ◦ h). Ex 1.14 (Linear Algebra). If A, B, C, are three n × n matrices, then (AB)C = A(BC). The proof is not easy. However, you may check some examples for n = 2. 10 CHAPTER 1. GROUPS AND SUBGROUPS 1.2.2 Tables When S is a finite set, a binary operation ∗ can be described by a table. Ex 1.15. Suppose S = {a, b, c}, then a binary operation ∗ on S may be represented by ∗ a b c a a∗a a∗b a∗c b b∗a b∗b b∗c c c∗a c∗b c∗c Ex 1.16 (Ex 2.14, p24). It is not commutative, not associative. The ∗ is commutative iff the table is symmetric. 1.2.3 Warning To be a binary operation, ∗ should be well-defined and closed on S. Ex 1.17 (Ex 2.19, 2.20, 2.21, p25). 1.2.4 I-2, p25 1.3 Homework 3, 4, 8, 26, 36 Isomorphic Binary Structures Ex 1.18. Compare the tables of binary operations (“+3 ”) on S1 = {1, 2, 0} and on S2 = {one, two, zero} respectively: + 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 and + one two zero one two zero one two zero one two If we make a one-to-one and onto correspondence: 1 ↔ one, 2 ↔ two, then S1 and S2 have the same structure. Ex 1.19 (3.1, 3.2 Tables, p29). 0 ↔ zero, zero one two zero 1.3. ISOMORPHIC BINARY STRUCTURES 11 Def 1.20. A binary algebraic structure (S, ∗) is a set S together with a binary operation ∗ on S. Def 1.21 (isomorphism). Let (S, ∗) and (S 0 , ∗0 ) be algebraic structures. An isomorphism of S with S 0 is a one-to-one function φ mapping S onto S 0 such that (There is a misprinted on the book.) φ(x ∗ y) = φ(x) ∗0 φ(y) for all x, y ∈ S (homomorphism property) If such a map φ exists, then S and S 0 are isomorphic binary structures, which we denote by S ' S 0 . ? Process To show that (S, ∗) and (S 0 , ∗0 ) are isomorphic: 1. Define a function φ : S → S 0 that gives an isomorphism of S with S 0 . 2. Show that φ is a one-to-one function. That is, prove that φ(x) = φ(y) iff x = y. 3. Show that φ is onto S 0 . That is, every element s0 of S 0 can be written as φ(s) for some s ∈ S. 4. Show that φ(x ∗ y) = φ(x) ∗0 φ(y) for all x, y ∈ S. Ex 1.22 (Ex 3.8, p30). Show that (R, +) and (R+ , ·) are isomorphic. Ex 1.23 (Top of p18). Show that (Rc , +c ) and (Rd , +d ) are isomorphic. There is no universal way to prove that two binary algebraic structures (S, ∗) and (S 0 , ∗0 ) are not isomorphic. However, if S and S 0 have different cardinalities, then surely there is no one-to-one and onto map between S and S 0 , and thus S and S 0 are not isomorphic. Ex 1.24. (Zn , +n ) and (Zm , +m ) are not isomorphic (But the previous example says that (Rn , +n ) and (Rm , +m ) are isomorphic). Def 1.25. A structural property of a binary structure is one that must be shared by any isomorphic structure. Ex 1.26 (Ex 3.11, p32). (Z, ·) and (Z+ , ·) are not isomorphic. Ex 1.27 (p 32). Examples of possible structural properties and possible not structural properties. 12 CHAPTER 1. GROUPS AND SUBGROUPS Def 1.28. Let (S, ∗) be a binary structure, an element e of S is an identity element for ∗ if e ∗ s = s ∗ e = s for all s ∈ S. Thm 1.29. A binary structure (S, ∗) has at most one identity element. The existence of an identity element is a structural property. Thm 1.30. Suppose φ : S → S 0 is an isomorphism between binary algebraic structure (S, ∗) and (S 0 , ∗0 ). If S has an identity element e for ∗, then φ(e) is the identity element for ∗0 in S 0 . Ex 1.31 (Ex 3.15, 3.16, 3.17, p 33). If two binary algebraic sets have some different structural properties, then they are not isomorphic. This provides one efficient way to prove that two binary algebraic sets are not isomorphic. 1.3.1 Homework (I-3, p34-p36) 3, 4, 5, 26, 27, 30. (Optional:) Some challenging problems: 18, 24, 25, 34