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8.6 Group Morphisms 281 EXERCISE 8.6.3 Let Z be the group of integers under addition, let n be a positive integer (n ≥ 2). Show that φ : Z → Zn defined by φ(k) = (n) + k is an epimorphism that is not an isomorphism. Example 8.6.4 Let G = (R + , ×, 1, −1 ) and H = (R, +, 0, −). Even though we have not discussed logarithms in this text, your work in pre-calculus reveals that φ(x) = ln x is an isomorphism. For ln x is a bijection from R + to R and satisfies φ(xy) = ln(xy) = ln x + ln y = φ(x) + φ(y). � Example 8.6.5 Let G and H be groups, and define φ : G → H by φ(x) = eH for all x ∈ G. Then φ is called the trivial morphism. � EXERCISE 8.6.6 Define addition in R 2 by (x1,y1) + (x2,y2) = (x1 + x2,y1 + y2) which is clearly a group. Let φ : R 2 → R be defined by φ[(x, y)] =x. Show that φ is an epimorphism that is not an isomorphism from the additive group R 2 to the additive group of real numbers. (This is called a projection morphism because every point in the xy-plane is mapped directly up or down to its x-coordinate.) EXERCISE 8.6.7 Let G be a group, and define φ : G → G by φ(g) = g −1 . Assuming φ is a bijection from G to itself, show that φ is an automorphism of G if and only if G is abelian. The morphic behavior of φ : G → H does not explicitly require that particular elements of G must map to particular elements of H. However, there are some restrictions of this sort inherent in the definition. Theorem 8.6.8 If φ : G → H is a group morphism, then: 1. φ(eG) = eH. 2. For all x ∈ G, φ(x−1 ) =[φ(x)] −1 . We will prove part (1) and leave part 2 to you as an exercise. Proof of Part (1). By cancellation, φ(eG) = eH. eH · φ(eG) = φ(eG) = φ(eG ∗ eG) = φ(eG) · φ(eG) (8.73) EXERCISE 8.6.9 Prove part (2) of Theorem 8.6.8.
282 Chapter 8 Groups By using part (1) of Theorem 8.6.8 as the root of an induction argument for n ≥ 0, then using part (2) to take care of n
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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Page 262 and 263: 7.7.2 Uniform Continuity and Compac
Page 264 and 265: P A R T I I I Basic Principles of A
Page 266 and 267: Groups 8 In its simplest terms, alg
Page 268 and 269: 8.1 Introduction to Groups 245 are
Page 270 and 271: 8.1 Introduction to Groups 247 Defi
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Page 274 and 275: Show that C × is an abelian group
Page 276 and 277: 8.2 Subgroups 253 G1 and G3 are aut
Page 278 and 279: 8.2 Subgroups 255 Notice how proper
Page 280 and 281: 8.2 Subgroups 257 Example 8.2.15 su
Page 282 and 283: EXERCISE 8.2.23 If G is a group and
Page 284 and 285: 8.3 Quotient Groups 261 [0] ={...,
Page 286 and 287: 8.3 Quotient Groups 263 is standard
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Page 290 and 291: (iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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Page 298 and 299: EXERCISE 8.4.12 Complete Table 8.64
Page 300 and 301: 8.5 Normal Subgroups 277 Theorem 8.
Page 302 and 303: 8.5 Normal Subgroups 279 point to b
Page 306 and 307: 8.6 Group Morphisms 283 Notice that
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Page 310 and 311: Rings 9 We can create algebraic str
Page 312 and 313: 9.1 Rings and Fields 289 (R9) Multi
Page 314 and 315: 9.1 Rings and Fields 291 we define
Page 316 and 317: 9.2 Subrings 293 In addition to pro
Page 318 and 319: 9.2 Subrings 295 p1 = q1 = 3. Verif
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Page 322 and 323: 9.3 Ring Properties 299 EXERCISE 9.
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Page 326 and 327: 9.4 Ring Extensions 303 Example 9.4
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Page 338 and 339: 9.8 Integral Domains 315 EXERCISE 9
Page 340 and 341: 9.8 Integral Domains 317 Corollary
Page 342 and 343: 9.9 Unique Factorization Domains 31
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Page 346 and 347: 9.10 Principal Ideal Domains 323 So
Page 348 and 349: 9.11 Euclidean Domains 325 next exe
Page 350 and 351: 9.11 Euclidean Domains 327 Exercise
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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