A Friendly Introduction to Abstract Algebra offers a new approach to laying a foundation for abstract mathematics. Prior experience with proofs is not assumed, and the book takes time to build proof-writing skills in ways that will serve students through a lifetime of learning and creating mathematics.
The author's pedagogical philosophy is that when students abstract from a wide range of examples, they are better equipped to conjecture, formalize, and prove new ideas in abstract algebra. Thus, students thoroughly explore all concepts through illuminating examples before formal definitions are introduced. The instruction in proof writing is similarly grounded in student exploration and experience. Throughout the book, the author carefully explains where the ideas in a given proof come from, along with hints and tips on how students can derive those proofs on their own.
Readers of this text are not just consumers of mathematical knowledge. Rather, they are learning mathematics by creating mathematics. The author's gentle, helpful writing voice makes this text a particularly appealing choice for instructors and students alike. The book's website has companion materials that support the active-learning approaches in the book, including in-class modules designed to facilitate student exploration.
Author(s): Ryota Matsuura
Series: AMS/MAA Textbooks, 72
Edition: 1
Publisher: MAA Press/American Mathematical Society
Year: 2022
Language: English
Pages: 404
Tags: Algebra; Abstract Algebra; Proofs; Sets; Symmetries; Permutations; Matrices; Groups; Isomorphisms; Homomorphisms; Cosets; Lagrange’s Theorem; Rings
Cover
Title page
Copyright
Contents
Preface
For the student
For the instructor
Note about rings
Road map
Acknowledgments
Unit I: Preliminaries
Chapter 1. Introduction to Proofs
1.1. Proving an implication
1.2. Proof by cases
1.3. Contrapositive
1.4. Proof by contradiction
1.5. If and only if
1.6. Counterexample
Exercises
Chapter 2. Sets and Subsets
2.1. What is a set?
2.2. Set of integers and its subsets
2.3. Closure
2.4. Showing set equality
Exercises
Chapter 3. Divisors
3.1. Divisor
3.2. GCD theorem
3.3. Proofs involving the GCD theorem
Exercises
Unit II: Examples of Groups
Chapter 4. Modular Arithmetic
4.1. Number system Z ₇
4.2. Equality in Z ₇
4.3. Multiplicative inverses
Exercises
Chapter 5. Symmetries
5.1. Symmetries of a square
5.2. Group properties of ?₄
5.3. Centralizer
Exercises
Chapter 6. Permutations
6.1. Permutations of the set {1,2,3}
6.2. Group properties of ?_{?}
6.3. Computations in ?_{?}
6.4. Associative law in ?_{?} (and in ?_{?})
Exercises
Chapter 7. Matrices
7.1. Matrix arithmetic
7.2. Matrix group ?(Z ₁₀)
7.3. Multiplicative inverses
7.4. Determinant
Exercises
Unit III: Introduction to Groups
Chapter 8. Introduction to Groups
8.1. Definition of a “group”
8.2. Essential properties of a group
8.3. Proving that a group is commutative
8.4. Non-associative operations
8.5. Direct product
Exercises
Chapter 9. Groups of Small Size
9.1. Smallest group
9.2. Groups with two elements
9.3. Groups with three elements
9.4. Sudoku property
9.5. Groups with four elements
Exercises
Chapter 10. Matrix Groups
10.1. Groups Z ₁₀ and ?₁₀
10.2. Groups ?(Z ₁₀) and ?(Z ₁₀)
10.3. Group ?(Z ₁₀)
Exercises
Chapter 11. Subgroups
11.1. Examples of subgroups
11.2. Subgroup proofs
11.3. Center and centralizer revisited
Exercises
Chapter 12. Order of an Element
12.1. Motivating example
12.2. When does ?^{?}=??
12.3. Conjugates
12.4. Order in an additive group
12.5. Elements with infinite order
Exercises
Chapter 13. Cyclic Groups, Part I
13.1. Generators of the additive group Z ₁₂
13.2. Generators of the multiplicative group ?₁₃
13.3. Matching Z ₁₂ and ?₁₃
13.4. Taking positive and negative powers of ?
13.5. When the group operation is addition
Exercises
Chapter 14. Cyclic Groups, Part II
14.1. Why negative powers are needed
14.2. Additive groups revisited
14.3. ⟨3⟩ behaves “just like” Z
14.4. Subgroups of cyclic groups
Exercises
Unit IV: Group Homomorphisms
Chapter 15. Functions
15.1. Domain and codomain
15.2. One-to-one function
15.3. Onto function
15.4. When domain and codomain have the same size
Exercises
Chapter 16. Isomorphisms
16.1. Groups Z ₁₂ and ⟨?⟩: Elements match up
16.2. Groups Z ₁₂ and ⟨?⟩: Operations match up
16.3. Elements with infinite order revisited
16.4. Inverse isomorphisms
Exercises
Chapter 17. Homomorphisms, Part I
17.1. Group homomorphism
17.2. Properties of homomorphisms
17.3. Order of an element
Exercises
Chapter 18. Homomorphisms, Part II
18.1. Kernel of a homomorphism
18.2. Image of a homomorphism
18.3. Partitioning the domain
18.4. Finding homomorphisms
Exercises
Unit V: Quotient Groups
Chapter 19. Introduction to Cosets
19.1. Multiplicative group example
19.2. Additive group example
19.3. Right cosets
19.4. Properties of cosets
19.5. When are cosets equal?
Exercises
Chapter 20. Lagrange’s Theorem
20.1. Motivating Lagrange’s theorem
20.2. Proving Lagrange’s theorem
20.3. Applications of Lagrange’s theorem
Exercises
Chapter 21. Multiplying/Adding Cosets
21.1. Turning a set of cosets into a group
21.2. Coset multiplication shortcut
21.3. Cosets of ?=5Z in Z revisited
Exercises
Chapter 22. Quotient Group Examples
22.1. Quotient group ?₁₃/? revisited
22.2. Quotient group ?₃₇/?
22.3. Quotient group ?/? (generalization)
Exercises
Chapter 23. Quotient Group Proofs
23.1. Sample quotient group proofs
23.2. Collapsing ? into ?/?
Exercises
Chapter 24. Normal Subgroups
24.1. How does the shortcut fail and work?
24.2. Normal subgroups: What and why
24.3. Examples of normal subgroups
24.4. Normal subgroup test
Exercises
Chapter 25. First Isomorphism Theorem
25.1. Familiar homomorphism
25.2. Another homomorphism
25.3. First Isomorphism Theorem
25.4. Finding and building homomorphisms
Exercises
Unit VI: Introduction to Rings
Chapter 26. Introduction to Rings
26.1. Examples and definition
26.2. Fundamental properties
26.3. Units and zero divisors
26.4. Subrings
26.5. Group of units
Exercises
Chapter 27. Integral Domains and Fields
27.1. Integral domains
27.2. Fields
27.3. Idempotent elements
Exercises
Chapter 28. Polynomial Rings, Part I
28.1. Examples and definition
28.2. Degree of a polynomial
28.3. Units and zero divisors
Exercises
Chapter 29. Polynomial Rings, Part II
29.1. Division algorithm in ?[?]
29.2. Factor theorem
29.3. Nilpotent elements
Big picture stuff
Exercises
Chapter 30. Factoring Polynomials
30.1. Examples and definition
30.2. Factorable or unfactorable?
Big picture stuff
Exercises
Unit VII: Quotient Rings
Chapter 31. Ring Homomorphisms
31.1. Evaluation map
31.2. Properties of ring homomorphisms
31.3. Kernel and image
31.4. Examples and definition of an ideal
31.5. Ideals in Z and in ?[?]
Big picture stuff
Exercises
Chapter 32. Introduction to Quotient Rings
32.1. From a quotient group to a quotient ring
32.2. Role of an ideal in a quotient ring
32.3. Quotient ring Z ₃[?]/⟨?²⟩
32.4. First Isomorphism Theorem for rings
Big picture stuff
Exercises
Chapter 33. Quotient Ring Z ₇[?]/⟨?²-1⟩
33.1. Division algorithm revisited
33.2. Another way to reduce in Z ₇[?]/⟨?²-1⟩
33.3. ?[?]/⟨?(?)⟩ is not a field
33.4. ?[?]/⟨?(?)⟩ is a field
Big picture stuff
Exercises
Chapter 34. Quotient Ring R [?]/⟨?²+1⟩
34.1. Reducing elements in R [?]/⟨?²+1⟩
34.2. Field of complex numbers
34.3. ?[?]/⟨?(?)⟩ is a field revisited
Exercises
Chapter 35. ?[?]/⟨?(?)⟩ Is/Isn’t a Field, Part I
35.1. Translate from ?[?] to Z
35.2. Translate (back) from Z to ?[?]
35.3. Proof of Theorem 35.1(b)
Big picture stuff
Exercises
Chapter 36. Maximal Ideals
36.1. Examples and definition
36.2. Maximality of ⟨?(?)⟩
Big picture stuff
Exercises
Chapter 37. ?[?]/⟨?(?)⟩ Is/Isn’t a Field, Part II
37.1. Maximal ideals and quotient rings
37.2. Putting it all together
37.3. Oh wait, but there’s more!
37.4. Prime ideals
Exercises
Appendix A. Proof of the GCD Theorem
Appendix B. Composition Table for ?₄
Appendix C. Symbols and Notations
Appendix D. Essential Theorems
Index of Terms
Back Cover
