Softcover ISBN:  9781470468811 
Product Code:  TEXT/72 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
eBook ISBN:  9781470470371 
Product Code:  TEXT/72.E 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
Softcover ISBN:  9781470468811 
eBook: ISBN:  9781470470371 
Product Code:  TEXT/72.B 
List Price:  $138.00 $103.50 
MAA Member Price:  $103.50 $77.63 
AMS Member Price:  $103.50 $77.63 
Softcover ISBN:  9781470468811 
Product Code:  TEXT/72 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
eBook ISBN:  9781470470371 
Product Code:  TEXT/72.E 
List Price:  $69.00 
MAA Member Price:  $51.75 
AMS Member Price:  $51.75 
Softcover ISBN:  9781470468811 
eBook ISBN:  9781470470371 
Product Code:  TEXT/72.B 
List Price:  $138.00 $103.50 
MAA Member Price:  $103.50 $77.63 
AMS Member Price:  $103.50 $77.63 

Book DetailsAMS/MAA TextbooksVolume: 72; 2022; 387 ppMSC: Primary 12; 13; 16; 20
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 proofwriting 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 activelearning approaches in the book, including inclass modules designed to facilitate student exploration.
ReadershipUndergraduate students interested in learning abstract algebra.

Table of Contents

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 𝐷₄

5.3. Centralizer

Exercises

Chapter 6. Permutations

6.1. Permutations of the set {1,2,3}

6.2. Group properties of 𝑆_{𝑛}

6.3. Computations in 𝑆_{𝑛}

6.4. Associative law in 𝑆_{𝑛} (and in 𝐷_{𝑛})

Exercises

Chapter 7. Matrices

7.1. Matrix arithmetic

7.2. Matrix group 𝑀(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. Nonassociative 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 𝑈₁₀

10.2. Groups 𝑀(Z ₁₀) and 𝐺(Z ₁₀)

10.3. Group 𝑆(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 𝑔^{𝑘}=𝜖?

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 𝑈₁₃

13.3. Matching Z ₁₂ and 𝑈₁₃

13.4. Taking positive and negative powers of 𝑔

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. Onetoone 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 ⟨𝑔⟩: Elements match up

16.2. Groups Z ₁₂ and ⟨𝑔⟩: 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 𝐻=5Z in Z revisited

Exercises

Chapter 22. Quotient Group Examples

22.1. Quotient group 𝑈₁₃/𝐻 revisited

22.2. Quotient group 𝑈₃₇/𝐻

22.3. Quotient group 𝐺/𝐻 (generalization)

Exercises

Chapter 23. Quotient Group Proofs

23.1. Sample quotient group proofs

23.2. Collapsing 𝐺 into 𝐺/𝐻

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 𝐹[𝑥]

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 𝐹[𝑥]

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[𝑥]/⟨𝑥²⟩

32.4. First Isomorphism Theorem for rings

Big picture stuff

Exercises

Chapter 33. Quotient Ring Z ₇[𝑥]/⟨𝑥²1⟩

33.1. Division algorithm revisited

33.2. Another way to reduce in Z ₇[𝑥]/⟨𝑥²1⟩

33.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is not a field

33.4. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field

Big picture stuff

Exercises

Chapter 34. Quotient Ring R [𝑥]/⟨𝑥²+1⟩

34.1. Reducing elements in R [𝑥]/⟨𝑥²+1⟩

34.2. Field of complex numbers

34.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field revisited

Exercises

Chapter 35. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ Is/Isn’t a Field, Part I

35.1. Translate from 𝐹[𝑥] to Z

35.2. Translate (back) from Z to 𝐹[𝑥]

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 ⟨𝑔(𝑥)⟩

Big picture stuff

Exercises

Chapter 37. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ 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 𝐷₄

Appendix C. Symbols and Notations

Appendix D. Essential Theorems

Index of Terms

Back Cover


Additional Material

Reviews

'A Friendly Introduction to Abstract Algebra' seems unusually pedagogically strong to me, and I think it would be terrific for any course that teaches introductory abstract algebra in an active manner. It might also be appropriate for an applicable algebra course. I think students would love it and could learn very well from it. Even if it is not a primary textbook, it could be a very useful supplement. It's chockfull of examples and concrete calculations and spends an extraordinary amount of time teaching students how to do proofs and come up with the idea for proofs.
Steven Strogatz, Cornell University


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 proofwriting 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 activelearning approaches in the book, including inclass modules designed to facilitate student exploration.
Undergraduate students interested in learning abstract algebra.

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 𝐷₄

5.3. Centralizer

Exercises

Chapter 6. Permutations

6.1. Permutations of the set {1,2,3}

6.2. Group properties of 𝑆_{𝑛}

6.3. Computations in 𝑆_{𝑛}

6.4. Associative law in 𝑆_{𝑛} (and in 𝐷_{𝑛})

Exercises

Chapter 7. Matrices

7.1. Matrix arithmetic

7.2. Matrix group 𝑀(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. Nonassociative 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 𝑈₁₀

10.2. Groups 𝑀(Z ₁₀) and 𝐺(Z ₁₀)

10.3. Group 𝑆(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 𝑔^{𝑘}=𝜖?

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 𝑈₁₃

13.3. Matching Z ₁₂ and 𝑈₁₃

13.4. Taking positive and negative powers of 𝑔

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. Onetoone 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 ⟨𝑔⟩: Elements match up

16.2. Groups Z ₁₂ and ⟨𝑔⟩: 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 𝐻=5Z in Z revisited

Exercises

Chapter 22. Quotient Group Examples

22.1. Quotient group 𝑈₁₃/𝐻 revisited

22.2. Quotient group 𝑈₃₇/𝐻

22.3. Quotient group 𝐺/𝐻 (generalization)

Exercises

Chapter 23. Quotient Group Proofs

23.1. Sample quotient group proofs

23.2. Collapsing 𝐺 into 𝐺/𝐻

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 𝐹[𝑥]

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 𝐹[𝑥]

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[𝑥]/⟨𝑥²⟩

32.4. First Isomorphism Theorem for rings

Big picture stuff

Exercises

Chapter 33. Quotient Ring Z ₇[𝑥]/⟨𝑥²1⟩

33.1. Division algorithm revisited

33.2. Another way to reduce in Z ₇[𝑥]/⟨𝑥²1⟩

33.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is not a field

33.4. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field

Big picture stuff

Exercises

Chapter 34. Quotient Ring R [𝑥]/⟨𝑥²+1⟩

34.1. Reducing elements in R [𝑥]/⟨𝑥²+1⟩

34.2. Field of complex numbers

34.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field revisited

Exercises

Chapter 35. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ Is/Isn’t a Field, Part I

35.1. Translate from 𝐹[𝑥] to Z

35.2. Translate (back) from Z to 𝐹[𝑥]

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 ⟨𝑔(𝑥)⟩

Big picture stuff

Exercises

Chapter 37. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ 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 𝐷₄

Appendix C. Symbols and Notations

Appendix D. Essential Theorems

Index of Terms

Back Cover

'A Friendly Introduction to Abstract Algebra' seems unusually pedagogically strong to me, and I think it would be terrific for any course that teaches introductory abstract algebra in an active manner. It might also be appropriate for an applicable algebra course. I think students would love it and could learn very well from it. Even if it is not a primary textbook, it could be a very useful supplement. It's chockfull of examples and concrete calculations and spends an extraordinary amount of time teaching students how to do proofs and come up with the idea for proofs.
Steven Strogatz, Cornell University