Solutions Manual — A First Course in Abstract Algebra, 8th Edition — John B. Fraleigh — Latest Update 2025/2026 — (All Chapters Covered I–IX)

A First Course in Abstract
Algebra 8th Edition
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SOLUTIONS
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MANUAL
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John B. Fraleigh
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Comprehensive Solutions Manual for
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Instructors and Students
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© John B. Fraleigh. All rights reserved. Reproduction or distribution without permission is

prohibited.
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©MEDCONNOISSEUR

, Table of Contents

Solution Manual for A First Course in Abstract Algebra (8th Edition)
Author: John B. Fraleigh
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Part I: Groups and Subgroups
Chapter 1: Binary Operations
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Chapter 2: Groups
Chapter 3: Abelian Examples
Chapter 4: Nonabelian Examples
Chapter 5: Subgroups
Chapter 6: Cyclic Groups
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Chapter 7: Generators and Cayley Digraphs

Part II: Structure of Groups
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Chapter 8: Groups of Permutations
Chapter 9: Finitely Generated Abelian Groups
Chapter 10: Cosets and the Theorem of Lagrange
Chapter 11: Plane Isometries
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Part III: Homomorphisms and Factor Groups
Chapter 12: Factor Groups
Chapter 13: Factor Group Computations and Simple Groups
Chapter 14: Group Action on a Set
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Chapter 15: Applications of G-Sets to Counting

Part IV: Advanced Group Theory
Chapter 16: Isomorphism Theorems
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Chapter 17: Sylow Theorems
Chapter 18: Series of Groups
Chapter 19: Free Abelian Groups
Chapter 20: Free Groups
Chapter 21: Group Presentations
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Part V: Rings and Fields
Chapter 22: Rings and Fields
Chapter 23: Integral Domains
Chapter 24: Fermat’s and Euler’s Theorems
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Chapter 25: RSA Encryption




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, Part VI: Constructing Rings and Fields
Chapter 26: The Field of Quotients of an Integral Domain
Chapter 27: Rings of Polynomials
Chapter 28: Factorization of Polynomials over a Field
Chapter 29: Algebraic Coding Theory
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Chapter 30: Homomorphisms and Factor Rings
Chapter 31: Prime and Maximal Ideals
Chapter 32: Noncommutative Examples
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Part VII: Commutative Algebra
Chapter 33: Vector Spaces
Chapter 34: Unique Factorization Domains
Chapter 35: Euclidean Domains
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Chapter 36: Number Theory
Chapter 37: Algebraic Geometry
Chapter 38: Gröbner Bases for Ideals
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Part VIII: Extension Fields
Chapter 39: Introduction to Extension Fields
Chapter 40: Algebraic Extensions
Chapter 41: Geometric Constructions
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Chapter 42: Finite Fields

Part IX: Galois Theory
Chapter 43: Automorphisms of Fields
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Chapter 44: Splitting Fields
Chapter 45: Separable Extensions
Chapter 46: Galois Theory
Chapter 47: Illustrations of Galois Theory
Chapter 48: Cyclotomic Extensions
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Chapter 49: Insolvability of the Quintic
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, 0. Sets and Relations 1




0. Sets and Relations

1. { 3, − 3}
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2. {2, –3}.
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
60, −60}
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4. {2, 3, 4, 5, 6, 7, 8}
5. It is not a well-defined set. (Some may argue that no element of Z+ is large,

because every element exceeds only a finite number of other elements but is exceeded
by an infinite number of other elements. Such people might claim the answer should
be ∅.)
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6. ∅
7. The set is ∅ because 33 = 27 and 43 = 64.
8. { r  r= a
for some a a Z + and some integer n  0}.
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9. It is not a well-defined set.
10. The set containing all numbers that are (positive, negative, or zero) integer
multiples of 1, 1/2, or 1/3.
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11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
12. a. This is a function which is both one-to-one and onto B.
b. This not a subset of A × B, and therefore not a function.
c. It is not a function because there are two pairs with first member 1.
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d. This is a function which is neither one-to-one (6 appears twice in the second
coordinate) nor onto B ( 4 is not in the second coordinate).
e. It is a function. It is not one-to-one because there are two pairs with second member 6.
It is not onto B because there is no pair with second member 2.
f. This is not a function mapping A into B since 3 is not in the first coordinate of any
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ordered pair.
13. Draw the line through P and x, and let y be its point of intersection with the line
segment CD.
14. a.  : 0,1 → 0, 2 where  ( x) = 2x

b.  : 1, 3 → 5, 25 where  ( x) = 2x + 3
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d −c
c.  : a, b → c, d  where  ( x) = c + ( x − a)
b−a

15. Let  : S → R be defined by  ( x) = tan( (x − 1 )).
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16. a. d.
b.
c.
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