- Merely said, the dummit and foote solutions manual abstract algebra 3rd edition is universally compatible bearing in mind any devices to read. Dummit and foote solutions manual OR Pierce, Benjamin A. Genetics, A Conceptual Approach (7th ed). Freeman (Looseleaf) + Solutions Manual + SaplingPlus access for.
- Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan F elix Abril 12, 2017 Section 8.1 Exercise 2 (a). Show that 13 is relatively prime to 20 and determine the inverse of 13 mod 20. We use the wonderful Euclidean algorithm.
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Access Free Dummit And Foote Abstract Algebra Solution Manual thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. Elements of Abstract Algebra A Course in Algebra Abstract Algebra Using various examples this monograph shows that algebra is one of the most.
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Chapter 0: Preliminaries
- §0.1: Basics
(#1)(#2)(#3)(#4)(#5)(#6)(#7) - §0.2: Properties of the Integers
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) - §0.3: Z/(n) – The Integers Modulo n
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16)
Chapter 1: Introduction to Groups
- §1.1: Basic Axioms and Examples
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17) (#18) (#19)(#20)(#21)(#22)(#23)(#24)(#25)(#26)(#27)(#28)(#29)(#30)(#31)(#32)(#33)(#34)(#35)(#36) - §1.2: Dihedral Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §1.3: Symmetric Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §1.4: Matrix Groups
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11) - §1.5: The Quaternion Group
(#1)(#2)(#3) - §1.6: Homomorphisms and Isomorphisms
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19)(#20)(#21)(#22)(#23)(#24)(#25)(#26) - §1.7: Group Actions
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11) (#12) (#13)(#14)(#15)(#16)(#17)(#18) (#19) (#20) (#21) (#22) (#23)
Chapter 2: Subgroups
- §2.1: Definition and Examples
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17) - §2.2: Centralizers and Normalizers, Stabilizers and Kernels
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14) - §2.3: Cyclic Groups and Cyclic Subgroups
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19)(#20)(#21)(#22)(#23)(#24)(#25)(#26) - §2.4: Subgroups Generated by Subsets of a Group
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §2.5: The Lattice of Subgroups of a Group
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20)
Chapter 3: Quotient Groups and Homomorphisms
- §3.1: Definitions and Examples
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) - §3.2: More on Cosets and Lagrange’s Theorem
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19)(#20)(#21)(#22)(#23) - §3.3: The Isomorphism Theorems
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10) - §3.4: Composition Series and the Hölder Program
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) - §3.5: Transpositions and the Alternating Group
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)
Chapter 4: Group Actions
Dummit Foote Solutions Manual Pdf
- §4.1: Group Actions and Permutation Representations
(#1)(#2)(#3)(#4) (#5) (#6)(#7)(#8)(#9)(#10) - §4.2: Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem
(#1)(#2)(#3) (#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13) (#14) - §4.3: Groups Acting on Themselves by Conjugation – The Class Equation
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) - §4.4: Automorphisms
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §4.5: The Sylow Theorems
(#1)(#2)(#3)(#4)(#5)(#6)(#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54) (#55) (#56) - §4.6: The Simplicity of A_n
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
Chapter 5: Direct and Semidirect Products and Abelian Groups
- §5.1: Direct Products
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17) (#18) - §5.2: The Fundamental Theorem of Finitely Generated Abelian Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) - §5.3: Table of Groups of Small Order
(#1) - §5.4: Recognizing Direct Products
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §5.5: Semidirect Products
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25)
Chapter 6: Further Topics in Group Theory
- §6.1: p-Groups, Nilpotent Groups, and Solvable Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) - §6.2: Applications in Groups of Medium Order
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) - §6.3: A Word on Free Groups
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Chapter 7: Introduction to Rings
- §7.1: Basic Definitions and Examples
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19)(#20)(#21)(#22)(#23)(#24)(#25)(#26)(#27)(#28)(#29)(#30) - §7.2: Examples: Polynomial Rings, Matrix Rings, and Group Rings
(#1)(#2)(#3)(#4)(#5)(#6)(#7) (#8) (#9)(#10)(#11)(#12)(#13) - §7.3: Ring Homomorphisms and Quotient Rings
(#1)(#2)(#3)(#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14)(#15)(#16)(#17)(#18)(#19)(#20)(#21)(#22)(#23)(#24)(#25)(#26)(#27)(#28)(#29)(#30)(#31)(#32)(#33)(#34)(#35)(#36)(#37) - §7.4: Properties of Ideals
(#1)(#2) (#3) (#4)(#5)(#6)(#7)(#8)(#9)(#10)(#11)(#12)(#13)(#14) (#15) (#16)(#17)(#18)(#19)(#20)(#21) (#22) (#23)(#24)(#25)(#26)(#27)(#28)(#29)(#30)(#31)(#32)(#33)(#34)(#35)(#36) (#37) (#38) (#39) (#40) (#41) - §7.5: Rings of Fractions
(#1) (#2) (#3) (#4) (#5) (#6) - §7.6: The Chinese Remainder Theorem
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)
Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
- §8.1: Euclidean Domains
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) - §8.2: Principal Ideal Domains
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §8.3: Unique Factorization Domains
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)
Chapter 9: Polynomial Rings
- §9.1: Definitions and Basic Properties
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §9.2: Polynomial Rings over Fields I
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) - §9.3: Polynomial Rings that are Unique Factorization Domains
(#1) (#2) (#3) (#4) (#5) - §9.4: Irreducibility Criteria
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §9.5: Polynomial Rings over Fields II
(#1) (#2) (#3) (#4) (#5) (#6) (#7) - §9.6: Polynomials in Several Variables over a Field and Gröbner Bases
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45)
Chapter 10: Introduction to Module Theory
- §10.1: Basic Definitions and Examples
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) - §10.2: Quotient Modules and Module Homomorphisms
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §10.3: Generation of Modules, Direct Sums, and Free Modules
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) - §10.4: Tensor Products of Modules
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) - §10.5: Exact Sequences – Projective, Injective, and Flat Modules
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28)
Chapter 11: Vector Spaces
- §11.1: Definitions and Basic Theory
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §11.2: The Matrix of a Linear Transformation
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) - §11.3: Dual Vector Spaces
(#1) (#2) (#3) (#4) (#5) - §11.4: Determinants
(#1) (#2) (#3) (#4) (#5) (#6) - §11.5: Tensor Algebras, Symmetric and Exterior Algebras
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14)
Dummit And Foote Solution Manual
Chapter 12: Modules over Principal Ideal Domains
- §12.1: The Basic Theory
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) - §12.2: The Rational Canonical Form
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) - §12.3: The Jordan Canonical Form
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54) (#55) (#56) (#57) (#58) (#59)
Chapter 13: Field Theory
- §13.1: Basic Theory of Field Extensions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §13.2: Algebraic Extensions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) - §13.3: Classical Straightedge and Compass Constructions
(#1) (#2) (#3) (#4) (#5) - §13.4: Splitting Fields and Algebraic Closures
(#1) (#2) (#3) (#4) (#5) (#6) - §13.5: Separable and Inseparabel Extensions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) - §13.6: Cyclotomic Polynomials and Extensions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17)
Chapter 14: Galois Theory
- §14.1: Basic Definitions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §14.2: The Fundamental Theorem of Galois Theory
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) - §14.3: Finite Fields
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) - §14.4: Composite Extensions and Simple Extensions
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §14.5: Cyclotomic Extensions and Abelian Extensions over mathbb{Q}
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §14.6: Galois Groups of Polynomials
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) - §14.7: Solvable and Radical Extensions: Insolvability of the Quintic
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) - §14.8: Computation of Galois Groups over mathbb{Q}
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §14.9: Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19)
Chapter 15: Commutative Rings and Algebraic Geometry
- §15.1: Noetherian Rings and Affine Algebraic Sets
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) - §15.2: Radicals and Affine Varieties
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) (#41) (#42) (#43) (#44) (#45) (#46) (#47) (#48) (#49) (#50) (#51) (#52) (#53) (#54) - §15.3: Integral Extensions and Hilbert’s Nullstellensatz
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) - §15.4: Localization
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) - §15.5: The Prime Spectrum of a Ring
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31)
Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
- §16.1: Artinian Rings
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §16.2: Discrete Valuation Rings
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) - §16.3: Dedekind Domains
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25)
Chapter 17: Introduction to Homological Algebra and Group Cohomology
- §17.1: Introduction to Homological Algebra – Ext and Tor
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) - §17.2: The Cohomology of Groups
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) - §17.3: Crossed Homomorphisms and H^1(G,A)
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) - §17.4: Group Extensions, Factor Sets, and H^2(G,A)
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
Chapter 18: Representation Theory and Character Theory
- §18.1: Linear Actions and Modules over Group Rings
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) - §18.2: Wedderburn’s Theorem and Some Consequences
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §18.3: Character Theory and the Orthogonality Relations
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Chapter 19: Examples and Applications of Character Theory
- §19.1: Characters of Groups of Small Order
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