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Chapter 8: p-Groups and the Sylow Theorems
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Chapter 8: p-Groups and the Sylow Theorems
by W. Keith Nicholson
Introduction to Abstract Algebra, Solutions Manual, 4th Edition
Cover
Title Page
Copyright
Chapter 0: Preliminaries
0.1 Proofs
0.2 Sets
0.3 Mappings
0.4 Equivalences
Chapter 1: Integers and Permutations
1.1 Induction
1.2 Divisors and Prime Factorization
1.3 Integers Modulo n
1.4 Permutations
Chapter 2: Groups
2.1 Binary Operations
2.2 Groups
2.3 Subgroups
2.4 Cyclic Groups and the Order of an Element
2.5 Homomorphisms and Isomorphisms
2.6 Cosets and Lagrange's Theorem
2.7 Groups of Motions and Symmetries
2.8 Normal Subgroups
2.9 Factor Groups
2.10 The Isomorphism Theorem
2.11 An Application to Binary Linear Codes
Chapter 3: Rings
3.1 Examples and Basic Properties
3.2 Integral Domains and Fields
3.3 Ideals and Factor Rings
3.4 Homomorphisms
3.5 Ordered Integral Domains
Chapter 4: Polynomials
4.1 Polynomials
4.2 Factorization of Polynomials over a Field
4.3 Factor Rings of Polynomials over a Field
4.4 Partial Fractions
4.5 Symmetric Polynomials
Chapter 5: Factorization in Integral Domains
5.1 Irreducibles and Unique Factorization
5.2 Principal Ideal Domains
Chapter 6: Fields
6.1 Vector Spaces
6.2 Algebraic Extensions
6.3 Splitting Fields
6.4 Finite Fields
6.5 Geometric Constructions
6.7 An Application to Cyclic and BCH Codes
Chapter 7: Modules over Principal Ideal Domains
7.1 Modules
7.2 Modules over a Principal Ideal Domain
Chapter 8: p-Groups and the Sylow Theorems
8.1 Products and Factors
8.2 Cauchy's Theorem
8.3 Group Actions
8.4 The Sylow Theorems
8.5 Semidirect Products
8.6 An Application to Combinatorics
Chapter 9: Series of Subgroups
9.1 The Jordan-Hölder Theorem
9.2 Solvable Groups
9.3 Nilpotent Groups
Chapter 10: Galois Theory
10.1 Galois Groups and Separability
10.2 The Main Theorem of Galois Theory
10.3 Insolvability of Polynomials
10.4 Cyclotomic Polynomials and Wedderburn's Theorem
Chapter 11: Finiteness Conditions for Rings and Modules
11.1 Wedderburn's Theorem
11.2 The Wedderburn-Artin Theorem
Appendices
Appendix A: Complex Numbers
Appendix B: Matrix Arithmetic
Appendix C: Zorn's Lemma
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7.2 Modules over a Principal Ideal Domain
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8.1 Products and Factors
Chapter 8
p-Groups and the Sylow Theorems
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