1Introduction to Groups
179Solved Example:: Practice Problem:
21.1 Definition of a Group
1807.5 Properties of Semidirect Products
31.2 Examples of Groups
1811. Subgroup Structure:
41.3 Properties of Groups
1822. Order of the Semidirect Product:
5Solved Examples and Practice Problems with Solutions:
1833. Embedding of Subgroups:
6Practice Problems:: Here are the solutions to the practice problems:
1844. Automorphism Preservation:
71.4 Subgroups
1855. Direct Product as a Special Case:
8Properties of Subgroups:
1866. Isomorphism Criteria:
9Solved Examples and Practice Problems with Solutions:: Practice Problems:
187Solved Example:: Practice Problem:
101.5 Group Homomorphisms
1887.6 Applications of Direct and Semidirect Products
11Properties of Group Homomorphisms:
1891. Group Theory:
12Solved Examples and Practice Problems with Solutions:
1902. Symmetry Groups:
13Practice Problems:: Solutions to Practice Problems:
1913. Representation Theory:
141.6 Isomorphisms
1924. Algebraic Topology:
15Properties of Isomorphisms:
1935. Physics:
16Solved Examples and Practice Problems with Solutions:: Practice Problems:
1946. Computer Science:
17Conclusion
195Solved Example:
18Cyclic Groups
196Practice Problem:
192.1 Definition of Cyclic Groups
197Practice Problem (From Section 7.5):
202.2 Properties of Cyclic Groups: Cyclic groups have several important properties:
198Practice Problem (From Section 7.6):
212.3 Generating Sets
199Conclusion
22Solved Examples and Practice Problems
200Finite Abelian Groups
23Solution to Practice Problem 1:: Solution to Practice Problem 3:
2018.1 Definition of Abelian Groups
242.4 Subgroups of Cyclic Groups
2028.2 Properties of Finite Abelian Groups
252.5 Cyclic Subgroups: Properties of Cyclic Subgroups:
2031. Existence of elements of finite order:
262.6 Applications of Cyclic Groups
2042. Existence of unique elements of a given order:
27Solved Examples and Practice Problems
2053. Cyclic structure:
28Solution to Practice Problem 1:
2064. Subgroup structure:: 5. Isomorphism classes:
29Solution to Practice Problem 2:
2078.3 Fundamental Theorem of Finite Abelian Groups: Theorem (Fundamental Theorem of Finite Abelian Groups):
30Solution to Practice Problem 3:: Examples of cyclic subgroups of G:
2088.4 Cyclic Decomposition: Theorem (Cyclic Decomposition of Finite Abelian Groups):
31Conclusion
2098.5 Applications of Finite Abelian Groups
32Permutation Groups
2101. Coding Theory and Cryptography:
333.1 Definition of Permutations
2112. Signal Processing and Fourier Analysis:
343.2 Cycle Notation: Properties of cycle notation:
2123. Crystallography and Solid-State Physics:
353.3 Permutation Groups
2134. Combinatorics and Enumeration:: 5. Number Theory and Algebraic Geometry:
36Subgroups of Symmetric Groups:
2148.6 Sylow Theorems
37Solved Examples and Practice Problems:
215Theorem (Sylow’s First Theorem):
38Solution to Practice Problem 1:
216Theorem (Sylow’s Second Theorem):
39Solution to Practice Problem 2:: Solution to Practice Problem 3:
217Theorem (Sylow’s Third Theorem):
403.4 Properties of Permutation Groups
2181. Determining the structure of finite groups:
411. Order of the Symmetric Group Sn:
2192. Existence of normal subgroups:
422. Even and Odd Permutations:
2203. Criteria for simple groups:
433. Simplicity of An for n ≥ 5:
2214. Proof of Cauchy’s Theorem:: Here are the solutions to the practice problems:
444. Transitivity and Primitivity:
222Conclusion
455. Cycle Structure:
223Group Presentations
466. Conjugacy Classes:: 7. Subgroup Structure:
2249.1 Definition of Group Presentations
473.5 Cycle Decomposition: Properties of Cycle Decomposition:
2259.2 Free Groups
483.6 Applications of Permutation Groups
2269.3 Generators and Relations
491. Combinatorics:
227Generators:
502. Algebraic Structures:
228Relations:: Here are the solutions to the practice problems:
513. Galois Theory:
2299.4 Tietze Transformations
524. Coding Theory:
230There are two main Tietze transformations:
535. Cryptography:
2311. Adding a new generator and a new relator:
546. Physics:
2322. Applying a substitution:
557. Computer Science:
233Solved Example:: Practice Problem:
568. Computational Biology:
2349.5 Applications of Group Presentations
579. Experimental Design:
2351. Computational Group Theory:
5810. Representation Theory:
2362. Combinatorial Group Theory:
59Solved Examples and Practice Problems:: Solution to Practice Problem 1:
2373. Low-Dimensional Topology:
60Conclusion
2384. Computational Geometry:
61Group Actions
2395. Cryptography:
624.1 Definition of Group Actions: Here’s an example of a group action:
2406. Formal Language Theory:
634.2 Orbits and Stabilizers
2417. Robotics and Kinematics:
64Orbits:: Stabilizers:
242Solved Example:: Practice Problem:
654.3 Burnside’s Lemma
2439.6 Word Problems
66Statement of Burnside’s Lemma:: Practice Problems:
2441. Rewriting Systems:
674.4 Cayley’s Theorem
2452. Automatic Groups:
68Solved Example:: Practice Problem:
2463. Geometric Methods:
694.5 Conjugacy Classes
2474. Decision Procedures:: Solved Example:
70Properties of Conjugacy Classes:: Solved Example:
248Conclusion
714.6 Applications of Group Actions
249Group Extensions
721. Symmetry in Geometry and Crystallography:
25010.1 Definition of Group Extensions
732. Permutation Groups and Combinatorics:
251Solved Example:: Practice Problem:
743. Galois Theory and Field Extensions:
25210.2 Split Extensions
754. Representation Theory:
253Solved Example:: Practice Problem:
765. Algebraic Topology:
25410.3 Central Extensions
776. Differential Geometry:
255Solved Example:
787. Coding Theory and Cryptography:
256Practice Problem 1:: Practice Problem 2:
798. Dynamical Systems and Chaos Theory:
25710.4 Schur Multiplier
80Solved Examples and Practice Problems:: Practice Problems:
258Solved Example:: Practice Problem:
81Conclusion
25910.5 Applications of Group Extensions
82Cosets and Lagrange’s Theorem
2601. Representation Theory:
835.1 Definition of Cosets
2612. Algebraic Topology:
84Solved Example:: Practice Problem:
2623. Galois Theory and Number Theory:
855.2 Properties of Cosets
2634. Gauge Theories and Physics:
861. Coset Multiplication:
2645. Combinatorial Group Theory:
872. Coset Equality:
2656. Cryptography:
883. Lagrange’s Theorem:
2667. Computer Science and Computational Group Theory:
894. Coset Decomposition:
267Solved Example:: Practice Problem:
905. Coset Partitioning:
2681. Representation Theory:
916. Normality and Cosets:
2692. Resolution of Group-Theoretic Problems:
92Solved Example:: Practice Problem:
2703. Algebraic Topology:
935.3 Lagrange’s Theorem
271Solved Example:
94Solved Example:
272Practice Problem:
95Practice Problem:
273Here are the solutions to the practice problems:
96Here are the solutions to all the practice problems in Chapter 5:
274Practice Problem (10.4 Schur Multiplier):: Practice Problem (10.5 Applications of Group Extensions):
97Practice Problem 1:: Practice Problem 2:
275Conclusion
985.4 Applications of Lagrange’s Theorem
276Solvable and Nilpotent Groups
991. Existence of Subgroups:
27711.1 Definition of Solvable Groups
1002. Order of Group Elements:
27811.2 Properties of Solvable Groups
1013. Fermat’s Little Theorem and Euler’s Theorem:
27911.3 Derived Series
1024. Sylow Theorems:
280Solved Examples:: Practice Problems with Solutions:
1035. Cauchy’s Theorem:
28111.4 Nilpotent Groups
1046. Coding Theory:
28211.5 Properties of Nilpotent Groups
1057. Cryptography:
283Solved Examples:: Practice Problems with Solutions:
1068. Representation Theory:
28411.6 Applications of Solvable and Nilpotent Groups
107Solved Example:: Practice Problem:
2851. Representation Theory:
1085.5 Normal Subgroups
2862. Galois Theory:
109Properties of Normal Subgroups:
2873. Lie Theory:
110Solved Example:: Practice Problem:
2884. Algebraic Topology:
1115.6 Quotient Groups
2895. Group Cohomology:
112Properties of Quotient Groups:
2906. Algorithms and Computational Group Theory:
113Solved Example:
2917. Finite Group Theory:
114Practice Problem:
2928. Number Theory:
115Here are the solutions to the practice problems:
293Conclusion
116Practice Problem 1:
294Simple Groups
117Practice Problem 2:: Practice Problem 3:
29512.1 Definition of Simple Groups
118Conclusion
29612.2 Properties of Simple Groups
119Homomorphisms and Isomorphisms
297Solved Examples:: Practice Problems with Solutions:
1206.1 Group Homomorphisms
29812.3 Classification of Finite Simple Groups
121Solved Examples:: Practice Problems:
2991. Cyclic groups of prime order:
1226.2 Properties of Homomorphisms
3002. Alternating groups A_n for n ≥ 5:
1231. Preservation of Identity:
3013. Groups of Lie type:
1243. Kernel and Image:
3024. Sporadic groups:
1254. Homomorphism Theorems:
303Solved Examples:: Practice Problems with Solutions:
126Solved Examples:: Practice Problems:
30412.4 Sporadic Simple Groups
1276.3 Kernel and Image
305Solved Examples:: Practice Problems with Solutions:
128Kernel of a Homomorphism:
30612.5 Applications of Simple Groups
129Properties of the Kernel:
3071. Representation Theory:
130Image of a Homomorphism:
3082. Algebraic Geometry:
131Properties of the Image:
3093. Conformal Field Theory and String Theory:
132Solved Examples:
3104. Moonshine Conjectures:
133Practice Problems:
3115. Error-Correcting Codes:
134Isomorphisms:
3126. Cryptography:
135Properties of Isomorphisms:
3137. Computational Group Theory:
136Solved Examples:
3148. Combinatorics and Graph Theory:
137Practice Problems:: Here are the solutions to the practice problems:
3159. Mathematical Physics:
1386.1 Group Homomorphisms
31610. Pure Mathematics:
139Practice Problem Solutions:
31712.6 Composition Series
1406.2 Properties of Homomorphisms
318Properties of Composition Series:
141Practice Problem Solutions:
319Solved Examples:: Practice Problems with Solutions:
1426.3 Kernel and Image
320Conclusion
143Practice Problem Solutions:
321Representation Theory
144Isomorphisms:: Practice Problem Solutions:
32213.1 Definition of Group Representations: Practice Problems:
1456.4 Isomorphisms
32313.2 Linear Representations: Practice:
146Solved Examples:: Practice Problems:
32413.3 Character Theory
1476.5 First Isomorphism Theorem
325Some key properties of characters:: Practice Problems:
148Consequences and Applications of the First Isomorphism Theorem:: Practice Problems:
32613.4 Induced Representations
1496.6 Applications of Homomorphisms and Isomorphisms
327Practice Problems:
1501. Classifying and Studying Finite Groups:
32813.5 Applications of Representation Theory
1512. Constructing New Groups from Existing Ones:
3291) Analyzing structure and representations of finite groups
1523. Representing Groups by Matrices or Permutations:
3302) Solving algebraic and differential equations
1534. Studying Group Actions and Symmetries:
3313) Quantum mechanics
1545. Coding Theory and Cryptography:
3324) Crystallography and solid-state physics
1556. Fourier Analysis and Signal Processing:
3335) Combinatorics
1567. Simplifying Computations and Proofs:
3346) Fourier analysis: Practice Problems:
1578. Studying Algebraic Structures and Their Relationships:
33513.6 Schur’s Lemma
158Solved Examples:
336Conclusion
159Practice Problems:: Here are the solutions to the practice problems:
337Lie Groups
1606.4 Isomorphisms
33814.1 Definition of Lie Groups
161Practice Problem Solutions:
33914.2 Matrix Lie Groups
1626.5 First Isomorphism Theorem
34014.3 Lie Algebras
163Practice Problem Solutions:
341Some important examples of Lie algebras include:: Solved Examples and Practice Problems:
1646.6 Applications of Homomorphisms and Isomorphisms
34214.4 Exponential Map
165Practice Problem Solutions:
343Solved Example:: Practice Problem:
166Conclusion
34414.5 Applications of Lie Groups
167Direct Products and Semidirect Products
3451. Classical Mechanics and Hamiltonian Systems:
1687.1 Definition of Direct Products
3462. Quantum Mechanics and Particle Physics:
1697.2 Properties of Direct Products
3473. General Relativity and Gravitational Physics:
1701. Order of the Direct Product:
3484. Differential Equations and Symmetry Methods:
1712. Commutativity of the Direct Product:
3495. Signal Processing and Control Theory:
1723. Associativity of the Direct Product:
3506. Representation Theory and Harmonic Analysis:: 7. Geometry and Topology:
1734. Existence of Subgroups Isomorphic to G and H:
35114.6 Compact Lie Groups
1745. Direct Product of Abelian Groups:
352Compact Lie groups have several important properties:
1756. Direct Product with the Trivial Group:: 7. Internal Direct Products:
353Compact Lie groups have numerous applications in various areas of mathematics and physics, including:: Solved Examples and Practice Problems:
1767.3 External Direct Products
354Conclusion
1777.4 Semidirect Products
355Glossary
178Properties of Semidirect Products:
356Index
