Syllabus for Part I: Groups
Nov 5: Lecture 1. Chapter 1.1–1.2(p.25)
Introduction to groups.Nov 8: Exercises
Suggested exercises: Chapter 1.1: 1,5,7,9,11,20,22,28,34Chapter 1.2: 10
Nov 12: Lecture 2. Chapter 1.2–1.4
Dihedral groups, symmetric groups, matrix groups.Nov 15: Exercises
Suggested exercises: Chapter 1.2: 3,7,15Chapter 1.3: 1,3,6,9,11,15
Chapter 1.4: 2,7
Nov 21: Lecture 3. Chapter 1.6,2.1–2.3, 2.5
Homomorphisms and isomorphisms, subgroups, centralizers, normalizers, cyclic groups.Nov 22: Exercises
Suggested exercises: Chapter 1.6: 2,4,9,17Chapter 2.1: 1,3,6,8
Chapter 2.2: 10,14
Chapter 2.3: 1,3
Nov 26: Lecture 4. Chapter 2.3–2.4, 3.1
Continuation on cyclic groups, generation. Cosets, normal subgroups and quotient groups.Nov 29: Exercises
Suggested exercises: Chapter 2.3: 23Chapter 2.4: 3,8,13
Chapter 3.1: 1,3,8,10,21
Dec 4: Lecture 5. Chapter 3.2, 1.7, 2.2 (p.51–52)
Group actions and Lagrange's theoremDec 5: Exercises
Suggested exercises: Chapter 3.1: 24,29,33,36Chapter 2.2: 7,13
Chapter 3.2: 1,2,16
Dec 11: Lecture 6. Chapter 3.3, 4.2, 4.3
Isomorphism theorems, orbits and stabilizers, the class equationDec 12: Exercises
Suggested Exercises: Chapter 3.3: 1,3Chapter 4.2: 2,8
Chapter 4.3: 2,3,5,6,10,13,25,30
Dec 19: Lecture 7. Chapter 4.4–4.5
The Sylow theorems and applications.Syllabus for Part II: Rings
Jan 20: Lecture 8. Chapter 7.1–7.2, 9.1
Rings, integral domains, fields. ExamplesJan 24: Exercises
Suggested exercises: Chapter 4.5: 6,7,13,14,16,18,23Chapter 7.1: 5,6,7,8,11,16
Chapter 7.2: 1,3,5
Jan 27: Lecture 9. Chapter 7.3–7.4
Homomorphisms, ideals, quotient rings, maximal ideals and prime ideals.Jan 30: Exercises
Suggested exercises: Chapter 7.3: 4,6,8,13,17,19,29Chapter 7.4: 5,7,8,10,15,17
Feb 3: Lecture 10. Chapter 7.5–7.6
Rings of fractions. The Chinese Remainder Theorem.Feb 6: Exercises
Suggested exercises: Chapter 7.5: 2,3Chapter 7.6: 5,6,7
Feb 10: Lecture 11. Chapter 8.1–8.3
Euclidean domains, principal ideal domains, unique factorization domains.Feb 13: Exercises
Suggested exercises: Chapter 8.1: 5,6,7,9Chapter 8.2: 5
Chapter 8.3: 5,10
Chapter 9.2: 3,9