Spring 2012
| Date | Topic | Homework | Turn in | Due date |
| Tuesday 1/17 | Properties of Z and Z/nZ | p.4 # 5, 6; p.8 # 6, 11; p.11 # 9 | ||
| Thursday 1/19 | Introduction to groups: Definitions and Examples. Dihedral groups. | Study the proof of Proposition 2 (p.20) | HW # 1: § 1.1 # 6, 11, 20, 24, 25, 31 § 1.2 # 1 | 1/26 |
| Tuesday 1/24 | Symmetric groups. Quaternion group. Homomorphisms and isomorphisms | § 1.3 # 3, 4, 6, 7, 18, 19 § 1.5 # 1 § 1.6 # 3, 4, 18, 20, 22 | HW # 2 §1.3 # 10, 13, 15 §1.6 # 1, 2, 17, 26 | 2/2 |
| Thursday 1/26 | Subgroups. Group actions. Normalizers and centralizers. | §1.7 # 11, 13, 16, 17, 18, 19 §2.1 # 3, 6, 9, 10a, 12 §2.2 # 5, 6, 11 | ||
| Tuesday 1/31 | Cyclic groups Generators Lattices of subgroups | § 2.3 # 1, 2, 3, 10, 11 § 2.4 # 5, 6 § 2.5 # 9 a, b | ||
| Thursday 2/2 | Cosets. Normality. | HW # 3 § 3.1 # 1, 36, 42 § 3.2 # 4, 8, 12, 18 | 2/9 | |
| Tuesday 2/7 | Quotient groups. The isomorphism Theorems | | ||
| Thursday 2/9 | Simple groups Alternating groups | Show that if G is a simple group, then any homomorphic image of G is either isomorphic to G or of order one. | HW # 4 § 3.3 # 3 § 3.4 # 1 § 3.5 # 3, 4 | 2/16 |
| Tuesday 2/14 | Group actions. Cayley’s theorem | § 4.2 # 1, 2, 14 | ||
| Thursday 2/16 | The class equation | HW # 5 Handout | 2/28 | |
| Tuesday 2/21 | The Sylow Theorems | |||
| Thursday 2/23 | Review | |||
| Tuesday 2/28 | Test # 1 | |||
| Thursday 3/1 | Direct product. | Show a group of order 36 is not simple. (Submit answer by email by Monday 3/5 at 12 noon.) | HW # 6 Handout on Sylow Theorems | 3/8 |
| Tuesday 3/6 | Finitely generated abelian groups. | pp. 156-15 7# 1, 18 a-c pp.165-166 # 1 a, b; 2 a-c; 3 a-c; 4 | ||
| Thursday 3/8 | Solvable groups | Problem # 8 on page 198. Handout on Solvable groups. pp. 173-174 # 1, 2, 4, 7, 10 For the solutions to Problem 3 on Test 1 click here. For a proof of the propositions presented in class click here. | HW # 7 Handout on Solvable groups | 3/22 |
| March 12-16, 2012 | SPRING BREAK | |||
| Tuesday 3/20 | Introduction to Rings | § 7.1 # 1, 2, 3, 4, 7, 9, 11, 15 | ||
| Thursday 3/22 | Ring homomorphisms. Ideals | § 7.3 # 16, 18, 22, 24a, 27, 28 | HW # 8 § 7.1 # 11, 15 § 7.3 # 22, 27, 28 | 3/29 |
| Tuesday 3/27 | Maximal ideals. Principal ideals | § 7.4 # 4, 5, 6, 8, 10, 19 | ||
| Thursday 3/29 | Principal Ideal Domains (PID) | § 8.2 | HW # 9 § 7.4 # 4, 5, 6, 10, 19 | 4/10 |
| Tuesday 4/3 | Polynomial Rings over Fields | § 9.1 § 9.2 § 9.4 | ||
| Thursday 4/5 | Review | |||
| Tuesday 4/10 | Test # 2 | |||
| Thursday 4/12 | Introduction to Modules | § 10.1 | ||
| Tuesday 4/17 | Presentations | HW # 10 §10.1 # 3, 5, 6, 8 §10.2 # 1, 3, 8, 9 | 5/2 | |
| Thursday 4/19 | Module Homomorphisms. Quotient modules | § 10.2 | ||
| Tuesday 4/24 | Generation of Modules. F[x]-modules | § 10.3 | HW # 11 §10.1 # 19 §10.3 # 4, 9, 11 | 5/3 |
| Thursday 4/26 | Direct product of modules. Cyclic modules | § 10.3 | For the module problems from previous prelims click here. | |
| Tuesday 5/1 | Simple modules | § 10.3 | ||
| Thursday 5/3 | Review | |||
| Tuesday 5/8 11 a.m. -1:30 p.m. | Final Exam |