|
|
|
|
|
|
Math 113 Course Outline |
|
|
|
|
|
|
Summer 2005, MTWT 4:00PM - 6:00PM, 6 Evans |
|
|
|
|
|
|
Instructor: Benjamin Johnson |
|
|
|
|
|
|
|
|
|
Group Work |
|
0. Sets and Relations |
| Week
1 |
|
Lecture Topics |
|
Homework |
|
| 1 |
20-Jun |
Mon |
Course Intro, 0 |
1.1; 1, 5, 9, 15, 17, 35, 43, 50, 57, 63, 64 |
|
|
I. GROUPS AND SUBGROUPS |
| 2 |
21-Jun |
Tues |
1, 2 |
1.2; 1, 3, 7, 15,
21. 1.3; 1, 3, 9, 15, 21, 27, 37, 43,
61 |
|
|
1. Introduction and Examples |
| 3 |
22-Jun |
Wed |
3,4 |
|
0,1,2 |
|
2. Binary Operations |
| 4 |
23-Jun |
Thurs |
5, Discussion |
|
Quiz |
|
3. Isomorphism of Binary Structures |
|
|
|
|
|
|
4. Groups |
| Week
2 |
|
|
|
|
5. Subgroups |
| 5 |
27-Jun |
Mon |
6,7 |
|
3,4,5 |
|
6. Cyclic Groups |
| 6 |
28-Jun |
Tues |
8, Discussion |
|
Quiz |
|
7. Generating Sets, and Cayley Digraphs |
| 7 |
29-Jun |
Wed |
8,9 |
|
6,7,8 |
|
| 8 |
30-Jun |
Thurs |
10, Discussion |
|
Quiz |
|
II. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS |
|
|
|
|
|
|
8. Groups of Permutations |
| Week
3 |
|
|
|
|
9. Orbits, Cycles, and the Alternating Groups |
| |
4-Jul |
Mon |
HOLIDAY |
|
|
|
10. Cosets and the Theorem of Lagrange |
| 9 |
5-Jul |
Tues |
11, Discussion |
|
8,9,Quiz |
|
11. Direct Products and Finitely Generated Abelian Groups |
| 10 |
6-Jul |
Wed |
13, Review |
|
10,11 |
|
| 11 |
7-Jul |
Thurs |
Midterm 1 (0-11) |
|
|
|
III. HOMOMORPHISMS AND FACTOR GROUPS |
|
|
|
|
|
|
13. Homomorphisms |
| Week
4 |
|
|
|
|
14. Factor Groups |
| 12 |
11-Jul |
Mon |
14,15 |
|
13 |
|
15. Factor-Group Computations and Simple Groups |
| 13 |
12-Jul |
Tues |
15, Discussion |
|
Quiz |
|
16. Group Action on a Set |
| 14 |
13-Jul |
Wed |
16,17 |
|
14,15 |
|
17. Applicationd of G-Sets to Counting |
| 15 |
14-Jul |
Thurs |
18, Discussion |
|
Quiz |
|
|
|
|
|
|
|
IV. RINGS AND FIELDS |
| Week 5 |
|
|
|
|
18. Rings and Fields |
| 16 |
18-Jul |
Mon |
18,19 |
|
16,17,18 |
|
19. Integral Domains |
| 17 |
19-Jul |
Tues |
20, Discussion |
|
Quiz |
|
20. Fermat's and Euler's Theorems |
| 18 |
20-Jul |
Wed |
21,22 |
|
18, 19, 20 |
|
21. The Field of Quotients of an Integral Domain |
| 19 |
21-Jul |
Thurs |
23, Discussion |
|
Quiz |
|
22. Rings of Polynomials |
|
|
|
|
|
|
23. Factorization of Polynomials over a Field |
| Week
6 |
|
|
|
|
24. Non-Commutative Examples |
| 20 |
25-Jul |
Mon |
24, Review |
|
21, 22, 23 |
|
25. Ordered Rings and Fields |
| 21 |
26-Jul |
Tues |
Midterm 2 (13-23) |
|
|
|
| 22 |
27-Jul |
Wed |
25,26 |
|
24 |
|
V. IDEALS AND FACTOR RINGS |
| 23 |
28-Jul |
Thurs |
27, Discussion |
|
Quiz |
|
26. Homomorphisms and Factor Rings |
|
|
|
|
|
|
27. Prime and Maximal Ideals |
| Week
7 |
|
|
|
|
| 24 |
1-Aug |
Mon |
29 |
|
25, 26,27 |
|
VI. EXTENSION FIELDS |
| 25 |
2-Aug |
Tues |
30, Discussion |
|
Quiz |
|
29. Introduction to Extension Fields |
| 26 |
3-Aug |
Wed |
31 |
|
29,30 |
|
30. Vector Spaces |
| 27 |
4-Aug |
Thurs |
33, Discussion |
|
Quiz |
|
31. Algebraic Extensions |
|
|
|
|
|
|
33. Finite Fields |
| Week
8 |
|
|
|
|
| 28 |
8-Aug |
Mon |
34, 35 |
|
31, 33 |
|
VII. ADVANCED GROUP THEORY |
| 29 |
9-Aug |
Tues |
36, Discussion |
|
Quiz |
|
34. Isomorphism Theorems |
| 30 |
10-Aug |
Wed |
37, Review |
|
34,35,36 |
|
35. Series of Groups |
| 31 |
11-Aug |
Thurs |
Final Exam (Cumulative) |
|
|
|
36. Sylow Theorems |
|
|
|
|
|
37. Applications of the Sylow Theory |
|
|
|
|
|
| Note: This outline is tentative and subject to
change. |
|
|
|
|
|
|
|
|