M A T H  5 8 1 / 4 2 9 : (A D V A N C E D )  G R O U P  T H E O R Y

Instructor & Office Hours 

Manish M. Patnaik
CAB 577
phone: 780 492 0570
email: lastname at ualberta dot ca

Office Hours: (in CAB 577)
   Wednesdays 1-2pm

Course Information

The course meets Tuesday & Thursday  2-3:20 PM in CAB 269
Please find the syllabus here

Late homework is not accepted. The grading scheme is as follows:

Math 429:

Midterm Exam (30%)
Homework (20%)
Final (50%)

Math 581:

Midterm Exam (25 %)
Homework (15 %)
Expository Paper (10 %)
Final (50%)

Expository Paper Information

Possible topics:

1. Subgroups of free groups are free (Schreier's theorem)
2. Group cohomology (past H^1, H^2); or an indepth study of central extensions
3. Infinite Galois theory, pro-finite completions, etc.
4. Artin-Brauer theory on induced representations
5. Finite-dimensional representations of compact groups
6. Coxeter groups, BN pairs, and the Bruhat Decomposition
7. Milnor's group K_2 and central extensions of matrix groups
8. SL(2, R), Fundamental domains, Siegel sets
9. Infinite dimensional representations of finite groups (monstrous moonshine)
10. Burnside's p^aq^b theorem
11. Representations of the Symmetric group

Exam Information

Midterm Exam: Thursday October 23 at 2PM (in classroom)

The midterm exam will cover material up to (and including) the material on solvable groups (Oct. 16 lecture). It will not cover any material pertaining to group representations, free groups, or the Schur-Zassenhaus theorem. The exam will be 5-6 questions (closed book). You can use any result from class or the homework so long as it is not exactly the question being asked.

Here is a link to the midterm from Fall 2012 (topics will be similar, but not identical).

Here are the solutions for this year's exam.

Final Exam: Wednesday December 10 from 2-5pm (CAB 269)

The final exam will be cover material from the entire course, but will focus on the material on representation theory and galois theory. There will be 6-8 questions on the exam.

Here is a copy of a previous exam (note: you will not be responsible for separable extensions)

Course Notes and Schedule

Lecture date & title Comments
1. Th Sep. 4: Introduction: Groups, subgroups, homomorphisms Subgroups: 2.4, 2.5
Symmetric groups: 1.3
Homomorphisms: 1.6
2. Tu Sep. 9: Direct products, semi-direct products Direct Products: 5.1, 5.4
Semi-direct Products: 5.5
3. Th Sep. 11: Quotients, group actions Cosets and Quotients: 3.1, 3.2
Isomorphism theorems: 3.3
Group actions & Class Equation: 4.1, 4.2, 4.3
Groups of order p^2: 4.3 (Theorem 8, Corollary 9)
4. Tu Sep. 16: Finite abelian p-groups, Free groups Cauchy's thorem: Proposition 21, p .102
p-groups: 6.1
5. Th Sep. 18: Free groups II Free groups: 6.3
Initial, terminal objects in categories: Appendix II
HWK 1 due
6. Tu Sep. 23: Sylow Theorems 1 Sylow theorems: 4.5
7. Th Sep. 25: Sylow Theorems 2 Sylow theorems: 4.5, Nilpotent groups
8. Tu Sep. 30: NO CLASS
9. Th Oct. 2: classification of groups of low order Guest Lecture by Dr. Anna Puskas
10. Tu Oct. 7: Schur-Zassenhaus Theorem, pt. 1 See notes here
11. Th Oct. 9: Schur-Zassenhaus Theorem, pt. 2 See notes here
12. Tu Oct. 14: Normal Series, Composition series Composition series: 3.4
HWK 2 due
Math 581 write up-projects must be approved (note: deadline extended from Oct. 7 as on Syllabus)
13. Th Oct. 16: Jordan Holder theorem, Solvable groups Composition series, Solvable groups: 3.4, 6.1
14. Tu Oct. 21: Intro to Group representations 18.1
15. Th Oct. 23: Midterm Exam (30 %) MIDTERM EXAM
16. Tu Oct. 28: Schur's Lemma, Characters and orthogonality 18.1, 18.3
17. Th Oct. 30: Characters an orthogonality, tensor products 18.3
18. Tu Nov. 4: Orthogonality ctd, Induction (first part) 18.3, 19.3
Project Outline Due (MATH 581)
19. Th Nov. 6:Induced (second part), Frobenius Reciprocity 19.3
20. Th Nov. 13: Frobenius reciprocity, examples 19.3
21. Tu Nov. 18:Review of field theory, splitting fields, normal extensions 13.1, 13.2, 13.4
Hwk 3 due
22. Th Nov. 23:Normal extensions, primitive element theorem, galois extensions 13.4, 14.1
23. Tu Nov 25:Proof of main theorem of galois theory 14.2
24. Th Nov 27:examples 14.2, 14.5, 14.6
25. Tu Dec 2: Hwk 4 due, Math 581 projects due

Homework


Assignment # Due Date Comments
Homework 1, pdf Thursday Sep. 18

Homework 2, pdf Thursday Oct. 9 updated version with question 6.3, #14 written out
Homework 3, pdf Tuesday Nov. 18
Homework 4, pdf Tuesday Dec. 2