Math 322: Group Theory

Fall Term 2014
Lior Silberman

General Information

This is the introductory course in algebra, intended for honours students. Students who wish to buy a single abstract algebra book should buy the book [1], which will serve you for both 322 and 323 this year, and also covers the material of 422 and 423 (an alternative is book [2], which is gentler and less terse). If you want a group-theory specific textbook, the best book in my opinion is Rotman's (reference [3] below). You can download a copy by following the link while on the UBC network. That said, any book titled "Group Theory" (topic-specific) or "algebra" or "abstract algebra" (wide-coverage) is fine.

References

  1. Dummit and Foote, Abstract Algebra
  2. Gallian, Contemporary Abstract Algebra
  3. Rotman, An Introduction to the Theory of Groups, also available from SpringerLink.

During the course, we will study three classical theorems by Sylow. They are, of course, discussed in detail in the textbooks. Sylow's original paper from 1872 (written in French) is available online from the Göttingen University Library.

Midterm Exam

Problem Sets

  1. Problem Set 1, due 11/9/2014. Solutions.
  2. Problem Set 2, due 18/9/2014 (typos corrected, further hints added). Solutions.
  3. Problem Set 3, due 25/9/2014. Solutions.
  4. Problem Set 4, due 2/10/2014 (typos corrected). Solutions.
  5. Problem Set 5, due 9/10/2014 (typos corrected). Solutions.
  6. Problem Set 6 (expanded) now due 23/10/2014. Solutions.
  7. Problem Set 7, due 30/10/2014 (problem 3(c) clarified, 6 starred). Solutions.
  8. Problem Set 8, due 6/11/2014 (notation in 1(b) clarified). Solutions.
  9. Problem Set 9, due 13/11/2014 (notation in 4(c) corrected). Solutions.
  10. Problem Set 10, due 20/11/2014. Solutions.

Lecture-by-Lecture information

Readings are generally from Dummit and Foote (sections marked "N" are in the lecture notes). Those reading Rotman can find the material there

Week Date Material Reading Notes
1 Th 4/9 Introduction
The Integers		  
§0.2		  
T 9/9 Modular arithmetic §§0.3,0.1  
Th 11/9 (continued)   PS1 due
2 T 16/9 Permutations §1.3  
Th 18/9 (continued)   PS2 due
3 T 23/9 Groups and subgroups §§1.1,1.2,1.5,2.1 Concepts to review
Th 25/9 Homomorphisms, Cyclic groups   PS3 due
4 T 30/9 Cosets and Lagrange's Theorem
Normal Subgroups		 §3.2  
Th 2/10 Quotient Groups §3.3 PS4 due
5 T 7/10 Isomorphism Theorems
Simplicity of A_n		 §3.3
§4.6		 Feedback form
6 Th 9/10 Group actions §1.7, §§4.1-4.3 PS5 due
T 14/10 Midterm Exam Midterm Midterm
7 Th 16/10 (continued)   Zagier's Trick
T 21/10 Examples: orbits, stabilizers    
8 Th 23/10 p-groups   PS6 due
T 28/10 pq-groups    
9 Th 30/10 (continued)
Sylow's Theorems		  
§4.5		 PS7 due
T 4/11 (continued) §4.5  
10 Th 6/11 Groups of medium order §6.2  
Th 13/11 (continued)    
11 T 18/11 Finite Abelian groups §6.1  
Th 20/11 Finitely generated abelian groups §5.2  
12 T 25/11 Solvable groups §6.1  
Th 27/11 Solvable groups §6.1  
  M 15/12 Final exam    
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Last modified Friday March 06, 2015