# Math 322: Group Theory

Fall Term 2017
Lior Silberman

## General Information

• Office: MATX 1112, 604-827-3031
• Email: "lior" (at) Math.UBC.CA (please include the course number in the subject line, if applicable)
• Office hours: by appointment

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 and also covers the material of 422 to an extent. The gentler and less-terse alternative is the book [2]. 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 GĂ¶ttingen University Library.

## Midterm Exam

• The midterm exam will take place during class on Tuesday, October 17th. The midterm will include calculational and abstract problems, and also one problem asking for the statement of a definition from class.
• Examinable material includes everything up to the isomorphism theorems and the simplicity of A_n, and all problem sets up to PS5.
• For comparison here are the midterms from Fall 2014 (grade statistics) and Fall 2015 (grade statistics).
• For the calculational part of the midterm, see Sections 1,2 of the practice problem document.
• Here are the 2017 exam and its solution.

## Homework

• There is both online homework and written homework (problem sets).
• The problem sets below are accessible to all; solutions are stored on a secure website; registered students can access them after first logging on to Connect.
1. Problem Set 1, due 14/9/2017. Solutions.
2. Problem Set 2, due 21/9/2017 (typo in 1(a) fixed). Solutions.
3. Problem Set 3, due 28/9/2017. Solutions.
4. Problem Set 4, due 5/10/2017. Solutions.
5. Problem Set 5, due 12/10/2017. Solutions.
6. Problem Set 6, due 26/10/2017 (citation to Prop. 176 fixed).
7. Solutions.
8. Problem Set 7, due 2/11/2017. Solutions.
9. Problem Set 8, due 9/11/2017. Solutions.
10. Problem Set 9, deferred to 21/11/2017 (typo in 5(b) fixed). Solutions.
11. Problem Set 10, due 28/11/2017 (P2 clarified, extra credit problem added).. Solutions.
12. Problem Set 11, not for submission. 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

1 Th 7/9 Introduction
The Integers

§0.2
Putnam Sessions
T 12/9 Modular arithmetic §§0.3,0.1 Relations
Th 14/9 (continued)   PS1 due
2 T 19/9 Permutations §1.3
Th 21/9 (continued)   PS2 due
3 T 26/9 Groups and homomorphisms §§1.1,1.2,1.5,2.1 Concepts to review
Th 28/9 Subgroups, Cyclic groups   PS3 due
4 T 3/10 Cosets and Lagrange's Theorem §3.2
Th 5/10 Normal Subgroups
Quotient groups
§3.3 PS4 due
5 T 10/10 Isomorphism Theorems
Simplicity of A_n
§3.3
§4.6
Feedback form
6 Th 12/10 Group actions §1.7, §§4.1-4.2 PS5 due
T 17/10 Midterm Exam
7 Th 19/10 Conjugation §4.3 Zagier's Trick
T 24/10 Orbits, stabilizers   Examples
8 Th 26/10 p-groups N4.1 Groups of order p^3
PS6 due
T 31/10 pq-groups N4.2
9 Th 2/11 (continued) N4.2 PS7 due
T 7/11 Sylow's Theorems §4.5
10 Th 9/11 Applications   PS8 due
T 14/11 Groups of medium order §6.2
Th 16/11 Finite Abelian Groups §6.1, §5.2
11 T 21/11 Review of PS9 §6.1 PS9 due
Th 23/11 Nilpotent groups §6.1
12 T 28/11 Solvable groups §6.1 PS10 due
Th 30/11 Review
T 5/12 Final exam: 8:30-11:30 at BUCH A203

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