Math 2233 - Group Theory
(Adapted from the Course Calendar) The main goal of this course is to introduce groups. We will look at the language of groups, as well as some of the theorems related to the structure of groups. Prerequisites: Math 2231, Math 1281.
News (Last Updated April 8, 2008)
Final Exam Information
The final exam will be on April 21, 2008 at 1:00PM in RC005
I have made an Exam Review
Sheet.
|
Below is a summary of what we did in class, plus any relevant
news and/or information.
- April 7, 2008 This was our last day of class (I cancelled
the next class). We did some review for the exam.
- April 4, 2008 Third day of presenations.
- April 2, 2008 Second day of presentations.
- March 31, 2008 First day of presentations.
- March 28, 2008 Class cancelled.
- March 26, 2008 Welcome back from your Easter Break.
This morning we did course evaluations. As well, I talked
about the program to classify finite simple groups.
- March 19, 2008 We characterized groups of order 2p when
p is a prime. I also talked about groups of order 8 and 12, and presented
a list of groups of order 1 through 15. See Section 8.5.
- March 17, 2008 We looked at Dihedral Groups and
groups of order p^2q where p and q are primes.
- March 14, 2008 We proved the thrid Sylow Theorem,
thus finishing Section 8.4. I then started to discuss
how to use the Sylow theorems to say some results about
groups of small order (Section 8.5). The latest HWA was also handed back.
- March 12, 2008 We proved the second Sylow Theorem.
- March 10, 2008 We proved the first Sylow Theorem.
- March 7, 2008 I laid the groundwork to prove
the first Sylow Theorem. On Monday, we will prove the
first Sylow Theorem (See Section 8.4).
- March 5, 2008 I introduced the three Sylow Theorems.
We will prove them over the next week. See Section 8.3 for more.
- March 3, 2008 We proved the Fundamental Theorem
of Finite Abelian Groups.
- Feb. 29, 2008 We started Section 8.2 on the Fundamental
Theorm of Finite Abelian Groups. Today's material will be needed
for Monday's class when we prove the main result of this section.
I also handed back the midterm and gave out the new homework assignment.
- Feb. 27, 2008 The midterm was today.
- Feb. 25, 2008 Today we started our discussion of Chapter 8.
In particular, I went over the material on Section 8.1 on direct products.
- Feb. 15, 2008 HWA 4 was due in class. I also talked about
the Rubik's Cube as a group.
- Feb. 13, 2008 I did some more review of Chapter 7,
touching upon some of the major ideas we have talked about.
- Feb. 11, 2008 Today, I reviewed some of the basic
definitions of group theory by looking at the group S_4. I also
handed back HWA 3.
- Feb. 8, 2008 We finished up Chapter 7 today. You don't need to
know the proofs in Section 7.10, but you should know the
statements of the theorems. HWA 3 was due, and I gave out
a new HWA which is due next week Friday. The first midterm
will cover up to today's material.
- Feb. 6, 2008 I added the Midterm Info Sheet to the web page.
- Feb. 6, 2008 I spent some more time talking about Section 7.9,
and the notation of cycles. In particular, I showed two disjoint cycles
commute, and every cycle can be written as a product of transpositions.
I also defined what it means for a permutation to be odd or even.
- Feb. 4, 2008 I finished up Section 7.8, and started Section 7.9
by introducing the cycle notation for permuations. I also handed
back HWA 2.
- Feb. 1, 2008 Today we discussed the connection between
quotient groups and homomorphisms. In particular, we proved the
First Isomorphism Theorem of Groups. See Section 7.8 for more.
- Jan. 30, 2008 We discussed some more properties of normal
groups, and spent some time discussing examples of quotient
groups. We have finished up 7.6 and started some of 7.7.
HWA 2 was due, and HWA 3 was given out.
- Jan. 28, 2008 I introduced normal subgroups and
discussed their properties. We then discusses how to use
normal subgroups to be the group of cosets. See Section 7.6 for more.
- Jan. 25, 2008 We discussed the problem
of classifying finite groups. We proved that the only groups of order
p are the cyclic groups of order p when p is a prime. We also
listed all the non-isomorphic groups of order 7 or less.
I also introduced the notion of left cosets in order to motivate
the definition of normal subgroups.
- Jan. 23, 2008 We finished Section 7.4 by discussing Cayley's
theorem. I then introduced the notion of a congruence, and the
idea of a right coset from Section 7.5. In the next class,
we will prove Lagrange's theorem.
- Jan. 21, 2008 We had the first of two lectures on
Isomorphisms and Homomorphisms (see Section 7.4). The first HWA was
due today, and the second assignment was given out.
- Jan. 18, 2008 We finished up Section 7.3 on subgroups. In
particular, we discussed the cyclic subgroups and their properties.
- Jan. 16, 2008 I finished Section 7.2 with some
more properties of the order of an element. I then started Section 7.3
on subgroups; I introduced the definition, a criterion to check if a subset
is a subgroup, and introduced the center of a group.
- Jan. 14, 2008 I started Section 7.2. You should know
about the order of an element in a group, and some of its properties.
- Jan. 11, 2008 I finished Section 7.1 by giving
a large number of examples of groups (symmetric groups, dihedral
groups, and groups from rings).
- Jan. 9, 2008 I went over the course handout, and introduced
permutations and the definition of groups.
- Jan. 7, 2008 Class cancelled.
- Dec. 11, 2007
I started working on this web page. Come back to see more!
Instructor:
Adam Van Tuyl
Office: RB 2015
Office Hours: M 2:30-3:30 and W 3:30-4:30
Email: avantuyl AT sleet.lakeheadu.ca
Place and Time:
Class: MWF 9:30-10:30 in RB 1045
Textbook
Abstract Algebra by T. Hungerford
Homework will be given out every Friday,
and will be due the following Friday in class.
Assignments are posted below.
Assignment 1 (Due: Jan. 21)
Section 7.1 -- 2, 4, 8, 15a, 21, 27, 28
Section 7.2 -- 1, 5, 7bc, 10de, 12, 21, 26, 27, 30
Assignment 2 (Due: Jan. 30)
Section 7.3 -- 3a, 4a, 6 (Ex 7.2 #26 may help) 15, 17, 25, 26, 31
Section 7.4 -- 5, 11, 15, 19, 21, 28ab
Assignment 3 (Due: Feb. 8)
Section 7.5 -- 2c, 9, 17, 18
Section 7.6 -- 6, 16, 18a, 21
Section 7.7 -- 5, 9, 15, 18
Assignment 4 (Due: Feb. 15)
Section 7.8 -- 6a, 7, 13, 10, 21a
Section 7.9 -- 1d, 3d, 5c, 10, 13, 17, 25
Section 7.10 -- 10
Assignment 5 (Due: March 10)
Section 8.1 -- 5, 8, 12, 14, 20, 22
Section 8.2 -- 1, 3b, 5c, 12, 14, 16, 19
Assignment 6 (Due: March 26)
Section 8.3 -- 2a, 5b, 7a, 9, 15a, 17
Section 8.4 -- 1c, 5, 7, 10, 13
Assignment 7 (Due: April 4)
Section 8.5 -- 1, 2, 7, 9 (Hint: Ex. 13 of 8.3), 15
All class handouts are available as
PDF files.
Course Information and Project Info
Handout for first day
Guide to Writing Proofs
Handout on writing proofs
Midterm Info Sheet
Handout about the midterm
Final Exam Info Sheet
Handout about the final exam
Your final mark is broken down as:
35% -- Homework
20% -- Midterm
10% -- Presentation
35% -- Final Exam
Jan. 7, 2008
Second semester classes begin
Feb. 18-22, 2008
Reading Break
Feb. 27, 2008
Midterm (subject to change)
March 21-24, 2008
Easter Break (no classes)
April 8, 2008
Second semester classes end
April 11-24, 2008
Final Exams
Lakehead University
LU Math Department
Adam's Home Page
Student Code of Conduct