Math 3GR3 -- Abstract Algebra
(Fall 2017)
This course is an introduction to groups and rings with an emphasis on concrete examples.
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Course Information
Instructor: Adam Van Tuyl
Office: Hamilton Hall 419
Office Hours: Monday 9:30-10:20, Wednesday 2:30-3:20
Email: vantuyl@math.mcmaster.ca
Place and Time:
Class C01: Monday, Wednesday, Thursday 1:30-2:20 in Hamilton Hall 305
Tutorial T01: Wednesday 12:30-1:30 in Hamilton Hall 305
TA: James Rooney (rooneyjp@mcmaster.ca)
Rooney will have office hours Tuesdays, 11:30-1:30 in HH 401.
Textbook:
-
Abstract Algebra: Theory and Applications
(2016 Edition)
by Tom Judson. This is a FREE online textbook. Hard copies can also be found in the bookstore. Note that we will be using the 2016 edition. -
Mathematical Writing (Optional)
by Franco Vivaldi. This book is optional, and will help you write proofs. An electronic copy is FREE through the McMaster library.
News (Last Updated: Dec. 20, 2017)
To see a current copy of the lecture notes, see
Below is a summary of what we did in class, plus any relevant news and/or information.
- Dec. 20, 2017 The final marks have been posted. Have a great break!
- Dec. 6, 2017 Our last class. We did some review, and I discussed some future courses that follow up from abstract algebra.
- Dec. 4, 2017 I showed some similarities between Z, the ring of integers, and F[x], a polynomial ring over a field.
- Nov. 30, 2017 We went over Section 17.3 on irreducible polynomials.
- Nov. 29, 2017 Today I proved the division algorithm for polynomial rings (see Section 17.2
- Nov. 27, 2017 Today we started looking at Chapter 17 on polynomial rings. You should know how addition and multiplication is defined. As well, you should know the division algorithm for polynomial rings in one variable.
- Nov. 23, 2017 We finished Chapter 16 by introducing maximal and prime ideals (see Section 16.4).
- Nov. 22, 2017 We proved the First Isomorphism Theorem for rings.
- Nov. 20, 2017 We continued to look at Section 16.3. We learned about ring homomorphisms and some of their properties.
- Nov. 16, 2017 Midterm 2 was today.
- Nov. 15, 2017 We looked at ideals, objects that play the role of normal subgroups in rings.
- Nov. 13, 2017 We learned about two special types of rings: integral domains and fields.
- Nov. 9, 2017 Our first lecture on rings! (see Section 16.1).
- Nov. 8, 2017 We finished up our discussion of groups by going over consequences of the isomorphism theorems.
- Nov. 6, 2017 Today we went over the First Isomorphism Theorem (see Section 11.2).
- Nov. 2, 2017 We started Chapter 11 by introducing homomorphisms between groups and some of their properties. I also introduced the kernel of a homomorphism.
- Nov. 1, 2017 We finished Chapter 10 on normal subgroups. We also talked about simple groups. We discussed the simplicity of the alternating groups.
- Oct. 30, 2017 I introduced normal subgroups, and we learned how to make quotient groups from a group and a normal subgroup.
- Oct. 26, 2017 We discussed internal direct products, and started our discussion of factor groups (Section 10.1).
- Oct. 25, 2017 We finished Section 9.1 by proving Cayley's theorem. We also started on Section 9.2 by defining external direct products.
- Oct. 23, 2017 Today's topic was Chapter 9.1 on isomorphisms.
- Oct. 19, 2017 We finished up Chapter 6 by discussing some consequences of Lagrange's theorem, and we went over Euler and Fermat's Theorems from number theory.
- Oct. 18, 2017 Midterm 1 was today.
- Oct. 16, 2017 Welcome back from the break. Today we started Chapter 6 on cosests. We went over some of the basic properties of cosets and I proved Lagrange's Theorem.
- Oct. 5, 2017 We finished up Chapter 5 with a discussion of the dihedral groups. The first midterm will cover up to the material of today.
- Oct. 4, 2017 Today we talked about odd and even permutations, and I introduced the alternating group (see Section 5.1).
- Oct. 2, 2017 We started Chapter 5 on permutation groups. Become comfortable with the cycle notation that I introduced in class.
- Sept. 28, 2017 We finished up Chapter 4 on cyclic groups by going over the circle group of the complex numbers. I also handed back the first assignment.
- Sept. 27, 2017 I introduced a number of properties about cyclic groups. In paricular, we learned about the order of an element. See Section 4.1 for more.
- Sept. 25, 2017 I finished up Section 3.3 on subgroups, and started a discussion on cyclic groups.
- Sept. 21, 2017 I went over some of the basic properties of groups, and I defined subgroups.
- Sept. 20, 2017 I introduced the formal definition of groups today.
- Sept. 18, 2017 We started Chapter 3 by discussing some examples of groups. The formal definition will be introduced in the next class.
- Sept. 14, 2017 I finished Chapter 2 today by proving the Fundamental Theorem of Arithmetic. I also gave some pointers on how to write proofs.
- Sept. 13, 2017 We went over Section 2.2 on properties of integers. In particular, I explained Euclidean Algorithm. The first quiz was during the lab.
- Sept. 11, 2017 Today we started Chapter 2. I did some examples of induction, discussed the well-ordering principle, and introduced the division algorithm for integers. Just a reminder that the first quiz is in the lab on Wednesday.
- Sept. 7, 2017 We finished our discussion of Chapter 1 by going over the properties of equivalence relations.
- Sept. 6, 2017 Today was our first class. I introduced the course and did a crash course on set theory (see Section 1.2).
- August 31, 2017 There is no lab during the first week of classes.
- August 10, 2017 More information added, included the course outline.
- July 31, 2017 I started to set up the website.
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Homework
There will be five homework assignments. Homework is given out every other Thursday, and will be due, at the beginning of class, the following Thursday. Assignments must conform to the guidelines in the course outline. Your homework will be graded as follows:
- For every assignment, 3 or 4 questions will graded in detail (e.g., you are required to write complete mathematical proofs). These questions will be graded out of 5 pts using the rubric described in the course handout.
- The remaining questions will be graded for completion (1pt each)
Assignment 1 (Due: Sept. 21)
Chapter 1: 2, 14, 21, 25c
Chapter 2: 10, 15b,16, 18
Work through the SAGE tutorials for
Chapter 1 and
Chapter 2.
Assignment 2 (Due: Oct. 5)
Chapter 3: 10, 26, 33, 38, 46, 53
Chapter 4: 1ad. 23ab, 30, 37, 44
Work through the SAGE tutorials for
Chapter 3
and
Chapter 4.
Assignment 3 (Due: Oct. 26)
Chapter 5: 3b, 13 [You can use Chap 6, Ex 14], 17, 26a, 34
Chapter 6: 5be, 14, 18, 19 21 [Hint: use Them 4.13]
Work through the SAGE tutorials for
Chapter 5
and
Chapter 6.
Assignment 4 (Due: Nov. 9)
Chapter 9: 5, 22, 25, 34, 46
Chapter 10: 5, 6, 8, 9, 11 [Hint: use Chap 3, Exercise 54]
Work through the SAGE tutorials for
Chapter 9
and
Chapter 10.
Assignment 5 (Due: Nov. 30)
Chapter 11: 2a, 4, 10, 13, 16
Chapter 16: 5b, 6, 11, 16, 27 [Hint: Binomial Theorem]
Work through the SAGE tutorials for
Chapter 11
and
Chapter 16.
For more information on proofs, the following notes may help:
-
Guide to writing proofs
A handout on writing proofs I wrote for algebra in 2010. -
Some remarks on writing mathematical proofs
A nice short guide to writing mathematics by John M. Lee.
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Quizzes
There will be five quizzes. Each quiz will be about 20 minutes, and will be held during tutorial. The dates of the quizzes are
- Quiz 1 - September 13
- Quiz 2 - September 27
- Quiz 3 - October 25
- Quiz 4 - November 8
- Quiz 5 - November 22
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Handouts
All class handouts are available as PDF files.
Course Information
Course handout from first day of class
Midterm 1 Review Sheet
Handout describing first midterm.
Midterm 2 Review Sheet
Handout describing second midterm.
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Grading Scheme
20% = Assignments (5 x 4%)
10% = Quizzes (5 x 2%)
30% = Midterms (2 x 15%)
40% = Final Exam
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Important Dates
Sept. 5, 2017
First semester classes begin
Oct. 9-13, 2017
Fall break (no classes)
Oct. 18, 2017
Midterm 1
Nov. 10, 2017
Last day for cancelling courses without
failure by default
Nov. 16, 2017
Midterm 2
Dec. 6, 2017
First semester classes end
Dec. 8-21, 2017
Final Exams
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Class Polices
1. Policy on Academic Ethics. You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity.
Academic dishonesty is to knowingly act or fail to act in a way that
results or could result in unearned academic credit or advantage. This
behaviour can result in serious consequences, e.g. the grade of zero
on
an assignment, loss of credit with a notation on the transcript
(notation reads: Grade of F assigned for academic dishonesty),
and/or suspension or expulsion from the university.
It is your responsibility to understand what constitutes academic
dishonesty. For information on the various types of academic
dishonesty
please refer to the Academic Integrity Policy, located at:
http://www.mcmaster.ca/academicintegrity/
The following illustrates only three forms of academic dishonesty:
- plagiarism, e.g. the submission of work that is not one's own or for which other credit has been obtained.
- improper collaboration in group work,
- copying or using unauthorized aids in tests and examinations.
2. Policy regarding missed work.
If you have missed work, it is your responsibility to take action.
If you are absent from the university for
medical and non-medical (personal) situations,
lasting fewer than 3 days, you may report your absence, once per term,
without documentation, using the McMaster Student Absence Form
(MSAF). See
Requests
for Relief for Missed Academic Term Work
Absences for a longer duration or for other reasons must be reported
to
your Faculty/Program office, with documentation, and relief from term
work
may not necessarily be granted.
In Math 3GR3, the percentages of the missed work will be
transferred
to the final examination.
Please note that the MSAF may not be used for term work worth 25% or
more,
nor can it be used for the final examination.
3. Student Accessibility Services.
Students who require academic accommodation must contact Student
Accessibility
Services (SAS) to make arrangements with a Program Coordinator.
Academic accommodations must be arranged for each term of study.
Student Accessibility Services can be contacted by phone
905-525-9140 ext. 28652 or e-mail sas@mcmaster.ca.
For further information, consult McMaster University's Policy for
Academic Accommodation of Students with Disabilities.
4. Important Message.
The instructor and university reserve the right to modify elements of
the course during the term. The university may change the dates and
deadlines for any or all courses in extreme circumstances. If either
type of
modification becomes necessary, reasonable notice and communication
with the students will be given with explanation and the opportunity
to
comment on changes. It is the responsibility of the student to check
their McMaster email and course websites weekly during the term and
to note any changes.
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