Math 333

(Last updated May 25, 2022 @ 3pm)

Materials

Syllabus
Textbook (Abstract Algebra by Thomas W. Judson, 2021 Edition)
Homework Guide
Slack Workplace

A place to ask questions and discuss course content (including homework).

Contact: You can always reach me via email at nicholas.vlamis@qc.cuny.edu or on in our Slack channel @nicholas.vlamis. I will generally respond within 24 hours.

End of the semester (5/16–5/23)

Monday's Class Notes

Proved Sun-tzu's theorem.

Final is cumulative and will be given Monday, May 23 from 1:45-3:45pm in our normal classroom, 320 Kiely Hall.
- At least half of the exam will be dedicated to material covered after Exam 2.
- The Final Exam will be in a similar format to the prior two exams, but will be about 50% longer.
- From the textbook, we have covered Chapters 1, 2, 3, 5, 9, and 11 in their entirety, and we have also covered Sections 4.1, 10.1, 16.1, 16.2, and 16.4.
- You may bring three standard-sized sheets of notes to the exam, and you may write on the front and back of each.

Office Hours: Thursday 3-4pm, Friday 11am-12pm (Virtual), and Monday (May 23) 12-1pm.
Solutions to Final Exam

Week 14 (5/9–5/13)

Reading: Read Chapter 16 in the textbook
Monday's Class Notes

Introduced the notion of a ring and went over basic examples.

Course Evaluations - Due by May 18 (Please fill them out!!)
Handed back Homework 9 and 10 in class on Wednesday, May 11
Assignment 12 (Not to be collected)
Wednesday's Class Notes

Discussed properties of fields and introduced the notion of a ring homomorphism.

Week 13 (5/2–5/6)

Reading: Read Chapter 11 in the texbook
Monday's Class Notes

Introduced the notion of homomorophisms and gave examples.

Proved the isomorphism theorems.

Week 12 (4/25–4/29)

Reading: Read Section 10.1 in the texbook
Monday's Class Notes

Introduced the notion of internal direct product and proved the equivalence, up to isomorphism, of internal and external direct products.

Handed back Exam 2 and Homework 8 in class on Monday, April 25.
Exam 2 Solutions
Assignment 10 (Solution to Graded Problem)
Wednesday's Class Notes

Introduced normal subgroups and factor groups.

Week 11 (4/11–4/15)

Monday's Class Notes

Introduced the notion of direct product of groups.

Exam 2 is on Wednesday! You may bring a sheet of notes to the exam, that is, one 8.5"x11" sheet of paper with notes handwritten by you on the front and back of the page. The exam covers Chapters 4, 5, 6, and Section 9.1 in the textbook.

Week 10 (4/4–4/9)

Reading: Read Section 1 of Chapter 9 in the textbook
Monday's Class Notes

Showed that the converse of Lagrange's theorem is false. Introduced the notion of isomorphism.

Handed back Homework 6 and Homework 7 in class on Monday, April 4
Exam 2 is on Wednesday, April 13. You may bring a sheet of notes to the exam, that is, one 8.5"x11" sheet of paper with notes handwritten by you on the front and back of the page. Covers Chapters 4, 5, 6, and Section 9.1 in the textbook.

Classified cyclic groups up to isomorphism and proved Cayley's theorem.

Week 9 (3/28–4/1)

Reading: Read Chapter 6 in the textbook
Monday's Class Notes

Computed the order of a permutation. Introduced the notion and basic properties of cosets.

Proved Lagrange's theorem, Euler's theorem, and Fermat's Little Theorem

Week 8 (3/21–3/25)

Reading: Read Chapter 5 in the textbook
Monday's Class Notes

Proved that every permutation can be written as a product of disjoint cycles and can also be written as a product of transpositions.

Define even and odd permutations. Introduce the alternating groups and dihedral groups.

Homework 5 returned on Wednesday, March 23

Week 7 (3/14–3/18)

Reading: Read Section 4.1 and Section 5.1 in the textbook
Monday's Class Notes

Proved several theorems about cyclic groups.

Gave formula for order of an element in cyclic group. Introduced permutations and symmetric groups.

Week 6 (3/7–3/11)

Reading: Read Section 4.1 in the textbook
Exam 1 on Wednesday, March 9!

Covers Chapters 1-3 in the textbook

Monday's Class Notes

Introduced the center of a group and cyclic subgroups, and we went over basic examples.

Sample Title Page for Exam
Homework 4 handed back Monday, March 7
Additional office hours: 1-2pm on Wednesday, March 9

Week 5 (2/28–3/4)

Reading: Read Section 3.3 in the textbook.
Homework 5 (Solution to Graded Problem)

Note: You have two weeks to complete this assignment (due the exam), but there are questions on the homework that you are meant to work through before the exam.

Homework 2 and Homework 3 returned Monday, Februray 28 (see solutions to the graded problems below).
Monday's Class Notes

Went over basic exampls and properties of groups. Defined subgroup and went over first examples.

Exam 1 is on Wednesday, March 9. You may bring a sheet of notes to the exam, that is, one 8.5"x11" sheet of paper with notes handwritten by you on the front and back of the page.
Wednesday's Class Notes

Discussed more examples of subgroups and proved two standard subgroups tests.

Week 4 (2/21–2/25)

Reading: Read Section 3.2 in the textbook.
Homework 4 (Solution to Graded Problem)
Wednesday's Class Notes

Created Cayley table for symmetries of a square. Defined a group and discussed a few examples.

Week 3 (2/14–2/18)

Reading: Pages 36-41 (in Chapter 3) in the course textbook.
Homework 3 (Solution to Graded Problem)
Monday's Class Notes

Went over basic properties of prime numbers, the Fundamental Theorem of Arithmetic, and modular equivalence

Homework 1 returned on Wednesday, Feb. 16.
Wednesday's Class Notes

Went over basic properties of modular arithmetic and worked out some Cayley tables.

Week 2 (2/7–2/11)

Reading: Read Chapter 2 in the course textbook.
Homework 2 (Solution to Graded Problem)
Monday's Class Notes

Went over the Well-Ordering Principle, Mathematical Induction, and the Division Algorithm

Wednesday's Class Notes

Introduced the gcd, proved it's the a linear combination, and went over the Euclidean Algorithm

Week 1 (1/31–2/4)

Reading: Read Chapter 1 in the course textbook.
Homewwork 1 (Solution to Graded Problem)
Monday's Class Notes (contains some material that we will cover on Wednesday)

Recalled basic set theory.

Wednesday's Class Notes (contains some material that we will cover on Wednesday)

Introduced equivalence relations and partitions.

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