# Math 301

(Last updated Monday, May 15, 2023 @ 9:20pm)

## Course Information and Materials

Instructor: Professor Nicholas Vlamis

Instructor Office: 507 Kiely

Instructor Email: nicholas.vlamis@qc.cuny.edu

Class Meeting: Monday/Wednesday 1:40–3:30pm in Kiely 277

Office Hour: Monday/Wednesday 4–5pm (or by appointment)

Textbook: Abstract Algebra by Thomas W. Judson, 2021 Edition (pdf | html)

Contact: You can always reach me via email or on Discord. I will try to respond within 24 hours.

## Week 16 (Week of May 15)

Exam 3 is on Wednesday, May 17 from 1:45pm to 3:45pm in Kiely 277 (our standard classroom). Exam 3 will cover the content form Homework assignments 8, 9, 10, 11, and 12. This content is from the following sections in the textbook: 6.1, 6.2, 6.3, 9.1, 9.2, 10.1, 11.1, 11.2, 16.1, 16.2, and 16.5. As on the previous exams, you may bring one sheet of notes.

There will be an office hour on Tuesday, May 16 at 3pm in my office (507 Kiely). I will also have Zoom open in case you would like to join virtually. Link: https://us02web.zoom.us/j/82858982300?pwd=UUNOeS9scHdHTUhGNTFacUZGSE1BUT09

## Week 15 (Week of May 8)

Helpful reading: Section 16.1, Section 16.2, and Section 16.5 with the caveat that we only talked about a relatively small portion of these sections.

Homework Assignment #12 (Not to be turned in, but relevant to Exam 3)

Monday's Class: Inroduced rings and went over basic examples.

Wednesday's Class: Introduced fields and some examples. Proved some basic facts about finite fields, including the order of a finite field is the power of a prime. Constructed the field with four elements using an irreducible polynomial over the field with two elements.

Exam 3 is on Wednesday, May 17 from 1:45pm to 3:45pm in Kiely 277 (our standard classroom). Exam 3 will cover the content form Homework assignments 8, 9, 10, 11, and 12. This content is from the following sections in the textbook: 6.1, 6.2, 6.3, 9.1, 9.2, 10.1, 11.1, 11.2, 16.1, 16.2, and 16.5. As on the previous exams, you may bring one sheet of notes.

## Week 14 (Week of May 1)

Helpful reading: Section 10.1, Section 11.2

Homework Assignment #11 (Due Wednesday, May 10)

Monday's Class: Stated the fundamental theorem of finite abelian groups. Introduced normal subgroups and gave several characterizations. Discusssed examples of normal subgroups.

Wednesday's Class: Introduced factor groups. Stated the First Isomorphism Theorem. Discussed examples.

## Week 13 (Week of April 24)

Helpful reading: Section 9.1, Section 9.2

Monday's Class: Proved that every finite group is isomorphic to a matrix group by embedding the symmetric group S_n into GL(n,R). Introduced the notion of direct product and explored several examples.

Wednesday's Class: class was cancelled due to me being sick. To make up for the lost class time, I have recorded a video lecture (in 3 parts). Please watch before class on Monday, May 1.

## Week 12 (Week of April 17)

Helpful reading: Section 6.3, Section 9.1

***Class on Wednesday will begin late at 2:30pm

Monday's Class: Proved Euler's theorem and Fermat's little theorem as corollaries to Lagrange's theorem. Introduced the notion of an isomorphism and isomorphic groups. Went over a detailed example showing that Z_4 and U(5) are isomorphic. Proved that every infinite cyclic group is isomorphic to the integers.

Wednesday's Class: Classified cyclic groups up to isomorphic. Proved Cayley's theorem.

## Week 11 (Week of April 3)

Helpful reading: Section 6.1, Section 6.2

Exam 2 was on Monday, April 3.

CUNY Spring Recess: Wednesday, April 5 through Thursday, April 13

## Week 10 (Week of March 27)

Helpful reading: Section 6.1, Section 6.2

***Exam 2 is on Monday, April 3. Exam 2 will cover the content from homework assignments 5, 6, and 7. This is the content from Section 3.3, Section 4.1, Section 5.1, and Section 5.2 , as well as content introduced through the homework itself (such as homomorphisms, kernels, centralizers, torsion subgroup, etc.)

Homework Assignment #8 will be posted on Monday (and will be due Wednesday, April 19).

Monday's class: Proved Lagrange's theorem.

Wednesday's class: Went over some corollaries of Lagrange's theorem and proved the converse if falsse by showing that A_4 fails to have a subgorup of order 6.

## Week 9 (Week of March 20)

Helpful reading: Section 5.1, Section 5.2

Exam 2 is on Monday, April 3. Exam 2 will cover the content from homework assignments 5, 6, and 7. This is the content from Section 3.3, Section 4.1, Section 5.1, and Section 5.2 , as well as content introduced through the homework itself (such as homomorphisms, kernels, centralizers, torsion subgroup, etc.)

Monday's class: Proved two main theorems about cyclic groups: (1) every subgroup of a cyclic group is cylic, and (2) we computed the order of elements in finite cyclic groups.

Wednesday's class: Computed the orders of elements in the symmetric group. Re-introduced the dihedral groups, computed their order, and gave a generating set with relations, and discussed how to realize a dihedral group as a permutation group.

## Week 8 (Week of March 13)

Helpful reading: Section 4.1, Section 5.1

Monday's class: Proved two main theorems about cyclic groups: (1) every subgroup of a cyclic group is cylic, and (2) we computed the order of elements in finite cyclic groups.

Wednesday's class: Introduced permutations, symmetric groups, and permutation groups. Discussed two ways of expressing permutations: two-row representations and cycle notation.

## Week 7 (Week of March 6)

Helpful reading: Section 3.3, Section 4.1

Monday's class: Discussed subgroup tests, the center of a group, the center of Dihedral groups, and spent time working on problems about 2x2 (real) matrices.

Wednesday' class: Introduced cyclic (sub)groups, generators of cylcic groups, cyclic subgroups generated by an element, and the order of an element. We discussed several examples where we found generators, computed the cyclic subgroups, and found orders of elements. Proved that cyclic groups are abelian.

## Week 6 (Week of February 27)

Helpful reading: Section 3.3

Exam 1 on Wednesday

Exam 1 will cover Chapter 2, Section 3.1, and Section 3.2 from the textbook as well as the definition of equivalence relations.

You may bring one sheet of notes written on the front and back on an 8.5"x11" of paper.

Monday's Class: Introduced the notion of subgroup and went over numerous examples.

## Week 5 (Week of February 20)

Helpful reading: Section 3.2

***Exam 1 is on Wednesday, March 1.***

Exam 1 will cover Chapter 2, Section 3.1, and Section 3.2 from the textbook as well as the definition of equivalence relations.

Homework Assignment #4 (This assignment will not be collected, but there will be a problem from this assignment on Exam 1.)

CUNY closed on Monday, February 20, but we have class on Tuesday, Febuary 21.

***We will not have lecture on Wednesday; however, I encourage you to still come to class and work on HW4 together. Despite what I said before, there will be no video. Use this time to work on the assignment and prepare for upcoming exam.

Tuesday's class: Re-introduced the definition of a group. We discusses numerous examples and proved several propositions regarding the basic properties of groups (e.g. uniqueness of the identity element, uniquness of inverses, left/right-cancellation laws).

Announcement: On Exam 1, you may bring one sheet of notes written on the front and back on an 8.5"x11" of paper.

## Week 4 (Week of February 13)

Helpful reading: Section 3.1

CUNY was closed on Monday.

Wednesday's Class: Introduced modular arithmetic and its basic properties.

Next week's schedule: No class Monday (CUNY closed), class on Tuesday (CUNY on Monday schedule), no class meeting on Wednesday (I will post a video).

## Week 3 (Week of February 6)

Helpful reading: Section 2.1, Section 2.2, and Section 3.1

Monday's Class: Proved the first principle of mathematical induction, went over a basic induction proof, and introduced the second principle of mathematical induction. Proved Euclid's lemma, showed that every natural number greather than 1 is either a prime number or a product of prime numbers, and that there are infinitely many prime numbers

Wednesday's Class: Proved the fundamental theorem of arithmetic. Introduced equivalence relations and equivalence modulo n.

## Week 2 (Week of January 30)

Helpful reading: Section 2.1 and Section 2.2.

Monday's Class: Defined a group and discusses several basic examples. Began our introduction to elementary number theory and began the proof of the division algorithm.

Wednesday's Class: Finished proving the division algorithm. Defined prime numbers, common divisors, and the greatest common divisor. Proved that the GCD is a linear combination. Introduced/proved the Euclidean algorithm for computing GCD, and discussed how it can be used to write the GCD as a linear combination.

## Week 1 (Week of January 23)

Wednesday's Class: Investigated "clock arithmetic" and rigid motions of an equilateral triangle; Defined binary operation.

If you need a refresher on some basic set theory notions, you can look over the Set Theory subsection of Section 1.2 in the course text.