# Math 301 (Spring 2024)

(Last updated Friday, May 17 @ 11:15am)

## Course Information

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 320

Office Hour: Monday 4–5pm and Wednesday 12–1pm (or by appointment)

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

## Exam Dates

Exam 1: Wednesday, March 6 in class

Exam 2: Wednesday, April 10 in class

Exam 3: Wednesday, May 22 1:45–3:45pm in Kiely 320

## Materials

A place to ask questions and discuss course content (including homework). An online version of LaTex, which is a mark-up language for writing mathematics (or any document once you become a convert). You can also run LaTex natively on your computer. I'm a Mac user, so I can point you to MacTex, but there are also options for Windows and Linux. I really enjoyed writing homework in LaTeX when I was a student, and I really do think it made me a better student. ## Homework Assignments

HW1 (TeX), HW2 (TeX), HW3 (TeX), HW4 (TeX), HW5 (TeX), HW6 (TeX), HW7 (TeX), HW8 (TeX), HW9 (TeX), HW10 (TeX), HW11 (TeX)

## Solutions to Graded Homework Problems

HW1 (TeX), HW2 (TeX), HW3 (TeX), HW4 (TeX), HW5 (TeX), HW6 (TeX), HW7 (TeX), HW8 (TeX), HW9 (TeX), HW10 (TeX), HW11 (TeX)

## Week 15 (Week of May 13)

Suggested reading: Section 16.1, Section 16.2

Course Evaluations (please fill out before the final exam)

Exam 3 is on Wednesday, May 22 1:45–3:45pm in Kiely 320. The exam will cover Week 10 through Week 15, which includes HW 9, 10, 11, 12. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exam that includes only definitions and theorem/proposition statements from class and the textbook.

Office hours for the week of May 20: Monday and Tuesday 2–3pm.

Monday's class: Discussed charactertic and orders of fields. Constructed the field of 4 elements using polynomials over Z_2.

Wednesday's class: Proved the Freshman's Dream and Sun-Tzu's theorem (previously known as the Chinese Remainder Theorem).

Wednesday was the last day of class.

## Week 14 (Week of May 6)

Suggested reading: Section 11.2, Section 16.1, Section 16.2

Homework Assignment #12. This assignment will not be collected, but the problems are fair game for Exam 3.

Monday's class: Proved the first isomorphism theorem and Cauchy's theorem for finite abelian groups.

Wednesday's class: Introduced rings and there various properties rings can have, which leads to the defition of a field.

## Week 13 (Week of April 29)

Suggested reading: Section 10.1, Section 11.1

Office hours on Wednesday will be 11am–12pm and 3:30–4:30pm. (I cannot attend my regular office hour.)

Homework Assignment #11. Due Wednesday, May 8.

Wednesday's class: Recapped normal subgroups and homomorphisms from the required reading. Introduced factor groups.

## Week 12 (Week of April 15)

Suggested reading: Section 9.2, Section 10.1

Required Reading: Notes on normal subgroups and homomorphisms (TeX). These notes are a replacement for our missed class meeting on Wednesday.

We will not have class on Wednesday, April 17. However, I encourage you use the time and classroom to meet your peers and discussed the required reading.

There will be no office hours this week on account of my travels. Please post questions to Discord.

Homework Assignment #10. Due Wednesday, May 1.

Monday's class: Introduced the notion of the internal direct product and discussed examples. Discussed, but did not prove, the fundamental theorem of finite abelian groups.

## Week 11 (Week of April 8)

Suggested reading: Section 9.2

Exam 2 is on Wednesday, April 10. (See Week 10 for more detail.)

Monday's class: Introduced the notion of a direct product, proved that Z_m x Z_n is isomorphic to Z_mn if and only if gcd(m,n) =1, and computed orders of elements in direct products.

Homework Assignment #9. Due Wednesday, April 17. (In reality, I will collect this homework on Wednesday, May 1, as we will not meet on April 17. However, I encourage you to complete the assignment on time so that you do not fall behind.)

Next week, we will not meet on Wednesday, April 17. I will write a document for you to read that will overlap with content from Section 11.1 in the textbook, and I will give you some problems to work on. You will be responsible for this material. So, despite class meeting officially being cancelled, I recommend you still come to class and work through the content and problems together and make use of the class time (instead of doing it on your own time).

## Week 10 (Week of April 1)

Suggested reading: Section 9.1, Section 9.2

Office hours Wednesday moved to 3:30–4:30pm. If this affects your ability to complete your homework, you may drop it in my mailbox or under my office door by end of day Friday.

Exam 2 is on Wednesday, April 10. The exam will cover the course material covered Week 5 through Week 9, which includes HW5, HW6, HW7, and HW8. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exam that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 6.3).

Homework Assignment #9. Due Wednesday, April 17.

Monday's class: Introduced the notion of an isomorpism, went over examples, discussed basic properties, and proved that every infinite cyclic group is isomorphic to the integers.

Wednesday's class: Proved the classification of cyclic groups. Proved Cayley's theorem. Proved that all finite groups are linear.

## Week 9 (Week of March 25)

Suggested reading: Section 6.1, Section 6.2, Section 6.3

Homework Assignment #8. Due Wednesday, April 3.

Monday's class: Introduced left and right cosets, established basic facts, and proved Lagrange's theorem.

Wednesday's class: Proved Fermat's little theorem and Euler's theorem. Introduced the notion of conjugation and discussed how all k-cycles are conjugate. We proved that A_4 has no order 6 subgroup, implying the converse of Lagrange's theorem is false.

## Week 8 (Week of March 18)

Suggested reading: Section 5.1, Section 5.2

Office hours Wednesday moved to 3:30–4:30pm. If this affects your ability to complete your homework, you may drop it in my mailbox or under my office door by end of day Friday.

Homework Assignment #7. Due Wednesday, March 27.

Monday's class: Gave a formula for the order of a permutation. Defined the notion of an even/odd permutation and showed that the notion is well defined. Defined the alternating group.

Wednesday's class: Computed the order of the alternating group A_n. Introduced the notion of a graph and graph automorphism. Defined the dihedral group D_n as a group of graph automorphism. Computed the order of D_n. Proved that D_n can be generated by two elements.

## Week 7 (Week of March 11)

Suggested reading: Section 4.1, Section 5.1

Homework Assignment #6. Due Wednesday, March 20.

Monday's class: Further discussed orders of elements in a group. Gave a formula for computing the order of an element in a cyclic group.

Wednesday's class: Introduced permutations and the symmetric group. Introduced the notion of a cycle and proved that every permutation can be expressed as a product of disjoint cycles.

## Week 6 (Week of March 4)

Suggested reading: Section 3.3, Section 4.1

Exam 1 is on Wednesday, March 6. The exam will cover the course material through Week 4, which includes HW1, HW2, HW3, and HW4. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exams that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 3.2).

Homework Assignment #5. Due Wednesday, March 13.

Monday's class: Continued out treatment of cyclic groups, proving that every subgroup of a cyclic group is cyclic. Introduced the notion of an order of an element of a group. Computed the order of elements in cyclic groups.

## Week 5 (Week of February 26)

Suggested reading: Section 3.3, Section 4.1

Exam 1 is on Wednesday, March 6. The exam will cover the course material through Week 4, which includes HW1, HW2, HW3, and HW4. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exams that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 3.2).

Homework Assignment #5. Due Wednesday, March 13.

Monday's class: Introduced the notion of a subgroup and saw a plethora of examples.

Wednesday's class: Introduced the notion of a cyclic group and cyclic subgroups.

## Week 4 (Week of February 19)

Suggested reading: Section 3.2

Note: We have class on Thursday, February 22, as CUNY will follow a Monday Schedule.

Homework Assignment #4. Due Wednesday, Februrary 28.

Monday's class: Introduced the notion of a group and discussed basic examples.

Wednesday's class: Introduced a more extensive list of exampls of groups, and we established some basic properties of groups. Our class was interrupted by a campus emergency, and so please make sure read Section 3.2. In particular, we were in the process of finishing the proof of Proposition 3.21 when class was interrupted. I was going to finish the lecture portion of the class with stating an immediate corollary of Prop 3.21, which is Prop. 3.22 in the book.

## Week 3 (Week of February 12)

Suggested reading: Section 3.1

Homework Assignment #3. Due Thursday, Februrary 22.

Note: We have class on Thursday, February 22, as CUNY will follow a Monday Schedule.

No class Monday.

Wednesday's class: Introduce equivalence modulo n. Established basic properties of modular arithmetic.

## Week 2 (Week of February 5)

Suggested reading: Section 2.1 and Section 2.2

Homework Assignment #2. Due Wednesday, Februrary 14.

Monday's class: Introduced the greatest common divisor (gcd) of two integers. Proved that the gcd is a linear combination. Established the Euclidean algorithm for finding the gcd and discussed how to write the gcd as a linear combination using the steps from the Euclidean algorithm.

Wednesday's class: Introduced prime numbers. We proved Euclid's lemma, we proved that every natural number other than one is a product of prime numbers (and stated the Fundamental Theorem of Arithmetic), and we proved there are infinitely many primes. We wrote down the principal of mathematical induction (which was proved on HW1).

## Week 1 (Week of January 29)

Welcome to Math 301! I will update this section with summaries of each of the week's classes plus any additional information.

Suggested reading: We will begin the course by covering Section 2.1 and Section 2.2 in the textbook, but not in the same order as presented in the book. I will take most of Section 1.1 and Section 1.2 as assumed knowledge, which includes the basic language of set and how to work with them. When it comes up, I will briefly discuss the definition of an equivalence relation, so depending on your background, you might find it useful to work through these sections now.

Homework Assignment #1. Due Wednesday, Februrary 7.

Monday's class: Worked on a work sheet to recall the definitions of function, injective, surjective, bijective, and function composition.

Wednesday's class: introduced what it means for one integer to divide another. Proved the division algorithm.