# Math 301 (Spring 2024)

(Last updated Wednesday, April 10 @ 5:15pm)

## 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)

## Solutions to Graded Homework Problems

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

## 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.

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.