# Math 301

(Last updated Monday, December 11, 2023 @ 9:50 pm)

## 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 320

Office Hour: Monday 12–1pm and Wednesday 11am–12pm (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.

Overleaf: 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 or Linux. I really enjoyed writing homework in LaTeX when I was a student, and I really do think made me a better student.

## Week 16 (Week of December 11)

Course Evaluations. Please complete before final exam.

Exam 3 is 1:45–3:45pm on Wednesday, December 20 in Kiely 320.

Exam 3 will cover the material from Week 11 through Week 16. In terms of homework assignments, the exam covers Assignments 8 through 11. In the book, this is sections 9.1, 9.2, 10.1, 11.1, 11.2, 16.1, 16.2, 16.3 (we did not cover everyhing in the Chapter 16 sections listed). You may bring a sheet of notes to the exam that contains only definitions and theorem statements.

Office Hours: Friday 11am–12pm and Monday 1–2pm.

Monday's class: Went over ring homomorphisms and isomoprhisms. Proved Sun-tzu's theorem (also known as the Chinese Remainder Theorem) and discussed the algorithm for solving a system of linear congruence equations.

## Week 15 (Week of December 4)

Suggested Reading: Section 16.1, Section 16.2, and Section 16.3

Course Evaluations. Please complete before final exam.

Homework Assignment #11. Assignment will not be collected, but the material will be on the final exam.

Note: Office hours for next week will be announced at a later date.

Monday's class: Defined a ring. Discussed many examples and the various types of rings, as shown in the diagram. Of particular interest for our short discussion on this topic are integral domains and fields. We proved that every finite integral domain is a field.

Wednesday's class: Defined the characteristic of a ring. Proved that the characteristic of an integral domain is prime. Proved that the order of a finite field is a power of a prime. Constructed the field of order 4 using an irreducible polynomial over the field with two elements.

## Week 14 (Week of November 27)

Suggested Reading: Section 11.1 and Section 11.2

Monday's class: Went over properties of homomorphisms and kernels. Stated the first isomorphism theorem.

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

## Week 13 (Week of November 20)

Suggested Reading: Section 11.1

Monday's class: Introduced homomorphisms. Worked through some examples and basic properties.

## Week 12 (Week of November 13)

Suggested Reading: Section 9.2, and Section 10.1

Wednesday's class will start late at 2pm, and Wednesday's office hours are cancelled.

Monday's class: Introduced the notions of internal and external direct products. We proved several results about direct products, including a computation of the order of an element in a direct product and charazterizing when the product of two cyclic groups is cyclic.

Wednesday's class: Introduced normal subgroups and factor groups.

## Week 11 (Week of November 6)

Suggested Reading: Section 9.1, Section 9.2, and Section 10.1

Monday's class: Proved Cayley's theorem stating that every group is isomorphic to a permutation group. Defined matrix groups, and we proved that every finite group is isomorphic to a matrix group.

## Week 10 (Week of October 30)

Suggested Reading: Section 6.2, Section 6.3, Section 9.1

Exam 2 will be held in class on Wednesday, November 8. The exam will cover material from Week 6 through Week 10. This corresponds to Homework Assignments #5 through #7. In the book, this corresponds to Sections 4.1, 5.1, 5.2, 6.1, 6.2, and 6.3.

Homework Assignment #8 will be posted early next week.

Monday's class: Discussed corollaries and applications of Lagrange's theorem. Showed that the converse of Lagrange's theorem is false; this entailed discussing conjugation in groups.

Wednesday's class: Introduced the notion of isomorphisms. Classified all cyclic groups up to isomorphism. Classified all groups of order 4 up to isomorphism.

## Week 9 (Week of October 23)

Change in office hours: Monday office hours will be held 11am–12pm.

Suggested Reading: Section 5.2, Section 6.1, Section 6.2, Section 6.3

Monday's class: Gave a formulat for the order of a permutation. Discussed the dihedral group D_n in detail; in particular, we computed its order, its center, and realized it as a subgroup of S_n.

Wednesday's class: Introduced the notion of left and right cosets of a subgroup H in a group G, showed that the number of left cosets of H equals the number of right cosets of H, and that every coset of H has the same number of elements as H. Proved Lagrange's theorem and discussed some immediate corollaries. Proved Fermat's Little Theorem.

## Week 8 (Week of October 16)

I will not be in class on Wednesday, October 18. However, I expect you to come to class and work together on the homework problems together.

Suggested reading: Section 5.1 and Section 5.2

Monday's class: Introduced the notion of even and odd permutations (and worked hard to show that the definition makes sense). Defined the alternating group, showed it is a subgroup of S_n, and computed its order.

Mathematical Discover Series: I encourage you all to go to the speical colloquium on Monday that is meant for undergraduate students. You will learn about the Mandelbrot set, one of the most famous fractals, from John Hubbard (Cornell University), one of the mathematicians who uncovered many of the basic properties of the Mandelbrot set! (Link to the poster)

## Week 7 (Week of October 9)

No class on Monday (CUNY closed). But we will have class on Tuesday at the regular time (CUNY follows a Monay schedule).

Office hours on Tuesday cancelled. (You can find me at the major/minor fair if you want to say hello.)

Suggested reading: Section 4.1, Section 5.1

Tuesday's class: Proved several properties of cyclic groups: they are abelian, every subgroup of a cyclic group is cyclic, defined the order of an element, and gave a formula for the order of an element in a cyclic group.

Wendesday's class: Introduced permutation groups and the symmetric group on n letters, S_n. Went over various way of representing permutation, including cycle notation.

## Week 6 (Week of October 2)

Suggested reading: Section 4.1

Exam 1 will be held in class on Wednesday, October 4. The exam will cover all material covered in class from Week 1 through Week 5. In the book, this corresponds to sections 2.1, 2.2, 3.1, 3.2, and 3.3. For homework assignments, this corresponds to Homework Assignments 1, 2, 3, and 4. Remember the majority of exam questions (on the order of 80%) will either be taken directly from a homework assignment or be slight variations on such problems. You may bring one sheet of notes that contains definitions and theorem statements, but nothing else.

Monday's class: Definded cyclic groups and subgroups. Discussed examples.

## Week 5 (Week of September 25)

Suggested reading: Section 3.3

There was no class on Monday, September 25: CUNY was closed.

Exam 1 will be held in class on Wednesday, October 4. The exam will cover all material covered in class from Week 1 through Week 5. In the book, this corresponds to sections 2.1, 2.2, 3.1, 3.2, and 3.3. For homework assignments, this corresponds to Homework Assignments 1, 2, 3, and 4. Remember the majority of exam questions (on the order of 80%) will either be taken directly from a homework assignment or be slight variations on such problems. You may bring one sheet of notes that contains definitions and theorem statements, but nothing else.

Homework Assignment #4. Due Wednesday, October 11. (**Note: the due date is the week after Exam 1; however, the content on the assignment is fair game for the exam.)

Wednesday's class: Introduced the notion of a subgroup, went over subgroup tests, and saw some examples.

## Week 4 (Week of September 18)

Suggested reading: Section 3.2

Monday's class: Worked through constructing several Cayley Tables, including for the set of symmetries of an equilater triangle with respect to function compositons. Defined a group and provided several basic examples.

Wednesday's class: Discuseed examples of groups. Established basic properties of groups.

## Week 3 (Week of September 11)

Suggested reading: Section 3.1

Monday's class: Proved that every natural number is a product of prime numbers. Proved there are infinitely many prime numbers (we gave Euclid's proof). Established mathematical induction as a consequence of the well-ordering principle. Worked in groups on two induction problems, the second of which established a more general form of Euclid's lemma.

Wednesday's class: Stated the Fundamental Theorem of Arithmetic. Introduced modular arithmetic and established basic properties.

## Week 2 (Week of September 4)

CUNY was closed on Monday.

Suggested reading: During the first two weeks of class, we will be covering Section 2.1 and Section 2.2 in the textbook, but not in the same order as presented in the book.

Continue working on Homework Assignment #1, whch is due Wednesday, September 13. Make sure you read the Homework Guide, so you know what the expectaton is.

Wednesday's class: Introduced the greatest common divisor. Proved that the GCD exists, is unique, and can be expressed as a linear combination. Established the Euclidean algorithm and discussed how to use the Euclidean algorithm to write the GCD as a linear combination. Introduced the notion of a prime number and proved Euclid's lemma.

## Week 1 (Week of August 28)

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

Suggested reading: During the first two weeks of class, we will be covering Section 2.1 and Section 2.2 in the textbook, but not in the same order as presented in the book.

Monday's class: Worked in groups investigating the definition of a function and the properties of being injective and surjective. We also talked through a problem about matrices and recalled some facts from linear algebra.

Wednesday's class: Introduced the well-ordering principle. Defined what it means for one integer to divide another. Proved the Division Algorithm. Defined quotient and remainder. Worked on the first couple of homework problems in groups.