# Math 333

(Last updated May 25, 2022 @ 3pm)

## Materials

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)

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.

## Week 14 (5/9–5/13)

Introduced the notion of a ring and went over basic examples.
Discussed properties of fields and introduced the notion of a ring homomorphism.

## Week 13 (5/2–5/6)

Introduced the notion of homomorophisms and gave examples. Proved the isomorphism theorems.

## Week 12 (4/25–4/29)

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

## Week 11 (4/11–4/15)

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)

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)

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)

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)

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)

• Exam 1 on Wednesday, March 9!

Covers Chapters 1-3 in the textbookIntroduced the center of a group and cyclic subgroups, and we went over basic examples.

## Week 5 (2/28–3/4)

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)

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

## Week 3 (2/14–2/18)

Went over basic properties of prime numbers, the Fundamental Theorem of Arithmetic, and modular equivalence
Went over basic properties of modular arithmetic and worked out some Cayley tables.

## Week 2 (2/7–2/11)

Went over the Well-Ordering Principle, Mathematical Induction, and the Division AlgorithmIntroduced the gcd, proved it's the a linear combination, and went over the Euclidean Algorithm