Instructor: Nathan Chen Office: 609

Lectures: Tuesdays and Thursdays 10:10am-11:25am in room 417 (math department)

Office hours: Tuesday 11:30-12:30pm and Saturday 11am-12pm

Email: nathanchen@math.columbia.edu

Textbook: Abstract algebra: theory and applications by Thomas W. Judson. Freely available online, exercises will be from the 2022 version.

First class: January 17, 2023. Last class: April 27, 2023. Zoom link for the class is here (but please try to show up in person if you can). I will be following these notes of Patrick Gallagher (with the order slightly changed).

Lecture 1: 1/17/2023. Algebra of sets, De Morgan's Laws. See Gallagher Lecture 4 and Judson Chapter 1.

Lecture 2: 1/19/2023. Inverse maps. Partitions of sets and equivalence relations. See Gallagher Lecture 5 and Judson Chapter 1.

Lecture 3: 1/24/2023. Divisors, greatest common divisor, prime numbers. See Gallagher Lecture 1 and Judson Chapter 2.

Lecture 4: 1/26/2023. Least common multiple, unique factorization. See Gallagher Lecture 2 and Judson Chapter 2.

Lecture 5: 1/31/2023. Binary operations, associativity, groups. See Gallagher Lecture 7 and Judson Chapter 3.1, part of 3.2.

Lecture 6: 2/2/2023. Products of subsets, translations, subgroups, Lagrange's theorem. See Gallagher Lecture 8 and Judson Chapter 3.3.

Lecture 7: 2/7/2023. Powers of elements, orders of elements, cyclic groups. See Gallagher Lecture 9 and Judson Chapter 3.2, 4.1.

Lecture 8: 2/9/2023. Multiplicative group of complex numbers and its subgroups (Judson Chapter 4.2). Direct product of groups (Judson 9.2). The group of invertible residues modulo n under multiplication. Notes from class.

Lecture 9: 2/14/2023. Symmetric group, permutations and cycles (Judson Chapter 5.1 up to and including Remark 5.11). Multiplication of permutations in cycle notation.

Midterm 1. 2/16/2023. Topics. Here are solutions.

Lecture 10: 2/21/2023. Continuation of Lecture 9: disjoint cycles, transpositions, even/odd permutations, alternating group, dihedral group (rest of Judson Chapter 5 up to dihedral groups). Notes from class for Lectures 9 and 10.

Lecture 11: 2/23/2023. Converse to Lagrange's theorem is false, cycles of same length (Judson Chapter 6.2), group isomorphisms and examples (Judson 9.1). Notes from class.

Lecture 12: 2/28/2023. Intersections and internal direct products of subgroups. See first half of Gallagher Lecture 10 and Judson Chapter 9.1.

Lecture 13: 3/2/2023. More on internal direct products of subgroups, group homomorphisms, kernels, images. See Gallagher second half of Lecture 10, second half of Lecture 11 (beginning with the definition of a group homomorphism) and Judson Chapters 9.2, 11.1.

Lecture 14: 3/7/2023. Normal subgroups, factor groups, First Isomorphism Theorem. See Gallagher Lecture 11 and Judson Chapters 10.1, 11.2.

Lecture 15: 3/9/2023. Second and Third isomorphism theorems, the quaternion group, simplicity of the alternating group. See Gallagher Lecture 12 (up to pg. 3) and Judson Chapters 11.2, example 3.15, 10.2. Video below.

Spring break!

Lecture 16: 3/21/2023. Review of isomorphism theorems, solvable groups. See Gallagher Lecture 12 and Judson Chapters 11.2, 13.2. Notes from class.

Lecture 17: 3/23/2023. Classification of finite abelian groups. Judson Chapter 13.1. Notes from class.

Lecture 18: 3/28/2023. Short lecture on completing classification of finite abelian groups.

Midterm 2. 3/30/2023. Topics. Here are the solutions.

Lecture 19: 4/4/2023. Short lecture on Cayley's theorem, recap of midterm 2.

Lecture 20: 4/6/2023. Group actions, orbits, stabilizer subgroups. See Gallagher Lecture 16, Judson Chapter 14.1.

Lecture 21: 4/11/2023. Class equation, conjugacy classes, groups of prime power order. See Gallagher 17 and Judson Chapter 14.2. Notes from class.

Lecture 22: 4/13/2023. Sylow theorems. Judson Chapter 15.1. Video below. Notes.

Lecture 23: 4/18/2023. Sylow theorems and applications - continuation of the previous lecture. See Judson Chapter 15.2. Notes.

Lecture 24: 4/20/2023. Sylow theorems and applications - continuation of the previous lecture. See Judson Chapter 15.2. Notes.

Lecture 25: 4/25/2023. Matrix groups, orthogonal group. Notes.

Lecture 25: 4/27/2023. Isometries of R^n and some finite subgroups. Notes.

Final Exam. 5/11/2023 (Thursday). 9:00am-12:00pm. Final Exam Topics. Here are some remarks on the practice final (see website of Khovanov).

Late policy for homework: 5 points will be deducted per late day. Please try to turn in homework on time, as this will help the TAs with grading.

• Problem set 1. Due Friday, January 27th.

• Problem set 2. Due Friday, February 3. Solutions by Zachary.

• Problem set 3. Due Saturday, February 11. Solutions by Anna.

• Problem set 4. Due Sunday, February 26.

• Problem set 5. Due Sunday, March 5.

• Problem set 6. Due Monday, March 20.

• Problem set 7. Due Monday, March 27.

• Problem set 8. Due Sunday, April 16.

• Problem set 9. Due Sunday, April 30.