# Math GU4041: Introduction to Modern Algebra I Columbia University Fall 2019

## 1:10 - 2:25pm Tuesdays and Thursdays in 207 Mathematics Instructor: Kyler Siegel (kyler@math.columbia.edu

This semester long course is an introduction to group theory. Historically, through the 18th century algebra was primarily about solving polynomial equations. During the 19th century, a fundamental change in perspective took place, driven by many influential mathematicians including Lagrange, Cauchy, Abel, Gauss, Galois, Jordan, and others. From the early 19th century onward, "modern algebra" has primarily focused on such abstract concepts as groups, rings, fields, modules, representations, and so on. Although these abstract concepts might have seemed foreign or unmotivated to the ancients, together they give rise to elegant and powerful results with innumerable applications.

## Prerequisites.

We will not assume any prior experience with group theory, but we will assume a strong comfort level with multivariable calculus and linear algebra. Much of this course will center around giving rigorous proofs based on abstract definitions. We will expect students to be able to give logically precise arguments, written in full sentences, wherein every step is fully justified. Students without prior experience with this style of mathematics may find many of the exercises quite challenging.

## Topics.

Some of the topics we will cover over the course of the semester include:

• Naive set theory
• Group definition and basic examples
• Subgroups
• Lagrange's theorem
• Homomorphisms and isomorphisms
• Quotient groups
• The isomorphism theorems
• Jordan-Hölder theorem
• Group actions
• Burnside's lemma
• Simple groups
• Direct and semidirect products
• Classification of finitely generated abelian groups
• Sylow's theorems
• Classification of groups of small order
• Further topics in group theory

## Textbook.

The textbook we will follow most closely is Abstract algebra by David Dummit and Richard Foote (the first six chapters). As a secondary text, I recommend A first course in abstract algebra by John Fraleigh (the first three chapters). My recommendation is to read the relevant sections currently (listed in the schedule below when relevant) with the lectures. You may find it helpful to reread some sections several times in order for it to better sink it. I also recommend doing as many additional exercises from these textbooks as possible. Note that not all of the material from the lectures is necessarily contained in these reference texts.

## Structure of the course.

In addition to the lectures, there will be about twelve weekly problem sets. These will be listed on this website and typically due on Friday by 5pm. The problems can be handed in either during lecture or in the homework box on the 4th floor of the mathematics building. Late homeworks will not be accepted. Graded problem sets can be found in the box on the sixth floor.

It is very important to do all of the problem sets to the best of your ability, as this is the most effective way to absorb the material. While you are welcome to collaborate with your peers, you must attempt all problems on your own and your submitted solutions must be written out individually. Submissions which are copied or suspiciously similar may be rejected. Note that many of the problems will ask for rigorous proofs. All steps must be fully justified for full credit. We will expect proofs to be written in full sentences and to be logically precise. Students are encouraged (but not required) to type their solutions, for example using LaTeX .

There will be two midterm exams and one final exam. The date of the final is determined by the university registrar (currently projected as 12/17).

## Teaching assistants:

• Graduate teaching assistant: Nguyen Dung (nguyendung@math.columbia.edu). Office hours: Wednesdays 10am-1pm.
• Undergraduate teaching assistant: Josh Zhou (zz2397@columbia.edu>). Office hours: Mondays and Tuesdays 2pm-3pm.
• Undergraduate teaching assistant: Anton Wu (aw3067@columbia.edu). Office hours: Fridays 11am-1pm.

Homeworks: 15%, midterm exams: 50%, final exam: 35%. We will drop the lowest homework grade.

• Each of the TA's will hold weekly hours in the math help room and you are encouraged to come to their hours with any questions or confusions you may have.
• Additionally, Kyler will hold weekly office hours 2:30-3:30PM on Tuesdays and Thursdays (i.e. right after class) in room 609 in the mathematics building.
• For anyone new to writing proofs, I recommend checking Intro to Proofs Workshop. Handouts: 1,2.
• ## Schedule:

This will be continuously updated as the course proceeds. Please check regularly for updates and for problem set assignments. Note: I decided to remove the corresponding Fraleigh references since the chapters seem to get shuffled around from one edition to another. I do still recommend following allow with the relevant chapters in Fraleigh, which you can find by their topic titles.
Date Material References Problem Set
Tuesday 9/3/19 Historical introduction. Basic definitions and examples. Dummit and Foote chapters 0,1,2. Problem set 1 (due on Friday 9/13/18 by 5pm). Solution to partition problem.
Thursday 9/5/19 More history, introduction to naive set theory. Dummit and Foote chapters 0,1,2.
Tuesday 9/10/19 More examples of groups, especially $\mathbb{Z}/(n\mathbb{Z})$ and $C_n$. Dummit and Foote chapters 0,1,2.
Thursday 9/12/19 More examples of groups, especially $\mathbb{Z}/(n\mathbb{Z})^{\times}$.Direct products. Subgroups. Dummit and Foote chapters 0,1,2. Problem set 2 (due on Friday 9/20/18 by 5pm). Updated version (as of 10/15/19).
Tuesday 9/17/19 Dihedral groups. Homomorphisms and isomorphisms. Examples of (non)isomorphic groups. DF chapters 1.2,1.4,1.6.
Thursday 9/19/19 Lagrange's theorem. Properties of cyclic groups. DF chapter 2.3,3.2. Problem set 3 (due on Friday 9/27/18 by 5pm).
Tuesday 9/24/19 Cycle decompositions. Normal subgroups. Quotient groups. DF chapters 1.3,3.1,3.2.
Thursday 9/26/19 Kernels and images of homomorphisms. More on normal subgroups. The alternating group. DF chapters 3.1,3.2,3.5. Problem set 4 (due on Friday 10/11/18 by 5pm). Updated version (as of 10/2/19 - 4b slightly tweaked and corrected statement of FIT.). Solutions.
Tuesday 10/1/19 More on the alternating group. First isomorphism theorem. Normalizers. Subsets of the form AB. Second isomorphism theorem. DF 3.2,3.3,3.5.
Thursday 10/3/19 More on second isomorphism theorem. Third and fourth isomorphism theorems. DF 3.3. Topic list for midterm 1. Format and sample problems. Solutions (1IV updated 10/8/19). Midterm 1 solutions.
Tuesday 10/8/19 Midterm 1 during class
Thursday 10/10/19 Introduction to group actions (guest lecturer: Aleksander Doan). DF 4.1,4.2,4.3.
Tuesday 10/15/19 More on the fourth isomorphism theorem and lattices of subgroups. Group presentation perspective on the dihedral group. The lattices for $Q_8$ and $D_8$. DF 3.2, 3.4. Problem set 5 (due on Friday 10/25/18 by 5pm). Solutions.
Thursday 10/17/19 Cauchy's theorem and Sylow's first theorem. Composition series and the Hölder program. Examples of composition series. A bit on simplicity of the alternating group. DF 4.1,4.2,4.3,4.4.
Tuesday 10/22/19 More on group actions. Groups acting on themselves by left multiplication and by conjugation. The class equation and applications. DF 4.1,4.2,4.3,4.4.
Thursday 10/24/19 More applications of group actions and the class equation. Conjugacy classes in the symmetric group and simplicity of the alternating group. DF 4.1,4.2,4.3,4.4. Problem set 6 (due on Friday 11/1/19 by 5pm). Solutions.
Tuesday 10/29/19 More on conjugacy classes in the symmetry group and simplicity of the alternating group. DF 4.6.
Thursday 10/31/19 Sylow's theorems and applications. DF 4.5, F 18,19. Problem set 7 (due on Friday 11/8/19 by 5pm). Solutions here and here.
Tuesday 11/5/19 No class (election day)
Thursday 11/7/19 More on Sylow's theorems and applications. DF 4.5. Topic list for midterm 2. Midterm 2 solutions.
Tuesday 11/12/19 More on Sylow's theorems and applications. DF 4.5.
Thursday 11/14/19 Midterm 2 during class Problem set 8 (updated 11/19/19; due on Friday 11/22/19 by 5pm).
Tuesday 11/19/19 Direct products. The classification of finitely generated abelian groups. DF 5.1, 5.2, 5.4.
Thursday 11/21/19 More on the classification of finitely generated abelian groups. DF 5.2,5.3,6.1. Problem set 9 (updated 11/28/19; due on Friday 12/6/19 by 5pm).
Tuesday 11/26/19 Semidirect products (guest lecturer: Aleksander Doan). DF 5.5.
Thursday 11/28/19 No class (Thanksgiving)
Tuesday 12/3/19 More on semidirect products and applications. DF 5.3,5.5.
Thursday 12/5/19
(last class)
Examples of semidirect products. Classifying groups of order 12. More on automorphism groups. DF 5.5
Tuesday 12/17/19 Final exam, 1:10-4:00pm in Mathematics 207
(alternate session: 12/16/19 9:10am-12pm in Mathematics 528)
Final exam topics list and extra practice problems