**Time: ** Tuesday-Thursday, 10:10-11:25, room 312

**Office Hours:** Room 521, Tuesday 12-1, Wednesday 11-12, and by appointment

**TA's:**

Zheheng Xiao (zx2377@columbia.edu, office hours 1-3 PM Thursdays, Mathematics 406)

Zhaocheng Dong (zd2292@columbia.edu, office hours 6-8 PM Wednesdays, Mathematics 405)

Mrudul Thatte (mrudul@math.columbia.edu, office hours 10-12 AM Fridays and by appointment)

This is the first part of the Modern Algebra sequence. Most of the course is devoted to proving the basic properties of groups, especially finite groups.

**Provisional syllabus**: Each of the topics listed below will occupy roughly 1-2 classes.

```
Sets, functions and equivalence relations
Modular arithmetic and residues
Basic definitions of groups, subgroups and homomorphisms
Basic properties of groups
Examples of groups: cyclic groups, cartesian products, permutations and symmetric groups
Lagrange's theorem and applications
Normal subgroups and quotient groups, alternating groups and conjugation
Isomorphism theorems
Classification of abelian groups
Group actions, orbits, conjugacy classes, and the class equation
Solvable and nilpotent groups, groups of p-power order
Sylow theorems
Classification of groups of small order, simple groups
Group actions, geometry, and the Platonic solids
```

**Some notes**

These *Cayley tables for some groups of small order* cayley-tables were prepared by Professor Robert Friedman. Bear in mind that the dihedral group he denotes D_3 is called D_6 in this course.

The presentation of the basic theory of symmetric groups will be based on notes-on-permutation-groups. The proof of simplicity of alternating groups, which we may not have time to cover in class, is treated in simplicityofan.

The main results on composition series are contained in the notes jordan-hoelder.

The symmetries of the cube and octahedron are explained in regular-polyhedra. The five regular polyhedra are discussed in the notes on platonic-solids.

**Prerequisites**: Multivariable calculus and linear algebra are the only formal prerequisites. However, students should have experience with methods of mathematical reasoning, including mathematical induction, and should be familiar with complex numbers. Students who have taken courses that involve writing proofs, such as Honors Mathematics A/B or Introduction to Higher Mathematics should be well prepared for this course.

**Grading**: There will be two midterms (in class). Grades will be computed as follows:

**Homework**: 20%

(There will be 12 homework assignments, the two lowest grades will be dropped)

Final exam: 40%

Midterms: 10% for the lower grade, 30% for the higher grade

**Midterms**: October 10, November 16

**PRACTICE MIDTERMS **

*First midterm*

practice-midterm1a midterm_practice_solutions_*1*

practice-midterm1b midterm_1_solutions2020

midterm1-2023 midterm1solutions2023

*Second midterm*

practice-midterm2a practice2solutions

practice-midterm2b)midterm_2_solutions

In the spring of 2020 the second midterm and the final were virtual and all courses were pass-fail. That is not the case this year, so the second midterm will **not** be an open book test.

midterm2-2023midterm_2_solutions2023

**PRACTICE FINALS**

Here are the practice final and the actual final from 2020. This year's final will be approximately the same length.

practice-final-2020

final2020final2020-solutions

Solutions to practice-final-2020 are on the 2020 course page. Please try not to look at the 2020 final solutions before December 16!

Again, this year's final will **not** be an open book test.

Weekly homework assignments will be posted below. Homework should be deposited in the course's mailbox on the 4th floor of the Mathematics building (in the corridor to the right when facing the elevator).

If you have a conflict with any of the exams, you must inform the instructor as soon as possible and at least one month before the exam. Make-up exams will not be given unless the student has two other exams scheduled the same day. Students with three exams scheduled on the same day should visit the Student Service Center in 205 Kent Hall to fill out a form which can then be submitted to each instructor or department. An attempt will then be made to arrange for one of the instructors to schedule a make-up exam on a different day. Students can only be excused from the exams because of serious illness or family emergency; documentation from your doctor or dean must be provided.

No electronic devices (laptops, calculators, telephones) are allowed during exams.

**Textbook**: No textbook is required, but a number of books are recommended.

The book *Abstract Algebra* by Dummit and Foote (on reserve in the math library) can be used as a reference.

**Some online resources**:

Abstract Algebra: Theory and Applications, by Thomas W. Judson

Notes on Modern Algebra I by Patrick Gallagher

Abstract Algebra. Introduction to Group Theory by Jim Howie.

**Class notes**

First week : first-week

Second week : second-week

Third week : third-week

Notes were not typeset for the fourth, sixth, or seventh weeks.

Fifth and eighth weeks (permutations) permutation-groups

Eighth/ninth week (isomorphism theorems) isomorphism-theorems

Ninth/tenth week (abelian groups) abelian-p-primary

Tenth week (group actions) group_actions

Eleventh week (Class equation and semidirect products) semidirect

Twelfth week (Solvable and nilpotent groups) solvable

Thirteenth week (Sylow theorems) sylow

Composition series composition-series

Simplicity of alternating groups simplicityofa5simplicityofan

**Homework assignments**

1st-week (due September 14) solutions1

2nd-week (due September 21) solutions2

3rd-week (due September 28) solutions3

4th-week (due October 5) solutions4

5th-week (due October 12) solutions5

6th-week (due October 24) solutions6

7th-week (due October 27) solutions7

8th-week (due November 2) solutions8

9th-week (due November 9) solutions9

10th-week (due November 20) solutions10

11th-week (due November 30) solutions11

12th-week (due December 7)