Michael Harris | MATHEMATICS GU4041, Spring 2020 (old)

Time: Tuesday-Thursday, 10:10-11:25, room 520 Mathematics

Office Hours: Room 521, Tuesday 2-3, Zoom ID 103-344-259
Wednesday 11-12, Zoom ID 225-196-537
and by appointment

TA's: Noah Ben Olander (nbo2104@columbia.edu), office hours Th 3-6
Anton Wu (anton.wu@columbia.edu) , office hours Th 1-3
Iris Rosenblum-Sellers (igr2102@columbia.edu), office hours MW 3-4, Zoom ID 977-093-897

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 one-two 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
Group actions and geometry

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.

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.

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.

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

Notes of online classes

Notes on the isomorphism-theorems

Classification of abeliangroups1

Classification of finite-abelian-p-primary-groups.

Notes on semidirect-products

Notes on group_actions

Notes on simplicity-of-a5

Notes on the cauchy-frobenius-burnside_theorem

Notes on solvable_groups

Notes on sylow_theorems

Notes on jordan-hoelder

Notes on simplicity-of-an

Notes on platonic-solids

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

       Homework:  20%  
          (There will be **thirteen** homework assignments, the two lowest grades will be dropped)

        Final exam:  40%

        Midterms:  20% each

Midterms: February 27, April 9

First practice midterm: practice-midterm-1
Solutions: practicemidterm-solutions1

First midterm solutions: midtermsolutions1

Second practice midterm: practice-midterm-2
Solutions: practicemidterm-solutions2

Second midterm solutions: midterm_2_solutions

Final: to be announced

Practice final: practice-final
Solutions: practicefinal-solutions

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.

Academic integrity: Students are encouraged to work together on homework but any assignments handed in should be the work of the person whose name appears at the top of the page. Collaboration during exams is considered cheating and is taken very seriously. Cheating during a midterm or final entails failing the course. Students are advised to consult the Columbia University Undergraduate Guide to Academic Integrity.

Disability services: In order to ensure their rights to reasonable accommodations, it is the responsibility of students to report any learning-related disabilities, to do so in a timely fashion, and to do so through the Office of Disability Services. Students who have documented conditions and are determined by DS to need individualized services will be provided an DS-certified ‘Accommodation Letter’. It is students’ responsibility to provide this letter to all their instructors and in so doing request the stated accommodations.

More information on the DS registration process is available online at https://health.columbia.edu/content/disability-services.

Homework assignments

(due January 30) solutions1

2nd-week (due February 6) solutions2

3rd-week (due February 13) solutions3

4th-week (due February 20) solutions4

5th-week (due February 28) solutions5

6th-week 6th-week (due March 5) solutions6

7th-week (due March 12) solutions7

8th-week (due March 31) solutions8

9th-week (due April 7) solutions9

10th-week (due April 16) solutions10