Abstract Algebra, Fall 2018, University of Colorado, Boulder

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# Math 3140: Abstract Algebra I -- Fall 2018

Lecture Time and Location. MWF 9:00--9:50pm in MUEN D439.
Final Exam Date, Time and Location. Wednesday, December 19, 1:30--4pm in MUEN D439.

Instructor: Dr. William DeMeo
Email: williamdemeo@gmail.com
Office: MATH, Room 202
Office hours: Mondays and Wednesdays, 10:15--11:30am

It is helpful, but not required, to send the instructor an email in advance to notify him of your intention to attend office hours.

## Introduction

You are now reading the main course webpage (which is simply a README.md file in a GitHub repository called math3140-fall2018).
The paragraphs below serve as the syllabus for Math 3140.

For this class the CU D2L system will be used only for recording grades.

## Overview

Basic theory of groups and other algebraic structures, with an emphasis on writing proofs.

• Introduction to algebraic and relational structures.
• Brief introduction to a few special structures: groupoids, semigroups, monoids, graphs, posets, lattices.
• Elementary theory of groups: subgroups, cyclic groups, permutation groups, cosets.
• Isomorphisms, quotient groups, homomorphisms; group isomorphism theorems.
• Group actions, Sylow theorems.
• Introduction to rings (time permitting).

## Learning Outcomes

Develop skill in communicating mathematics. Develop competence in writing proofs. Construct proofs about the following mathematical structures and concepts: groups, lattices, homomorphisms, quotient groups, group actions, rings.

## Prerequisites

The most important prerequisite for this class is Math 2001. If you did well in Math 2001, you should be able to do well in this course. Although you are not expected to be highly skilled at constructing your own original proofs upon entering this course, you should be familiar with basic methods of proof---such as induction, proof-by-contradiction---and what it means to prove a proposition or to prove the negation of a proposition; you should also know what words like "converse" and "contrapositive" mean.

During the first few lectures, we will review some of the required background. If this early material seems unfamiliar and difficult to you, then it is likely that you will find the course very challenging.

## Textbook Information

Abstract Algebra: theory and applications by Tom Judson

## Homework

Homework is worth 30% of the course grade and will be assigned slightly less than once per week. Students must solve many homework problems in order to do well in the course.

Homework is announced in lecture approximately once per week, and is usually due the week after it is announced (with exact due dates specified in lecture).

I will do my best to keep our homework page updated with a list of the problems that are due each week, but it is the students' responsibility to show up for lecture to stay current with the course agenda.

Although many homework problems will be assigned, only a limited number from each assignment will be graded. Therefore, solutions to many of the homework problems will be made available, with the expectation that students will read the correct solutions, compare them with their own, and understand the differences. Students are not required to turn in corrected solutions, but would do well to keep up with the corrections since these can be very useful when preparing for exams.

## Exams

Two Midterm Exams each worth 20% of the course grade will be given during the semester; the exact dates will be announced soon after the course begins.

The Final Exam is worth 30% of the course grade. In accordance with university policy, the final exam must be taken by all students at the scheduled time. There are no make-up exams.

• MIDTERM EXAM 1 (focus: Ch 1--5)
DATE: Wednesday, October 10
LOCATION: MUEN D439

• MIDTERM EXAM 2 (focus: Ch 6, 9--11, 19)
DATE: Monday, November 12
LOCATION: MUEN D439

• FINAL EXAM (cumulative)
DATE: Wedensday, December 19
TIME: 1:30--4pm
LOCATION: MUEN D439

## Make-up Policy

No late homework will be accepted for any reason. If you fail to submit homework on time, this will not necessarily hurt your final course grade since the lowest homework score will be dropped.

Generally speaking, there are no make-up exams. However, if you must miss an exam for one of the legitimate reasons listed below, and if you contact the professor at least five days before the exam date, then you might be able to take a make-up exam before the scheduled exam time.

To request a make-up exam, a student must provide documented evidence of one of the following:

• Documented medical excuse - student's own medical emergency.
• Documented medical excuse - a member of the student's family has a medical emergency.
• Armed forces deployment (military duty).
• Officially mandated court appearances, including jury duty.
• A conflict with another exam or if you have three or more final exams on a given day. (In each case the exam with the fewest students must arrange the make-up exam.)

If you miss an exam due to some unforeseen circumstance, you must contact the professor within one class meeting after the missed test and provide an explanation. If your excuse is accepted, the missed test score may be replaced with 80% of your final exam score. For example, if your excuse is accepted and you score a 90% on the final, then you will receive a 72% for the missed test (0.80*0.90 = 0.72).

## Computing

No prior programming experience is required for this course, and the amount of computing done in this class will be left up to the students. I encourage, but do not require, all students to use the computer to help develop intuition about the algebraic structures that we will study, and I will provide some guidance in this regard.

Sage: One of the best ways to develop a deeper understanding of many mathematical subjects (abstract algebra in particular) is to use the computer to experiment with and apply the theory. For this purpose I recommend the open source math software called Sage. Sage essentially provides a nice browser-based interface to a vast array of well developed and powerful open source mathematical software.

Getting started with Sage is very easy. You don't even need to install any software. By using Sage though a web browser you can and do all your computing, and store all your Sage worksheets, in the cloud.

It's also possible to download and install Sage on your own computer. Sage is free and open source, and it is typically not too hard to install.

Here are some things you might try to get started with Sage:

1. Go to http://www.sagemath.org/ and click Try Sage Online.
2. Check out the web page at http://abstract.ups.edu/sage-aata.html. It is a Sage companion to our textbook!
3. Browse some of the Sage Thematic Tutorials

If you've used Sage before and just need a quick refresher, you might check out the Sage Quick Start Guides.

## Questions and Online Discussions

You are welcome to ask lots of questions in this class (during lecture, in office hours, via email, etc.). You are especially encouraged to use a public forum to ask questions so that all students can benefit. The discussion forum that we will use for this class has yet to be determined, but will likely be Piazza.

Please consider posting to the class wiki pages when you want to ask a question, or raise an issue, or start an online dialog with the prof and/or your classmates. If you have any trouble editing the class wiki pages, please notify the professor.

## Disabilities Statement

If you believe that you have a disability that qualifies under the Americans with Disabilities Act and Section 504 of the Rehabilitation Act and requires accommodations, you should contact the Student Disability Resources Office for information on appropriate policies and procedures. The next step is to talk to the instructor who will be happy to assist with accommodations, but will not provide them retroactively (so file the appropriate requests and paperwork before the first exam!)

## Other References

Our textbook should cover everything we need for this course. However, there are many other books that provide alternative expositions of the same topics, and some are particularly good and might be worth a look. Below are a few of your instructor's favorites, along with some comments about them, but first, here's a list of some free online resources:

Jacobson, Basic Algebra I -- covers roughly the same material as our textbook, but at a higher level of sophistication; used, for example, as a textbook for advanced undergraduates and beginning graduate students.

Herstein, Topics in Algebra -- covers roughly the same material as our textbook, but at a slightly higher level of sophistication; used, for example, in an undergraduate honors abstract algebra course.

Rose, A Course on Group Theory -- We will spend a significant portion of our time in this class studying group theory, and this book is an excellent (and inexpensive) introduction to the subject. Our textbook provides a sufficient treatment of group theory for our purposes, and Rose covers much more than we will. However, you might find Rose's book useful if/when you are not satisfied with our textbook's (or professor's) presentation of a particular topic.

Dixon, Problems in Group Theory -- our homework assignments should give you plenty of practice solving problems in group theory; if you want more, I highly recommend the book by Dixon.

Abstract Algebra, Fall 2018, University of Colorado, Boulder

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