Math 113: Abstract algebra

UC Berkeley, Spring 2003


Michael Hutchings
Office phone: 510-642-4329.
Office: 923 Evans.
Office hours: Tuesday 9:30 - 11:30, 1:00 - 2:00 (subject to change or temporary rescheduling), or by appointment.
Office hours the week of 5/19 to 5/23: Tuesday, Thursday, and Friday, 3:00 to 5:00 PM.

Graduate student instructor

In addition to my office hours, Eli Grigsby will hold office hours for all Math 113 students, in room 891 Evans, on Wednesdays from 10-3pm and Thursdays from 8-11am and 4-6pm.

Spacetime coordinates

The course meets in 70 Evans on Tuesdays and Thursdays at 8:00 AM.

Course goals

In previous courses you have seen many kinds of algebra, from the algebra of real and complex numbers, to polynomials, functions, vectors, and matrices. Abstract algebra encompasses all of this and much more, and delves deeply into questions of ``what is going on'' with algebra in general. Roughly speaking, abstract algebra studies the structure of sets with operations on them. We will study three basic kinds of "sets with operations on them", called Groups, Rings, and Fields.

A group is, roughly, a set with one "binary operation" on it satisfying certain axioms which we will learn about. Examples of groups include the integers with the operation of addition, the nonzero real numbers with the operation of multiplication, and the invertible n by n matrices with the operation of matrix multiplication. But groups arise in many other diverse ways. For example, the symmetries of an object in space naturally comprise a group. The moves that one can do on Rubik's cube form a very interesting example of a group. After studying many examples of groups, we will develop some general theory which concerns the basic principles underlying all groups.

A ring is, roughly, a set with two binary operations on it satisfying certain properties which we will learn about. An example is the integers with the operations of addition and multiplication. Another example is the ring of polynomials. A field is a ring with certain additional nice properties. At the end of the course we will have built up enough machinery to prove that one cannot trisect a sixty degree angle using a ruler and compass.

In addition to the specific topics we will study, which lie at the foundations of much of higher mathematics, an important goal of the course is to develop facility with mathematical reasoning and proofs in general, as a transition to more advanced mathematics courses, and for logical thinking outside of mathematics as well.


The textbook for this course is John B. Fraleigh, A first course in abstract algebra , 7th edition, Addison-Wesley. This book is very readable, has been well liked by students in the past, and contains lots of good exercises and examples.

Most of the lectures will correspond to particular sections of the book, and studying these sections should be very helpful for understanding the material. However, please note that in class I will often present material in a different order or from a different perspective than that of the book. We will also occasionally discuss topics which are not in the book at all. Thus it is important to attend class and, since you shouldn't expect to understand everything right away, to take good notes.

There are many other abstract algebra texts out there, and if you want some additional perspectives you might try browsing through these in the library or your favorite bookstore.


It is recommended that you obtain a 3x3 Rubik's cube, if you do not already have access to one, e.g. from This will be used to illustrate some group-theoretic concepts in a fun, hands-on way.

More about Rubik's cube

Rubik's cube has been extensively studied by cube enthusiasts over the last 25 years, and I recently discovered this webpage which has links to enormous amounts of amazing information about it. For example, section 6 of that webpage includes links to computer programs which in a matter of days can solve a scrambled cube with the absolute minimum number of turns, and which within seconds can come within a couple of turns of the minimum! As far as I know the minimum is always at most 22 or 24 turns (depending on whether you count a 180 degree turn of a face as one turn or two), and usually a bit less, much shorter than human algorithms for solving the cube which may require 60 turns or more. There is a beautiful trick using a little group theory (namely cosets, which we will learn about) which allows the computer to quickly find a nearly minimal solution, and also to dramatically prune the search tree in order to find a minimal solution in a reasonable amount of time.

Notes on how to write proofs

I put together some old notes which give a very basic introduction to mathematical proofs. Here they are: pdf postscript.

Approximate syllabus

The following is a rough, tentative plan for the topics we will cover. What we actually end up doing in class will be listed below after each lecture.

What we actually did in class

Homework assignments

Homework policy

Homework will be due on Thursdays at 8:10 AM sharp. You can either bring it to class or slide it under my office door. (If it doesn't fit under the door, please be more concise!) Assignments will be posted below on the preceding Thursday night. No late homeworks will be accepted for any reason, so that we can go over the homework problems at the beginning of Thursday's class. However it is OK if you miss the deadline once or twice, because your lowest two homework scores will be dropped.

When preparing your homework, please keep the following in mind:

1) You are encouraged to discuss the homework problems with your classmates. Mathematics can be a fun social activity! Perhaps the best way to learn is to think hard about a problem on your own until you get really stuck or solve it, and then ask someone else how they thought about it. However, when it comes time to write down your solutions to hand in, you must do this by yourself, in your own words, without looking at someone else's paper.

2) All answers should be written in complete English sentences which explain the logic of what you are doing, with mathematical symbols and equations interspersed as appropriate. For example, instead of writing "x^2 = 4, x = 2, x = -2", write "since x^2 = 4, it follows that x = 2 or x = -2." Results of calculations and answers to true/false questions etc. should always be justified. Proofs should be complete and detailed. The proofs in the book provide good models; but when in doubt, explain more details, and resist the temptation to use phrases such as "it is easy to see that...". You can of course cite theorems that we have already proved in class or from the book.

Exams and grading

There will be in-class midterms on 2/27 and 4/10, and a final exam on 5/24.

Make-up exams will be allowed only in extreme circumstances, and may be in a different format such as an oral exam.

Please note that exam grades cannot be changed unless there is an egregious error such as adding up the points incorrectly. Every effort is made to grade all exams according to the same standards, and regrading one student's exam would not be fair to everyone else.

The course grade will be determined as follows: homework 30%, midterms 15% each, final 40%.