Math 366: Introduction to Abstract Algebra



Renzo's class



MWF 12 – 12:50 am



ENGRG  E 105  // in the oval with good weather









Office Hours






Abstract Algebra: Theory and Applications
 by Thomas W Judson

Publisher: Virginia Commonwealth University Mathematics

available freely from:








If you want to know why you should study abstract algebra, I think the best answer is: to train your brain to think! Just how when you do push-ups at the gym, your goal is to get stronger, not just to perform *the perfect push-up*. However, if you stick with abstract math long enough (a lot longer than this one semester course) – then you will find that there are powerful applications of abstract algebra. Below is a small compilation of a few of them, if you are curious!


Applications of Group Theory




Worksheet on Rings (Tex file)


First midterm coming up Oct 4th!




PDF file

LaTeX source file



Office hours : the official office hours for this class are right after class. However, if these times are not convenient for you, you are very welcome to make an appointment and come ask questions, make comments, or just chat. You can also try showing up at my door anytime. But I might tell you to come back at another time if I am immersed into something else.

The tables of the law for this class are contained in the following document:


Homework:  math is not a sport for bystanders. Getting your hands dirty  is important  to make sure that things sink in and you are not just spending a semester assisting to my creative rambling. Homework will be due pretty much every Friday (but see below for the up-to-date information). I will NOT grade all the problems I assign. If one of the problems you feel unsure about doesn’t get graded, please don’t go “Whew! Lucky one!” but rather come ask me about it. The point is you understanding, not pretending to!!

Important: the number of exercises and pages are mismatched between the electronic and paper versions of the book. I will always refer to the electronic version available following the link above (latest edition), since not everyone owns the paper version of the book!



Wednedsday, Sep 4

READ CHAPTER 1 in the book and HAVE QUESTIONS for class. Wednesday’s class may be entirely devoted to answering basic questions.

Friday, Sep 6

Homework 1 (Tex File)

Friday, Sep 13

1) Write down in full detail  and paying great care to your exposition two different proofs for the identity:

1+2+3+…+n= ½ n(n+1)

2) Compute the sum of the first n squares in two different ways:

First, make a wild guess that the answer is a degree 3 polynomial in n

1+4+9+…+n^2= An^3+Bn^2+Cn +D

Plug in values for n to determine A,B,C,D by solving a linear system of 4 equations in 4 variables. Then prove the formula by induction.

In the second case, use the identity

(n+1)^3= n^3+3n^2+3n+1

to generalize to this case one of  the proofs seen in class for the sum of  the first n integers.

This week’s hwk is only two problems, so please put in a good effort to attempt to do them well!! This means that your writeup should be complete, tidy, well expressed, and in one of the languagues I can understand!


Friday, Sep 20

Exercise 12 page 33

Homework 2 (TeX file)

Friday, Sep 27

Exercises 18, 19, 23, 25, 31 page 34

Friday, Oct 4th

No homework. Study for the exam!

Friday, Oct 11th

Write up the answers to the following problems from the handout on Rings


Question 4 Sec 2.5

Theorem 2 Sec 3.3

Theorem 3 Sec 3.3

Theorem 5 Sec 4.1 (Careful! Make sure not to use theorem 2 in the proof of theorem 5!)


Friday Oct 18th

Write up the answers to the following problems from the handout on Rings


Problems 6,7,8 Section 4.2

Problems 10,11 Section 4.3

Friday Nov 1st

Exercises 7, 24, 30, 38, 44, 45, 47, 49, 54 pages  53-55

Friday Nov 8th

Exercises 14, 23, 26, 29, 30, 31, 37. Pages 71-74.


(1)   Give a formula for the order of a general element [m] in Z/nZ. Proceed by first experimenting with small values of n until you understand what is going on. Then write a general proof.

(2)   Complete and prove the following statement. There exists a group homomorphism f: Z/nZ à  sending [1]à g if and only if the order of g…(fill in). If the order of  g …(fill in), then f is an isomorphism between Z/nZ and <g>.



Monday Dec 2nd

Cayley’s Thm (TeX file)

Friday Dec 6th

Exercises 4, 5, 6, 8, 14 pages 166-167


Challenge: 7,9 page 167.





  • First Midterm: Oct 4th           
  • Second Midterm: Nov 13th             
  • Final: Dec 19th 4-6pm