Math 366: Introduction to Abstract Algebra



Renzo's class



MWF 10 – 10:50 am



ENGRG  E 103  // in the oval with good weather









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





Exercises in preparation for baby exam (pdf) (tex file here if you want)


Rings and Fields


Exercises in preparation for exam 2 (pdf) (tex file here if you want)




Baby Exam

Solution to Problem 6




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!



Jan 27

Exercises 1, 2, 7, 9, 18, 19 pages 18-20.

Additional exercise: prove that, given two sets A,B

A is a subset of B if and only if the intersection of A with the complement of B is empty.

Write a paragraph to explain how this fact relates to proving a theorem of the form
If hypoteses H are verified, then thesis T follows                                                             by contradiction.

Solution to problem 19

Feb 3

Exercises 22, 26 pages 18-20 are recommended as preparation for the exam, together with the worksheet.

Baby Exam

Feb 10

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.




Feb 17

Exercises  8, 12 page 32-33

1) Prove by induction the following statement: the sum of all internal angles of a convex polygon with n-sides is (n-2)180 degrees.

(note: in this case the statement makes sense only for n greater than or equal to 3, and therefore your base case should be n=3).

2) Let X be a set with n elements. Given three integers a,b,c such that a+b+c=n, how many ways can you subdivide X into three disjoint subsets with a,b and c elements?

(I am looking for an answer in terms of products and quotients of factorials, just like in case of the binomial coefficient).

3) Challenge: can you generalize problem 2) the following way: X a set with n elements. a_1, a_2, …, a_k  k integers adding up to n. How many ways can you subdivide X into k subsets S_i such that the cardinality of S_i is a_i. This number is called the multi-nomial coefficient n choose a_1, …, a_k.  As an extra challenge: can you figure out how Newton’s binomial theorem generalizes to a “ Newton multi-nomial theorem”?

(If you tackle the extra challenge. First make a guess to what the formula should be. Verify it in for some small values of n and k. Then try to prove it either by induction or just by explaining why the coefficients are what they are).


Feb 24

Exercises 18, 23, 25, 31 page 34

Mar 2

Answer Questions 1,2,3,4,5 on the handout “Rings and Fields

Mar 9

Study for the midterm! Midterm today!

Mar 23

1)If you are born in month “x < 12”, do problem x on the handout “Rings and Fields”. If you are born in December, then do the challenge (section 4.4)

2) If you are born in day x, then do problem [x] mod 11. If that turns out to be the same number as in 1), then pick a second problem of choice.

Mar 30

Exercises 7, 10, 24, 29, 30, 38, 44, 45, 47, pages  53-55.

Apr 6

On the online version of the book exercises:  1-(c),(d) [to make this question non-tautological, you need to insert the adjective non-trivial between “every” and “subgroup”],(e), 2 – (a),(b), 11, 14, 23, 29, 30. 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 G  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>.


Apr 13

On the online version of the book exercises: 1, 2, 8, 9, 18, 23, 27, 31, 33, 35 pages 90-93.


Apr 27

Exercises 1(a),(b),(e), 5, 6, 14 (a),(c),(d) pages 163-164.





  • Baby Exam: Feb 3rd
  • First Midterm: Mar 9th
  • Second Midterm: Apr 18th  Note the date has changed! This is the new and correct date for the last midterm!                      
  • Final: May 10th, 7:30-9:30 am.