**Course Description** -- MTHT 435 (Section Numbers 23022 and 23025) -- Spring 2014

**Instructor:** Louis H. Kauffman

**Office: **533 SEO

**Phone: **(312) 996-3066

**E-mail: ** kauffman@uic.edu

**Web page: **http://www.math.uic.edu/~kauffman

**Office Hours**: 2:00PM to 3:00PM on Tuesdays or by appointment.

**Course Hours**: 4:30PM to 7:15PM on Tuesdays in 636 SEO.

This is a course on abstract algebra and its applications.

See Text. The textbook is "Algebra-Abstract and Concrete" by Fred Goodman. It is available by download from his website.

**First Assignment:** Obtain a copy of the textbook. Read Chapters 1.,2.,3. Read for the ideas. We will start working with
this material in the second week. Your first written assignment is to organize the material that we covered in class during the
first class, and to write an account of it in complete sentences. You should have some Theorems and proofs in your account. Write it
all in such a way that someone else could read and understand it. If you do not type this 'essay', then please write it neatly and
legibly.

**Second Assignment:** Keep reading Chapters 1.,2.,3.
See Notes1. This is a set of notes and, on the last page,set of problems
for you to write up and hand in on Tuesday, January 28,2014. These problems, on page 17 of the notes, are all you need to hand in. See also
Complex Numbers . This is a longer set of notes on complex numbers and solving
cubic equations and also recursions.

**Third Assignment:** Keep reading Chapters 1.,2.,3.
See Notes2. This is a set of notes and problems. Please hand in problems
1,2,3,4,5 on Tuesday, February 4,2014.
See also
Groups. This is an excerpt about groups from the book "The Skeleton Key
of Mathematics" by D. E. Littlewood. And see Linear Algebra
These are notes on Linear algebra and matrix algebra.

**Fourth Assignment:** Keep reading Chapters 1.,2.,3.
See Notes3.
This is a set of notes and problems. Please hand in problems
1,2,3,4,5,6 (pages 14,15) on Tuesday, February 11,2014.
Reading assignment:
Rotations. This is an expository paper about symmetries an rotations.
Please read the first 8 pages for this week.

**Fifth and Sixth Assignment:** See Notes for Weeks 5 and 6.
These notes contain the problems listed below and a further problem about permutations in relation
to the structure of the multiplication table of a finite group. The notes contain a revision of our
conventions about diagrammatic permuatations so that the order of products is in line with function
composition.
Keep reading Chapters 1.,2.,3.
Use Goodman, Version 2.6 availble from the web.
These problems are overlapping what we have already done.
Some of it is new. Please read relevant parts of the book for more
information if you need it.
Problems Part I:
Page 10: 1.3.1,1.3.2, 1.3.3;
Page 15: 1.4.1;
Page 23: 1.5.3,1.5.4,1.5.5,1.5.6,1.5.8;
Page 73: 1.10.2,1.10.3,1.10.10.
Problems Part II: See the notes above.
Please hand in problems
on Tuesday, February 25,2014.
Reading assignment:
Rotations. This is an expository paper about symmetries an rotations.
Please read the first 8 pages and continue looking at the rest.
Galois. This is an epository account of part of the Galois Theory of
Equations (from Littlewood's book "The Skeleton Key of Mathematics"). It begins an explanation of the
relationship of group theory and solvability of equations by radicals. We will look at this theory
in more detail later in the course.

**Seventh and Eighth Assignment:** See Homework Problems.
These are due Tuesday, March 11, 2014.
See Notes for Weeks 7 and 8. These notes explain how to associate matrices
and permutations to a group multiplication table. The last problem in the assignment is given
on the last page of these notes.
Read Chapter Two of Goodman. We will assign problems from for the ninth week.
Reading assignment continues to include:
Rotations. This is an expository paper about symmetries an rotations.
Please read the first 8 pages and continue looking at the rest.
Galois. This is an epository account of part of the Galois Theory of
Equations (from Littlewood's book "The Skeleton Key of Mathematics"). It begins an explanation of the
relationship of group theory and solvability of equations by radicals. We will look at this theory
in more detail later in the course.

**Ninth Assignment and Tenth Assignment:**
See Notes for Week 9. These notes contain the first part of the problems.
Hand in problems 1 to 7 from pages 3,11 and 12 of the notes. Continue the reading from
the last assignment. ERRATUM: In exercise 4, page 11, it should read "Show that p prime implies that
phi(p^{n}) = p^{n} - p^{n-1}." Here phi is the Euler phi function as described in the problem.
We add the following to the assignment:
(a) Read page 66 and some before and after to allow you to understand the
proofs of Corollary 1.9.17 and Proposition 1.9.18 (whose proof is mostly
given in the middle of page 66). Once you have understood this, write a
self-contained explanation of these two results in your own words. Include
applications to the analysis of some examples of U_{n} when n is
composite.
(b) Solve exercises 1.9.10, 19.11, 1.9.12 on pages 68 and 69.
(c) Read pages 75-79 and in reading Proposition 1.11.7, look up the
earlier proof of the Chinese Remainder Theorem (Proposition 1.7.9). Take
note of the definitions of Ring and Field. Do the following exercises on
page 79: 1.11.1, 1.11.6, 1.11.11.
(d) Start systematically reading Chapter 2 (pages 85 - 134). Understand
the proofs and constructions. Take notes. Work on examples. Compare what
is happening in this chapter with the many examples of groups that you
already know. There is no request to write up and hand in problems
relative to this chapter yet. The assignment is to work through the
chapter on your own and then in April we will work on it together.
The due date of the present assignment is Tuesday, April 1,
2014.

**Eleventh Assignment:**
(a) Read page 66 and some before and after to allow you to understand the
proofs of Corollary 1.9.17 and Proposition 1.9.18 (whose proof is mostly
given in the middle of page 66). Once you have understood this, write a
self-contained explanation of these two results in your own words. Include
applications to the analysis of some examples of U_{n} when n is
composite. In particular, answer the following questions.
(i) Show that U_{18} is a cyclic group of order 6.
(ii) Determine the number of elements in the group U_{221}.
(b) Read the entire set of notes about finite subgroups of the rotation group.
Rotations.
(c) Start systematically reading Chapter 2 (pages 85 - 134). Understand
the proofs and constructions. Take notes. Work on examples. Compare what
is happening in this chapter with the many examples of groups that you
already know. There is no request to write up and hand in problems
relative to this chapter yet. The assignment is to work through the
chapter on your own and then we will work on it together.
The due date of the present assignment is Tuesday, April 8,
2014.

**Twelveth Assignment:**
(a) Here is a set of notes on quaternions from my book "Knots and Physics".
Quaternions.
Please read pages 403-409, 414-416.
Note Proposition 10.4 is what we proved in class about representing
rotations by quaternions.
Skip 416-423 or read for fun. We will discuss it in class!
Start reading again at bottom of 423 where it says "Return to Proposition
10.4". And read 423- 425. This covers what we did in class and you will
see an example of the composition of two 90 degree rotations at the end of
this reading. You can read the rest of the notes for fun. We'll discuss
them in class. Note that I use e^{u theta} to mean cos(theta) + sin(theta)u where u is a
unit length pure quaternion u = u_1 i + u_2 j + u_3 k where u_1, u_2, u_3
are real numbers with the sum of their squares equal to 1.
Now do the problems in the following problem set.
QuaternionProblems.
(b) Read the entire set of notes about finite subgroups of the rotation group.
Rotations.
(c) Start systematically reading Chapter 2 (pages 85 - 134). Understand
the proofs and constructions. Take notes. Work on examples. Compare what
is happening in this chapter with the many examples of groups that you
already know. There is no request to write up and hand in problems
relative to this chapter yet. The assignment is to work through the
chapter on your own and then we will work on it together.
The due date of the present assignment is Tuesday, April 15,
2014.

**Thirteenth Assignment:**
(a) Here is a set of notes on quaternions from my book "Knots and Physics".
Quaternions.
Please read pages 403-409, 414-416.
Note Proposition 10.4 is what we proved in class about representing
rotations by quaternions.
Skip 416-423 or read for fun. We will discuss it in class!
Start reading again at bottom of 423 where it says "Return to Proposition
10.4". And read 423- 425. This covers what we did in class and you will
see an example of the composition of two 90 degree rotations at the end of
this reading. You can read the rest of the notes for fun. We'll discuss
them in class. Note that I use e^{u theta} to mean cos(theta) + sin(theta)u where u is a
unit length pure quaternion u = u_1 i + u_2 j + u_3 k where u_1, u_2, u_3
are real numbers with the sum of their squares equal to 1.
Now do the problems in the following problem set. This continues the last assignment.
QuaternionProblems.
(b) Read the entire set of notes about finite subgroups of the rotation group.
Rotations.
The due date of the present assignment is Tuesday, April 22,2014.

**Fourteenth Assignment:**
DiagramsMatrices. These are rough notes on diagrams, matrices,
crossproducts and the associativity of quaternion multiplication.
AssignmentNotes. This is your assignment for this week, plus some notes
related to last class and the next class.
The due date of the present assignment is Tuesday, April 29,2014.

BELOW THIS POINT ARE LINKS TO TOPICS OF INTEREST POSSIBLE INTEREST.

See Peano Arithmetic for an approach to axiomatic arithmetic via the Peano axioms for the natural numbers. In these notes we use nested marks (right angle brackets) to represent natural numbers. These notes will evolve. The present version just presents the axioms and some commentary.

See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. We will discuss the problems in class after learning more about induction.

See Geometry This is a one page summary of some remarks that we made about geometry in class.

See Conway's Sprouts. This is an excerpt from the book "Winning Ways for Your Mathematical Plays" by Conway, Berlekamp and Guy. It contains information about the game of sprouts.

See Proof This is a remark about proofs in mathematics.

See Four Knights Here is a Java Applet for an interesting puzzle.

See ContFrac. This is a calculator on the web that converts numbers to their continued fracations. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See PenroseTriangle This is an excerpt from an article by Penrose and some other writing, for class discussion.

See Proof of Pythagorean Theorem. This is a new proof recently published in the American Mathematical Monthly.

See The Library of Babel for a short story by Jorge Luis Borges.

See UMG. This is the website for the Undergraduate Mathematics Journal. It a good source of interesting articles, and the NEXT time we teach this course SOME people will submit papers to this journal! See Old Hats for a good example of a paper published in the Journal.

See Conway's Army for a short article explaining a solitaire game and a proof about its limitations.

See Existence for an existence proof and a story related to the question whether existence proofs should exist.

See Desargues for a short proof of the three-circles theorem using the Desargues configuration. This is an elegant sample of projective geometry.

See Euler's Mathematics This is an excerpt from a recent book introducing proofs and mathematical ideas.

See Euler's Formula. This is a 2-page description of Euler's formula V - E + F = 2 for connected plane graphs. Euler's formula is interesting in its own right and it is a powerful tool for solving many combinatorial problems.

**Recommended Reading: ** Logicomix, Bloomsbury Press, by Apostolos Doxiadis and
Christos H. Papadimitriou. This book is available via
Amazon. Logicomix is a graphic novel about the search for foundations of mathematics at the hands of
Bertrand Russell, Gotlob Frege, David Hilbert, Kurt Goedel, Ludwig Wittgenstein and others, set in
the context of a political lecture by Bertrand Russell at the beginning of World War II.

Recommended Reading: The No Sided Professor. This is a topological short story by Martin Gardner.

Recommended Reading: The Feeling of Power. This is a short story by Isaac Asimov.

Recommended Reading: InfiniteHotel. A cartoon story about an infinite hotel that can seemingly accomodate any new influx of guests.

Recommended Reading: Math SciFi. "And He Built a Crooked House", by Robert Heinlein.

See Boolean Notation. A short article about Boolean Algebra notation.

See Peirce. A excerpt from the writings of Charles Sanders Peirce, American philosopher and mathematician (1839-1914). In this article, Peirce gives his "sign of illation", a symbol for implication that is a combination of the Boolean negation sign and the Boolean plus sign. He also has a worthwhile discussion of the nature of logic.

See Boolean Algebra. A short article about Boolean Algebra and some ideas related to it, including switching circuits, Laws of Form, recursions and circuits that count.

See Set Theory. This is an appendix from the book "Topology and Geometry" by Glen Bredon. The appendix is a concise but complete synopsis of basic set theory and goes beyond what is in Eccles. In particular, you will find discussion of well-ordering of sets, and in Theorem B18 the equivalence of a number of set theoretic principles such as well-ordering and the axiom of choice. You will also find a proof of the Cantor-Schroeder-Bernstein Theorem (B15 and B20). You do not need to read this whole appendix, but I DO ask you to look at Theorem B27. There you will find a beautiful proof that the real numbers have the same cardinality as P(N) where N is the natural numbers. The proof includes a proof that the set of finite subsets of N is countable (can you give an independent proof of this fact?). The proof of Theorem 27 uses the concept of continued fractions.

See Wang Algebra for a clever approach to graphs and spanning trees.