Math 120

Instructor: Prof. Ralph L. Cohen

Office: 383X

Phone: 723-1862

E-mail: ralph@math.stanford.edu

Office hours: Mondays and Wednesdays, 1:15 - 2:30

Course assistant: Bob Hough

Office: 380-G

E-mail: rdhough@math.stanford.edu

Office hours: Mondays, 3:00 - 5:00 pm, Tuesdays, 12:15 - 2:15 pm, Wednesdays, 9:00 - 1:00am.

Class time and location: TTh 9:30 - 10:45 in 380W

Textbook: Dummit and Foote, "Abstract Algebra", 3rd edition (required).

The syllabus is here.

Course Description:

This course introduces basic structures in abstract algebra. They include groups, rings, and fields. Within group theory we will discuss geometric symmetries via group actions, permutation groups, finite abelian groups, p-groups, and the Sylow theorems. Within ring theory we will discuss polynomial rings, principal ideal domains, and unique factorization domains.

Homework policy:

There will be weekly homework assignments. Homework assignmets and their solutions will be posted below. Homework assignments are due in class on each Thursday. No late homework will be accepted.

Writing in the Major:

This course is a Writing in the Major class. This course will emphasize both exposition in communicating mathematics and the structure of proofs. Part of your grade on each assignment and on the exams will be on your exposition of your solutions to problems.

Here is the first writing assignment.

The assignment is due on Tuesday, April 29, in class. Remember that this assignment emphasizes writing, so that the main focus should be on clearly communicating the ideas of the proof clearly and concisely.

Here is the second writing assignment.

The assignment is due on Thursday, May 29, in class.

To help you along in the writing assignments, our class Course Assistant, Bob Hough, who is a PhD candidate in the mathematics department,will be available for help on the assignments, either for mathematical or writing issues. Contact him by email, to arrange for a meeting. Bob is also willing to look at rough drafts of your writing assignments, and to make comments. If you would like him to do that, get your rough draft into him at least a week before the due date of the assignment, so he has time to review it, and so that you will have time to make any changes for your final draft.

It is recommended (but not required) that you type your writing assignments. You can use whatever mathematical word processing program you like, but the standard one, that is used throughout mathematics, statistics, and many areas of science and engineering, is called LaTeX.
This is a version of the “TeX” typesetting program developed by Prof. D. Knuth, here at Stanford. There are various good primers for LaTeX; for example, The Not so Short Introduction to LATeX. Implementations of LaTeX are available for free for all operating systems (Windows, Mac, Unix) . I use a Mac, and my favorite implementation is TeXShop but there are many others. Good luck!

Exam schedule:

The midterm is on Tuesday, May 6, in class.

The exam will cover through section 4.3 of the book, excluding section 3.4 The exam will be closed book.
Make sure you know the definitions of the key concepts discussed in class, such as group, subgroup, normal subgroup, abelian, homomorphism, kernel, group action, stabilizer, quotient group, coset, cyclic group, subgroup of a group G generated by a set S, order of a group, order of an element of a group.
Also know the statements of the main results proved in the class and in the book. When you use them to solve a problem on the exam, you will need to cite the result you are using.

Here is a practice exam

Here are the solutions to the midterm.

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Here is the Final Exam. It is a take home exam. The instructions are on the exam itself. The completed exam should be turned in to Bob Hough at his office, 380-G, by 9:30 am, Monday, June 9.

No late exams will be accepted, and there will be no make-up final exams.

Grading policy

Your final grades will be computed based on the following weighting of your work during the term:
Homework - 20%
WIM Assignments - 20%
Midterm - 20%
Final - 40%

Problem sets

Unless otherwise noted, the page numbers refer to the textbook.

• Problem Set 1, due Thursday, April 10,
  p. 22, #6, 10, 20, 24, 25, 30, 31, 34 p. 27, #2, p. 33 #14, 15, p. 40, #2, 17, 20, p.44 #4, 19
   

• Problem Set 2, due Thursday, April 17,
  p.60, #10, 13 p. 65, #6, 14, 15 p. 71 #7, 12 p. 85 #1, 3
   

• Problem Set 3, due Thursday, April 24,
  p.85, #22a, 33 p. 95, #4, 8, 11, 14 p.101, #3
  Also do the following non-text problem:
  In a perfect shuffle of a 6 card deck, the cards 1,2,...,6 are separated into two equal piles: 1,2,3 and 4,5,6, and then the cards are interlaced, so that the new arrangement is 4,1,5,2,6,3. That is, card 1 moves to position 2, card 2 mioves to postion 4, etc. It is useful to describe this shuffle as a permutation of the nonzero integers mod 7: the map is given by f(j)=2j. Thus for example card 5 goes to position 2x5=3 (mod 7). More generally, in a deck of 2n cards, the shuffle is the permutation of the nonzero numbers mod (2n+1) by the map f(j)=2j.
  (i) Express the perfect shuffle as a product of cycles for decks of 2, 4, 6, and 8 cards.
  (ii) For each number k, what is the largest deck of cards that will return to its original arrangement after k perfect shuffles? (Hint: performing k perfect shuffles of a deck of 2n corresponds to the map that sends j to (2^k)j (mod 2n+1).  

• Problem Set 4, due Thursday, May 1,
  p.111, #7, p. 116 #4, p. 121, #2.
   

• Problem Set 5, due Thursday, May 15,
  p.137-8, #6, 8, 9, p. 146-148, #5, 14, 24, 30, 46.
   

• Problem Set 6, due Thursday, May 22,
  p.156-7, #11, 18.
  Non-text problems to turn in:

  1. How many abelian groups have order 8,100?

  2. Suppose p and q are distinct primes. Suppose G is an abelian group with (p^2)q elements and that G is NOT cyclic.
  (a) What is the largest order of any of the elements of G?
  (b) How many elements of each order are there in G?
  Hint: use the fundamental theorem for finitely generated abelian groups.

  3. Prove that (1 2 3) is not the cube of any element in Sn.

  4. Prove that the group Q of rationals (under addition) is NOT isomorphic to any product AxB (unless A or B is trivial.) In other words, if Q is isomorphic to AxB, then either A or B must have just one element.

  5. (a) Let G be the group of positive rationals under multiplication. Prove that G is isomorphic to the product AxB of two nontrivial groups. (b)* Show that in fact G is isomorphic to GxG.

   

• Problem Set 7, due Thursday, May 29,
  pp.230-233, #11, 13, 29 pp. 247-250, #10, 26