Math 120 
Spring 2006

Announcements:

  • Additional office hours have been scheduled for the days predating the final: Sunday June 11th 5-7pm and Monday June 12th 3-5pm. Adrian will be available in 382 J.
  •  Ju-jong has kindly agreed to write solutions for the practice questions set. They can be found here.
  • Adrian will not hold his Tuesday & Thursday office hours next week. Additional office hours are to be scheduled for Sunday (June 11th) and Monday (June 12th). Times are to be announced. 
  • The final exam is scheduled for Tuesday June 13th, 7-10pm in room 380W. Practice questions are available here. These questions concern the second half of the course material. They will be discussed during the review sessions to be held in class on June 6th. For the topics of the first half, one should review the midterm practice questions
  • For the week of May 22nd only, Yu-Jong's office hours will be Tuesday and Wednesday 7-9pm.
  • Extra office hours are scheduled for the days predating the midterm: Adrian (Monday 5-7pm), Yu-Jong (Sunday 5-6pm and Monday 11-noon).
  • The midterm is scheduled for Tuesday May 9th, in class. Practice questions are available here (solutions for #13 and #15 are available here).
  • Graded assignments can be collected from the tray located next to the door of office 382J.
  • Note that the midterm date has been changed. The test will take place on Tuesday, May 9th, during the regular class time
  • The first day of class is Tuesday, April 4th. The class meets in room 380W located in the basement of the building 380.

Course Description

This course provides an introduction to the basic structures in algebra: groups, rings, and fields. The main topics to be discussed are: elements of group theory: permutation groups, finite abelian groups, p-groups, Sylow theorems, polynomial rings, principal ideal domains and unique factorization domains.

 

This course emphasizes both exposition in communicating mathematics and the structure of proofs. Math 120 satisfies therefore the writing in the major course requirement.

Instructor

Name

Office

E-mail

Phone

Office Hours

Adrian Clingher

380-382J

clingher@math.stanford.edu

723-2438

Tuesday 4-5pm, Thursday 5-6pm

Course Assistant

Name

Office

E-mail

Office Hours

Yu-Jong Tzeng

380-L

yjt@math.stanford.edu 

Tuesday 4-6pm, Wednesday 1-3pm

Course Time and Location

Tuesday and Thursday 1:15 - 2:30pm, 380-380W.

Text

Abstract Algebra by  David S. Dummit and Richard M. Foote

Exams and Homework 

There will be a midterm (Thursday, May 4th, during class time) and a final exam (Tuesday, June 13th, 7-10pm). The course grade is to be computed on a cummulative basis (30% midterm + 30% homework + 40% final). Weekly homework assignments will be posted here and are due each Thursday. The first assignment is due April 13th.

Homework Assignments

Assignment 1 (due April 13th):  page 22, questions 9,12 and 15; page 28, questions 6 and 15;  Solutions.

 

Assignment 2 (due April 20th):  page 33, questions 1 and 2; page 34, questions 20; page 40, questions 4,5 and 6.  Solutions.

 

Assignment 3 (due April 27th):  page 48, questions 2, 3, 4 and 10; page 60, questions 1, 2 and 3 ( in question #3 on page 60, |x| means the

first positive power of x which realizes the neutral element)  Solutions.

 

Assignment 4 (due May 4th):  page 44-45, questions 2,3 and 16; page 60, questions 10, 11; page 95, questions 8. (Hint for the last question: use the fact that if (m,n)=d, then there exists integers a, b such that d=an+bm.) Solutions.

 

Assignment 5: No assignment for this week.

 

Assignment 6 (due May 18th): page 85, question 3; page 88, questions 22 and 24; page 96, question 18. Solutions.

 

Assignment 7 (due May 25th): page 101, questions 3 and 7; page 165, question 2(a)(c), page 166 questions 4(a)(b) and 9. Solutions.

 

Assignment 8 (due on or before Tuesday June 6th): Write a short expository essay (2-3 pages, typed or handwritten) on one of the following three topics:

  • Lagrange's Theorem

  • Burnside's Lemma

  • Sylow's Theorems.

 

For the first topic, you should discuss first the notion of action as well as the associated concepts (orbit, stabilizer group, etc). Present

then the statement and detailed proof of Lagrange's Theorem and discuss its importance in group theory. For the second topic, you should

give the exact statement and proof of Burnside's Lemma and discuss in details at least one counting application of it. For the third topic,

you should start by introducing the notions of p-subgroups and Sylow p-subgroups. State the three Sylow theorems and present a

detailed proof for one of them. Give then an example of an application of the Sylow's theorems to the classification of finite groups.     

 

This is a writing assignment, so the main focus should be on clearly communicating the ideas in the proof. I recommend looking at the

textbook (or your favorite mathematics texts) and trying to emulate their style.