The text will be Dummit and Foote Abstract Algebra, Third edition. We will cover group theory (through the Sylow theorems), and beginning ring theory. Although groups are more "basic" algebraic objects, rings are also pervasive and useful even in thinking about rings. I will talk about rings from an early stage, and I recommend that you read Section 7.1 in the book early.
I'll be having office hours Monday and Tuesday December 10-11 from 11AM - 2PM.
Daniel Bump (383U on third floor) is the Professor. My office hours (subject to change)
Adva Wolf (381L on the first floor) is the Course Assistant for Math 120. Her office hours are:
Since the writing assignment is coming up, here are some notes about writing mathematics. They use the above writing sample Normal Subgroups and Homomorphisms as an example.
You may discuss the writing assigment with each other, or with me or Adva Wolf, but you may not show another student your written work.
Much of this is in the book, and particularly Proposition 2 on page 114 is relevant. Proposition 1 below is closely related to it, but more precise than what Dummit and Foote prove so you can't just copy their argument. And you will find proofs of the other facts in the book (or in the lectures). However write your own proofs, striving for clarity.
Read the above essay on Writing Mathematics, then prepare your project including the following three results. You may (optionally) include some expository material explaining why they are important, and then include proofs of the following results. Define all terms that you need in the proof. In particular, define the terms orbit, stabilizer and transitive. Also define the following term: if $G$ is a group acting on sets $X$ and $Y$ then the actions are equivalent if there is a bijection $\phi:X\to Y$ such that for $x\in X$ and $g\in G$ we have $\phi(g\cdot x)=g\cdot\phi(x)$.
A particular action: let $H$ be a subgroup of $G$ (not necessarily normal) and let $G/H$ denote the set of left cosets $x H$. Then $G$ acts on $G/H$ as follows: $g\cdot x H= gx H$.
Theorem 1. (The Orbit-Stabilizer Theorem). Let $G$ be a finite group acting transitively on a set $X$. Let $a\in X$ and let $H$ be the stabilizer of $a$. Then the action on $X$ is equivalent to the action of $G$ on $G/H$.Now let $G$ be a finite group acting on itself by conjugation. Using Theorem 1, prove:
Proposition 2. If $G$ is a finite group and $x\in G$ then the number of conjugates of $x$ in $G$ divides the order of $G$.
Let $p$ be a prime. A group $G$ is called a $p$-group if $|G|$ is a power of $p$. Use Proposition 2 to prove:
Proposition 3. Assume that $G=p^k$ for $k>0$ and let $Z$ be the center of $G$. Then $|Z|$ is a multiple of $p$.
The final exam will be on December 12 from 12:15-3:15 in our usual classroom 380W. Mainly the final will cover material from the sections where homework was assigned, plus two topics from Sections 9.3 and 9.4: Gauss's Lemma and Eisenstein's criterion. This material will be touched on in the final and is essential for Math 121.
Know the statements and logical relationships of Section 9.3: Proposition 5, Corollary 6, Theorem 7 and Corollary 8; and in Section 9.3, know the statments of Propositions 9 and 10 (both of which are actually obvious but worth thinking about) and Proposition 13 (Eisenstein's criterion).
There was no homework in Chapter 9, but as practice problems you might try 9.3 #2 and 9.4 #2.
Homework will be due Wednesdays. (This is not a class day.) You may turn in written homework or (optionally) you may email it to me if you use latex. If you want to email me homework, use the address bump at math dot stanford dot edu and put Math 120 in the title of the email. Send the pdf file. Please do not send me scans of handwritten work.
|Homework||Dummit and Foote||Solutions|
|HW1 (due Wed. October 3)||
|HW 1 Solutions|
|HW2 (due Wed. October 10)||
|HW 2 Solutions|
|HW3 (due Wed. October 17)||
|HW 3 Solutions|
|HW4 (due Thursday. October 26)||
|HW 4 Solutions|
|HW5 (due Wednesday October 31)||
|HW 5 Solutions|
|Week of November 8: WIM Project|
|HW6 (due Wednesday November 14)||
|HW 6 Solutions|
|HW7 (due Wednesday November 28)||
|HW 7 Solutions|
You are encouraged to use
If you follow the notes, you will create a file called
Homework that is turned in after I've given the homework to the grader each week might not get graded. If you have a valid reason to miss a homework (e.g. illness) please let me know.
The midterm will by Tuesday October 23 (in class). It will cover Dummit and Foote through Section 3.2.
Grading will be based on Midterm (30%) WIM project (25%) Final (35%) Homework (10%).
There will be one in-class midterm, the final, and a Writing in the Major (WIM) project. I will grade the writing project myself. It will be assigned right after the midterm. I will read your projects and write comments over the weekend. After I return them to you on Tuesday May 11 you will have 8 days to revise them and return them to me.
|Midterm||Tuesday October 23|
|WIM Project Assigned||Thursday October 23|
|WIM Project Due||Thursday November 8|
|WIM Projects Returned||Tuesday November 13 (I hope)|
|Revised Projects Due||Tuesday November 27|
|Last Lecture||December 6|
|Final Exam||will be posted here when announced by registrar.|