# Math 120: Groups and Rings (Fall 2018)

This class will meet 10:30 AM on Tuesdays and Thursdays in room 380W.

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.

## Finals Week Office Hours

I'll be having office hours Monday and Tuesday December 10-11 from 11AM - 2PM.

## Office Hours

### Daniel Bump

Daniel Bump (383U on third floor) is the Professor. My office hours (subject to change)

• Mondays at 1:30-2:30
• Tuesday at 3:30
• Wednesday at 2:30-3:30
• Thursdays at 12 noon. (Not on Sept. 27 or October 4)
(Other times if I'm free and in my office.)

Adva Wolf (381L on the first floor) is the Course Assistant for Math 120. Her office hours are:

• Tuesday: 4:30 - 5:30, except on December 4.
• Tuesday December 4: 5:30 - 6:30.
• Thursday: 1:15 - 4:15

## Writing Mathematics

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.

## Writing Assignment (Due Thursday November 8)

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 $p$.

## Final Exam

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

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.

HomeworkDummit and FooteSolutions
HW1 (due Wed. October 3)
 Section 1.1 #6,12,15,22,25; Section 1.2 # 1,2,3; Section 7.1 # 3,5.
HW 1 Solutions
HW2 (due Wed. October 10)
 Section 1.2 # 9; Section 1.3 # 2, 5, 13, 20; Section 1.4 # 7; Section 1.5 # 2; Section 1.6 # 1.
HW 2 Solutions
HW3 (due Wed. October 17)
 Section 1.7 # 11, 17, 19; Section 2.2 # 7, 10; Section 2.3 # 17, 25; Section 3.1 # 9, 32, 33;
HW 3 Solutions
HW4 (due Thursday. October 26)
 Section 3.1 # 34, 36, 42; Section 3.2 # 8; Section 3.3 # 2, 3; Section 3.5 # 1, 7;
HW 4 Solutions
HW5 (due Wednesday October 31)
 Section 4.1 # 1; Section 4.2 # 2, 9; Section 4.3 # 4, 8, 28, 34;
HW 5 Solutions
Week of November 8: WIM Project
HW6 (due Wednesday November 14)
 Section 4.4 #2, 13 Section 4.5 #13, 25 Section 7.3 #25, 26, 34 Section 7.4 #4, 5 Section 7.5 #3
HW 6 Solutions
HW7 (due Wednesday November 28)
 Section 5.4 # 12 Section 5.5 # 7,8 Section 7.4 # 37 Section 8.2 # 1,4,5 Section 8.3 # 2
HW 7 Solutions

## Latex

You are encouraged to use latex to do your writing assignment and homework.

If you follow the notes, you will create a file called conjugation.tex, which you can download from the link or type it yourself. Running pdflatex on this input will produce a pdf file called conjugation.pdf. Once you know how to do this you are ready to use latex. If you want further instructions, you may find some tutorials at http://www.latex-project.org/guides/.

## Late Homework

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.

## Midterm

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%).

## Timetable

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.