## Math 516 Fall 2007

CRN 13724,  11:00-11:50 MWF Adams Hall 306
Office Hours: MWF 10-11
Bhama Srinivasan , 526 SEO, 3-2160, srinivas@uic.edu
Department of Mathematics, Statistics, and Computer Science

Revised August 15, 2007, by BS

## Description

516 Abstract Algebra I
4 Hours.

Prerequisite: Math 330 and Math 425 or equivalent, or permission of the instructor.
Text: Dummitt and Foote, Abstract Algebra, Prentice Hall

This is the basic graduate algebra course, the first course in a sequence of two courses.
The courses are designed to cover the basic topics in algebra which would be needed
for a student who intends to pursue research not only in algebra but in various other areas.
For example, elements of commutative algebra which are needed for algebraic geometry
are covered in the second course.

The topics to be covered are:
Groups: Group actions, Sylow theorems, normal series, presentations
Chapters 1-5, 6.3 (omit 2.5, 4.6, 6.2, and parts of 5.5, 6.1)

Rings and Modules: Ideals, chain conditions, polynomial rings, modules over principal
ideal domains
Chapters 7-12

Grading: There will be homework assignments every week, as well as a final exam.

Homework solutions are posted on the web. See http://uic.docutek.com/eres/ and follow the

In the homework problems, use only the material covered so far. The course is a
"narrative", covering the material in a certain order. Using material not covered
creates confusion, to say the least. The text is a crutch, not gospel. It is useful to point
out what has been covered so far, by referring to the text.

The sections covered so far are: Chapter 0: 0.1-0.3

## Homework

Practice: p.22, #21, 22, 33; p.28, #6; p.33, #10, 11, 15;  p.35, #1,2,3
Homework 1 (Due 8/31) p.28, #4, 5; (Due 9/5): p.33, #14; p.35, #7; Extra problem on Induction;

Homework 2 (Due 9/7): p.35, #11 (omit (c) ); p.66, #9 (Hint: Linear algebra, echelon form)

Homework 3 (Due 9/14): p.48, #6; p.61, #12, 13, 26 (Practice: ; p.49, #16)

Hint for p.48, #6: Group of  invertible 2x2 matrices

More on p.66, #9: To complete the discussion in class today, we need to show

the diagonal matrix D with entries a, a^{-1} on the diagonal is a product of elementary

matrices. I did this only for a=-1, sufficient for F_3, and asked you to think of the

general question. Here is a hint.  D is a product of 4 elementary matrices, lower triangular, upper

triangular, lower triangular, upper triangular. Try it!

Homework 4 (Due 9/21): p.65-66, #12, 18; p.86-89, #21, 35

Practice: p.96, #14 (Use conjugacy classes of S_n, p.125)

Homework 5 (Due 9/28):  p.96, #18; p.101, #9

Extra Problem: Let G=H(F_p), p a prime (the Heisenberg group). Then G is a group of order p^3.
(i) Find Z(G), G' =[G,G]  (commutator subgroup of G, see p.193)
(ii) Show that G/Z(G) is isomorphic to Z_p x Z_p
(iii) Hence show G has p+1 subgroups containing Z(G) (excluding Z(G), G)

Homework 6 (Due 10/5):

Extra: Find an injective homomorphism of D_8 into GL(2, C). (Hint: Think of the

elements of D_8 as linear transformations of the plane.)

P.106, #5, 8

Homework for Monday 10/8: Sylow Theorems (from handout)

Homework 7 (Due 10/12): p.111, #5; p.130, #9 (add: The centralizer has

a subgroup isomorphic to D_8) ; p.174, #7 (Prelim problem; hint: G/Z(G) )

Hint for p.111, #5: Take your transposition as (12) and the p-cycle as (a_1, a_2, …, a_p).

You can even choose a_1=1.

Homework 8: Extra sheet. Hint for #3: Action of G as in Cayley’s Theorem.

Homework 9 (Due Friday 11/2): 7.1, p.232, #26, 27; 7.3, p.250, #29; 7.4,p.256, #3

Practice: 7.1: #13, 25; 7.2: #6, 12; 7.3: #21, 25, 30, 31, 37

Homework 10 (Friday 11/9) : Continuation of p.232, #26: Let p in R be such that

v(p)=1. Then show that any non-zero ideal I of R is of the form I=(p^n) for some

non-negative n.

7.4, #31, 32; 7.6, #5.

Homework 11 (Friday 11/16):     8.2, #5; 8.3, #6

For your final homework see http://www.math.uic.edu/~srinivas/finhw516.pdf

Homework 12 (Friday 11/30): #4 and #7 on Extra Sheet (above url)

Homework 13 (Wednesday 12/5): #8, 9, 10 (i) on Extra Sheet

For Friday 12/7, try #12 on Extra Sheet

Final Exam on Thursday 12/13, 8-10 in the classroom, AH 306

Review for the final exam:

Groups: Group actions, Sylow theorems, Methods to determine if a finite group

is simple/abelian, p-groups, free groups, presentations. Study basic definitions,

statements of  theorems such as Sylow theorems.

Rings: Integral domains, pid, ufd., Chinese Remainder Theorem. Again, study

basic definitions, statements of  theorems such as CRT.

Modules: Direct sums and products, Tensor products, free and projective modules, finitely generated modules over a pid,

Structure of a finitely generated abelian group given a presentation, invariant factors, elementary

divisors, rational form of a matrix.  Again, study

basic definitions, statements of  theorems such as modules over a pid.

The calculations that we did to find the rational form of a matrix (invariant factors) are explained

on pages 479-488 of the text, with some examples. This might be useful.

Study all the homework problems.