**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**

**This page is http://www.math.uic.edu/~srinivas/math516.htm/**

**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

links to Math 516, using the course password algebra516.

**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**

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)

(For (i), see also p.174, #20; for (iii), see also
p.155 (3) and p.157 #10)

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.