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/
Revised August 15, 2007, by BS516 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.