Miscellaneous Information

• The syllabus for Math 7200 can be found  here:  Math 7200 Syllabus
• Textbook: William A. Adkins and Steven H. Weintraub. Algebra: An Approach via Module Theory, Springer-Verlag New York, 1992. ISBN: 0-387-97839-9  For those who do not yet have the textbook, here is a copy of Chapter 1:  Chapter 1
• A description of  criteria for grading homework can be found here: Grading Criteria
• Here are some useful dates:
  Midterm Exam I:   September 29, 2006
  Midterm Exam II: November 3, 2006
  Final Exam:      Tuesday, December 12, 2006, 12:30 - 2:30 PM
• Homework assignments and brief summaries of material covered (or to be covered) in each week are included in the following list, arranged by week. 
• The graduate assistant for the course is Jean Bureau.  He will be available for assistance outside of class (Email: jbureau@math.lsu.edu) and he will be the grader for the course. Jean will also conduct an office hour at 2:30 P.M. Thursdays in 341 Lockett.
  

Weekly Summaries and Homework Assignments

Read Sections 1.1-1.3.  This consists of the basic definition of a group, examples (some of which will be presented in class) of groups, subgroups, order of groups and elements, the fundamental theorem on cyclic groups, cosets and Lagrange's theorem. A nice historical survey of the origin of group theory can be found here: Group Theory Survey.  The treatment of cyclic groups on Page 11 (Theorem 2.12)  is somewhat terse. A more detailed treatment is here: Cyclic Group Supplement. This supplement gives information about orders of elements of a cyclic group, the number of generators of a cyclic group, and what all the subgroups actually look like.

Exercise Set 1 (Due: September 1)
Exercise Set 1 (Solutions) Courtesy of Jean Bureau
 

We will continue with Section 1.3 and start with the applications of groups acting on sets in Section 1.4. 

Exercise Set 2 (Due: September 8)
Exercise Set 2 (Solutions) Courtesy of Jean Bureau
 

We will cover the basic properties of permutation groups in Section 1.5 and then return to the concept of groups acting on sets in Section 1.4, where we will prove the Sylow Theorems, which are concerned with counting the number of subgroups of prime power order of a group  G.

Exercise Set 3 (Due: September 15)
Exercise Set 3 (Solutions)
 

We will complete the discussion of Sylow's Theorems and see how they are used for the classification of groups of small order.  

Exercise Set 4 (Due: September 22)
Exercise Set 4 (Solutions)
 

We have now covered most of chapter 1, with the exception of semidirect products, which we will touch on briefly on Monday, before going on to start the theory of rings in Chapter 2. Note that some of the examples of groups on low order in Section 7 have been discussed as applications of Sylow theory.   The first exam will be on Friday, September 29.  The exam will test basic understanding of the definitions and theorems covered in class and on the homework assignments.  The solutions for exercise set 4 will be posted on Monday (September 25) after the homework is collected. For your convenience, I am posting here  a Supplemental Exercise Set  consisting of some representative problems of the type that you might expect on the exam.  Although the exams themselves cover different material, I am also posting copies of some old exams from previous incarnations of this course:  Final Exam 1995, Midterm Exam 1999. Alternate Midterm Exam 1999, Final Exam 1999. Their purpose is just to let you have an idea of the length and type of questions that might be expected.  In each of these courses the syllabus and placement of exams was different than in the current course.

You should be reading the first 4 sections of Chapter 2

Here are the solutions for the first midterm exam: Midterm Exam  I Solutions

Exercise Set 5 (Due: October 13)
Exercise Set 5 (Solutions)
 

This week we finished section 2.4 and started the theory of factorization of elements in integral domains.  This material is contained in sections 2.5 and 2.6.  It will be completed next week. 

This week we completed  the study of factorization in integral domains by proving Gauss's lemma and the theory of unique factorization in polynomial rings.

We will start with module theory this week, covering at least the basic language in the first three section of chapter 3, and possibly the basic facts in section 4 on free modules.  There will be an exam on November 3, covering the chapter on ring theory and sections 3.1 to 3.4 in chapter 3.

We will continue with our study of module theory from Chapter 3 this week, starting with direct sums, exact sequences, and the basic idea of free modules.  The second exam will be on Friday, November 3.  The exam will test basic understanding of the definitions and theorems covered in class and on the homework assignments. The syllabus for the exam will consist of the material from ring theory (Chapter 2) and Sections 3.1 to 3.3 in Chapter 3. For your convenience, I am posting here  a Supplemental Exercise Set  consisting of some representative problems of the type that you might expect on the exam. As with the first exam the problems are representative of what you can expect, although there will naturally be a much smaller number of exercises on the exam, as was the case with the first exam.  

Here are the solutions for the second midterm exam: Midterm Exam  II Solutions

This week we will be concentrating on completing the proof of the main theorems on finitely generated modules over principal ideal domains (PID's). Last week we proved the main general facts concerning free modules over an arbitrary ring.  This is the content of Section 3.4.  We will not have time to cover the material in Section 3.5, so you should skip that.  The main results are in Sections 3.6 and 3.7.  The theorem proved on Friday, November 10, was Theorem 6.2.  We only did the case of finite rank, which is Case 1 in the proof.  We will not need the more general result in Case 2, so you should skip that.  The next major goal is Theorem 6.23 (Invariant factor theorem for submodules).  This is a more precise version of Theorem 6.2, which allows for a choice of basis of the ambient module M, so that a basis of the submodule N can be written down explicitly.  Then Section 3.7 contains the study of torsion submodules.

This week we will complete the proof of the main theorems concerning the structure of finitely generated modules over Euclidean domains. This material is in Section 3.7 of the text.  Applications addressed in the exercises concern the solution of linear equations over the integers, determination of the structure of finitely generated abelian groups  when they are given by means of generators and relations, and determination of all isomorphism types of finite abelian groups of a given order.  After the Thanksgiving holidays, the remainder of the semester will be devoted to applications of this theory to the structure theory of a single linear operator on a finite dimensional vector space.

Exercise Set 9(Due: December 1)
Exercise Set 9 (Solutions)

This week and the final week will be devoted to applying the structure theorem for finitely generated modules over a principal ideal domain to analyzing the structure of a single linear transformation on a finite dimensional vector space.  The relevant sections are Sections 4.3 (representation of operators by matrices), Section 4.4 (canonical form theory), and Section 4.5 (examples of computations).  Exercise set 10 will be the last exercise set.  Note in particular that problem 2 is a long list of deriving information about a linear transformation from a portion of the relevant data.  These exercises can be rather long, so choose a couple of  2 (a)--(f) and do those.  

Exercise Set 10(Due: December 8)
Exercise Set 10 (Solutions)

The final exam will be devoted to the module theory and linear algebra that we have covered in the last part of the course since Exam II.  For your convenience, here is a copy of the final exam from Fall 2004, which covered a selection of material similar (but not necessarily exactly the same) to that covered this semester: Final Exam 2004