2005 - 06 Fall

MATH 367: Abstract Algebra

Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos. --- E. T. Bell
Mathematics, Queen and Servant of Science, New York, 1951, p 164.
(taken from http://www-history.mcs.st-andrews.ac.uk/Quotations/Bell.html)

• Announcements:
  
  • (15 Dec)CONTENTS OF THE SECOND EXAM
    Section no. # Topics # Exercises
    3.1 # all # all
    3.2 # all # all except 3.28.
    3.3 # all # all except 3.37.
    3.4 # all # all except 3.50(iii),3.51.
    3.5 # Skip everything after example 3.19 (page 255) except the definition of irreducible element in a PID. # all except 3.52,3.54,3.60,3.63.
    3.6 # Only Theorem 3.52 # all
    3.7 # Skip the definitions on page 268, 269 and 273; Proposition 3.58, Corollary 3.64. # all except 3.77.
    3.8 # all # all except 3.83, 3.87, 3.90. (3.85 will be included in the final)
    6.1 # All plus Zorn's Lemma plus the existence of maximal ideals. # all except 6.12.
    6.2 # all # all
  • (31 Oct) Exercise 5.18 is added to the list of exercises.
  • (27 Oct) CONTENTS OF THE FIRST EXAM:
    Chapter 2: Everything in 2.2, 2.3, 2.4, 2.5,
    2.6 (except Corollary 2.46, Theorem 2.47, Theorem 2.48, Corollary 2.64, Proposition 2.65),
    2.7 (only Theorem 2.66, Proposition 2.68, Proposition 2.77 (without proof), Proposition 2.78, Exercises: 2.93, 2.94 (ignore Q), 2.96, 2.97, 2.98. ).
    Section 5.1: Pages 385-387 (inclusive) and the lecture notes (especially the Fundamental Theorem of Finite Abelian Groups, primary form, dividing form and examples.)
    Exercises: 5.2(iii), 5.3(ii) and (iii), 5.4, 5.5, 5.7, 5.8.
    Section 5.2: Definition of a Sylow p-subgroup (p. 398), definition of conjugate (p. 399), Lemma 5.15(i), Theorem 5.16 (without the proof), Corollary 5.17, Theorem 5.18 (without the proof), Example 5.4, Theorem 5.19 (without the proof) and all the examples done in the lectures and recitations.
    Exercises: 5.9, 5.10, 5.14, 5.16, 5.18 (ignore 300), 5.19, 5.20.
  • Recommended books
  • 367 lectures will start on 19 September 2005, Monday.
  • Join the e-mail group.
  • Visit this page frequently for announcements.
• Lectures: Mon 10:40-11:30, Wed 10:40-12:30, Thu 10:40-12:30 (M-13)
  (After the third week on, there will be recitation hours every week mostly on Thursdays.)
• Main Text: A first course in abstract algebra by Joseph J. Rotman, 2nd edition, Prentice Hall, 2000. (QA162 R68 2000)
• Exams:
  • Exam 1:
    Date and Time: 9 November 2005 (Wed) 17:40
    Place: M-102, 103, 104, 105
    Contents: Groups
  • Exam 2:
    Date and Time: 21 December 2005 (Wed) 17:40
    Place: M-102, 103, 104, 105
    Contents: Rings and fields
  • Final:
    Date: 29 December 2005 (Thu) 13:30
    Place: M-05, 06, 07, 102
    Contents: Everything
    
• Course Plan (Revised on 21 Nov 2005)
  • Week 1. Basics of group theory, examples (including matrix, dihedral and permutation groups). (2.3)
  • Week 2. More on permutations, subgroups. (2.2, 2.4)
  • Week 3. Cosets, Lagrange's Theorem, homomorphisms. (2.4, 2.5)
  • Week 4. Normal subgroups, quotient groups, direct products, isomorphism theorems, the correspondence theorem. (2.5, 2.6)
  • Week 5. Permutation groups revisited, Cayley's Theorem, finite abelian groups. (2.2, 2.7, 5.1) (I did not follow 5.1, instead I used Goodman's book to prove the structure theorem for finite abelian groups.)
  • Week 6. Sylow Theorems. (5.2)
  • Week 7. Introduction to rings and fields, fraction fields. (3.1, 3.2)
  • Week 8. Review of group theory, the first exam.
  • Week 9. Polynomials, homomorphisms, ideals, GCD in F[x]. (3.3, 3.4, 3.5)
  • Week 10. PID's, Euclidean rings, irreducibility. (3.5, 3.7)
  • Week 11. Quotient rings, finite fields. (3.8)
  • Week 12. Prime ideals, maximal ideals. (6.1)
  • Week 13. UFD's. (3.6, 6.2)
  • Week 14. Review of rings and fields, the second exam.