MATH 367 Abstract Algebra (Fall 2005)
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://wwwhistory.mcs.standrews.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 385387 (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 psubgroup (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
email group.
 Visit this page frequently for announcements.
 Lectures: Mon 10:4011:30, Wed 10:4012:30, Thu 10:4012:30
(M13)
(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: M102, 103, 104, 105
Contents: Groups

Exam 2:
Date and Time: 21 December 2005 (Wed) 17:40
Place: M102, 103, 104, 105
Contents: Rings and fields
 Final:
Date: 29 December 2005 (Thu) 13:30
Place: M05, 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.
Last updated on 15 December 2005.