MATH
322 – Abstract Algebra
Fall 2010
Professor:
Dr. Janet Vassilev
Office: Humanities 467
Office
Hours: MWF 11 am-12 pm and by appointment.
Telephone: 
(505)
277-2214
email: jvassil@math.unm.edu
webpage:
http://www.math.unm.edu/~jvassil
Text : A
First Course in Abstract Algebra, 7th Edition By John
Fraleigh.
Course
Meetings: The course lectures will be held in Mitchell 208 on Mondays,
Wednesdays and Fridays at 1-2:50 pm.
Topics: Groups, rings, homomorphisms, permutation
groups, quotient structure, ideal theory, fields.
Daily
Quizzes (100 points): The first 5 minutes of everyday (excluding
exam days) will consist of an open notebook quiz on the concepts of the
previous lecture. The quizzes will be
worth 5 points each. I will drop your 4
lowest quizzes and average the remaining quizzes to obtain a score out of 100.
Homework
(200 points): Homework will be assigned weekly on Wednesdays and will
be collected the following Wednesday at the beginning of class. Homework
will not be graded unless it is written in order and labeled
appropriately. The definitions and theorems in the text and
given in class are your tools for the homework proofs. If the theorem has a name, use it. Otherwise, I would prefer that you fully
describe the theorem in words that you plan to use, than state “by Theorem
3”. Each week around 4 of the assigned problems will be graded. The
weekly assignments will each be worth 20 points. I will drop your lowest two homework scores
and the remaining homework will be averaged to get a score out of 200.
Exams (400
points): I will give two midterms (100 points) and a final (200
points). There are no make up exams. If a test is missed,
notify me as soon as possible on the day of the exam.
For the midterms only, if you have a legitimate and documented
excuse, your grade will be recalculated without that test. The Midterms
are tentatively scheduled for Monday September 27 and Monday, November 8.
The Final is on Friday, December 17, from 12:30 pm-2:30 pm.
Grades: General guidelines
for letter grades (subject to change; but they won't get any more strict):
90-100% - A; 80-89% - B; 70-79% - C; 60-69% - D; below 60% - F. In
assigning Final Grades for the course, I will compare your grade on all course
work (including the Final) and your grade on the Final Exam. You will
receive the better of the two grades.
Tentative
Schedule (for Dr. Vassilev’s Abstract Algebra):
Date
|
Section
|
Topic
|
Homework
|
8/23
|
2
|
Binary
Operations
|
|
8/25
|
3
|
Isomorphic
Structures
|
Section 2: 3, 5, 6, 9, 12, 24, 26, 28, 36
Section 3: 1, 3, 4, 12, 27
|
8/27
|
4
|
Groups
|
|
8/30
|
4
|
Groups
|
|
9/1
|
4, 5
|
Groups and
Subgroups
|
Section 4: 1, 3, 5, 7, *12, *18, 19, 20, 24, 25, 29,
31, 32, 35
|
9/3
|
5
|
Subgroups
|
|
9/8
|
5, 6
|
Division
Algorithm and Euclidean Algorithm
|
Section 5: 2, 4, 6, 14, 15, 20, 28, 31, 36, 39, 41,
43, 45, 47, 51, 54
|
9/10
|
6
|
Cyclic groups
|
|
9/13
|
6
|
Cyclic Groups
|
|
9/15
|
6,7
|
Cyclic groups
and Finitely generated groups
|
Section 6: 3,
7, 11, 18, 20, 22, 32, 33, 36, 46, 49, 55
Section 7: 4,
5, 7, 12, 13
|
9/17
|
7,8
|
Finitely generated
groups and dihedral groups
|
|
9/20
|
8
|
Dihedral groups
|
|
9/22
|
8
|
Groups of
permutations
|
Section 8: 1,
4, 6, 16, 30, 33, 35, 36, 52
|
9/24
|
|
Review
|
|
9/27
|
|
Midterm 1
|
|
9/29
|
9
|
Orbits, cycles
and alternating groups
|
Section 8: 40,
41, 46, 49
Section 9: 1,
3, 7, 8, 13, 14, 16
|
10/1
|
9
|
Orbits, cycles
and alternating groups
|
Quiz 15: What
is the orbit of 3 for the element
1 2 3 4 5 6 7 8
2 1 4 6 7 8 3 5
?
|
10/4
|
10
|
Cosets and
Lagrange
|
Quiz 16: Write out
a=(1234)(567) as a product of transpositions and determine if a is odd or
even.
|
10/6
|
10
|
Cosets and
Lagrange
|
Quiz 17:
Determine the left cosets of <4> in Z12
Section 9: 23, 29, 33, 34
Section 10: 1,
4, 6, 7, 20, 26, 27, 28, 30, 32
|
10/8
|
11
|
Direct Products
and Finitely generated abelian groups
|
Quiz 18: What
are the possible orders of subgroups in a group of order 20?
|
10/11
|
11
|
Direct Products
and Finitely generated abelian groups
|
Quiz 19: Give
the definition of a direct product of groups (Gi, *i
)give the binary operation associated to this new group.
|
10/13
|
12
|
Plane
Isometries
|
Quiz 20: Is Z2
x Z10 isomorphic to Z20?
Section 10: 12, 14, 15, 19, 34, 39, 45
Section 11: 3,
6, 11, 14, 16, 32, 36, 46, 47, 49
|
10/18
|
13
|
Homomorphisms
|
Quiz 21: List
two plane isometries of infinite order
|
10/20
|
14
|
Factor Groups
|
Quiz 22: Give
the definition of the kernel of a homomorphism.
Section 11: 50, 51
Section 13: 2, 4,
5, 8, 9, 17, 20, 23, 32, 37, 39, 44, 49, 50
|
10/22
|
14
|
Factor Groups
|
Quiz 23: Give
an example of a normal subgroup in a nonabelian group.
|
10/25
|
15
|
More on Factor
Groups
|
Quiz 24: What
is a criterion on H for G/H to be a group?
|
10/27
|
16
|
Group Actions
|
Quiz 25: Let G be Z2 x Z6 and
H=<(1,3)>. What is the order of
G/H?
Section 14: 2, 6, 10, 14, 22, 23, 30, 31, 34
Section 15: 4,
12, 19, 34, 35, 36
|
10/29
|
16, 17
|
Group Actions
and Counting
|
Quiz 26: Give the
definition of a group acting on a set.
|
11/1
|
17
|
Group Actions
and Counting
|
Quiz 27: Give
an example of a group which acts on the diagonals of a square.
|
11/3
|
18
|
Rings
|
Quiz 28: State
Burnside’s Theorem
Section 16: 1, 3, 8, 11, 12, 14
Section 17: 1,
2, 4
|
11/5
|
|
Review
|
Quiz 29: Give
the definition of a ring
|
11/8
|
|
Midterm 2
|
|
11/10
|
18
|
Rings and
Fields
|
Section 18: 2,
6, 8, 10, 16, 18, 21, 24, 33, 37, 38, 39, 44, 46, 49 (due next Friday)
|
11/12
|
19
|
Integral Domains
|
Quiz 31: What
is a ring homomorphism?
|
11/15
|
20
|
Fermat’s and
Euler’s Theorems
|
Section 19: 6,
10, 17, 23, 25, 29
Section 20: 4, 7, 10, 12, 14, 23, 27
Quiz 32: Give
an example of a zero divisor and the ring it lives in.
|
11/17
|
21
|
Field of
Quotients
|
Quiz 33: What
is phi(100)?
|
11/19
|
22
|
Polynomial
Rings
|
Quiz 34: What
is the set we used to construct the field of fractions of a domain D?
|
11/22
|
22
|
Polynomial
Rings
|
Quiz 35: What
is the difference between a power series and a polynomial?
|
11/24
|
23
|
Factorization
of Polynomials
|
Quiz 36: If R
is a domain, is R[x] a domain?
|
11/29
|
23
|
Factorization
of Polynomials
|
Quiz 37: If a
is a zero of f(x) then tell me a factor of f(x).
|
12/1
|
24, 26
|
Noncommutative
rings and Homomorphisms and Factor
Rings
|
Quiz 38: State Gauss Lemma.
Section 21: 1, 4, 12
Section 22 5, 8, 11, 12, 23, 27, 28
Section 23: 2,
9, 10, 12, 14, 19, 25, 35
|
12/3
|
27
|
Prime and
Maximal Ideals
|
|
12/6
|
28
|
Groebner Bases
|
|
12/8
|
|
Review
|
|
12/10
|
|
Review
|
|
12/17
|
|
Final exam
|
|