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