# 3124/13875 Modern Algebra Spring 2008

## Syllabus

Instructor:
| Henning S. Mortveit | Email:
| henning@vt.edu |

Office I:
| 1111 RBXV, Corporate Research Center | Phone I:
| (231-5327) |

Office II:
| 419 McBryde Hall | Phone II:
| - |

Class hours:
| MWF 1:25-2:15PM | Room:
| 134 McBryde Hall |

Prerequisite:
| Math 3034 | ||

Office hours:
| MW 2:15-3:15PM in Office II and by appointment in Office I |

**Text/Syllabus:**John Durbin,

*Modern Algebra*, 5

^{th}edition, John Wiley and Sons. We will cover selected material from chapters 1-6.

**Course goals:**Learn basic aspects of mathematical structures such as groups, rings and fields. Increase fluency in the reading and writing of mathematical proofs and arguments.

**Exams:**There will be two in-class exams tentatively scheduled for Wednesday February 27 and Friday April 11. The two-hour final exam (Section 13M) is on May 2 from 1:05PM to 3:05PM. The final exam will take place in McBryde 134 unless stated otherwise. If you cannot take an exam at the scheduled time, please let me know as soon as possible and

*before*the exam. A make-up exam will be given for reasons that in my judgment are acceptable.

**Homework:**The course has 12 assignments. Generally assignments will be announced on Fridays, and will be due in class the following Wednesday. The exact schedule can be found here. It is subject to change - changes will also be announced in class. Late homework will only be accepted if handed in the first class following the due date, but only for half the credit.

*Very important:*The assignments are an integral part of this course. The 12 assignments should be considered a minimal effort, and working through additional problems is strongly encouraged.

**Attendance:**Will be taken, and will be kept for Mathematics Department records. Attendance may be used to adjust the final grade.

**Grading:**Is on a curve. However, 90% will be at least an A-, 80% will be at least a B-,70% will be at least a C-, and 60% will be at least a D-. Each assignment is worth 10 points, each in-class exam 40 points, and the final exam 50 points.

**Honor system:**The University Honor System is in effect for assignments and exams (see http://www.honorsystem.vt.edu). Discussion of class topics among students is encouraged, but the solutions to assignments that you hand in must be your own. All exams are closed-book, closed-notes.

**Students with special needs:**Students with disabilities, special needs or special circumstances should meet with the instructor during the first week of classes to discuss accommodations.

**General Notes:**Falling behind in this course is

*dangerous*, so turn in assignments on time, come to class prepared and take advantage of the office hours. (Also: read Professor Bud Brown's hints for success - available at http://www.math.vt.edu/people/brown/hints.html.)

**Supplementary literature:**Some other book on algebra are listed below.

- John B. Fraleigh,
*A First Course In Abstract Algebra*, Addison Wesley. (This book is a little more advanced than Durbin's book.) - P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul,
*Basic Abstract Algebra*, Cambridge University Press. (More advanced than Durbin's book. Has many good examples.) - David S. Dummit and Richard M. Foote,
*Abstract Algebra*, John Wiley. (This is the course book for 4124 Abstract Algebra.) - Thomas W. Hungerford,
*Algebra*, Springer Verlag. (A classic reference. A demanding book.)

## Exams

Solution notes and comments will be posted here.### In-class exam 1: Wednesday, February 27

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### In-class exam 2: Friday, April 11

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### Final exam: Friday, May 2

## Assignments

Assignments, solution notes and comments will be posted here.### Assignment 1: (Due Wednesday, January 23)

- Section 1: 1, 3-6, 19-21, 28
- Section 2: 8, 16, 21-24

### Assignment 2: (Due Wednesday, January 30)

- Section 3: 1, 2, 7, 18, 24, 29
- Section 4: 1, 4, 13 (You may use the result of problem 12 without proof.)
- Problem: Is m*n = (mn)/2 a binary operation on the set of positive rational numbers? (Hint: Yes, but you must argue why.) Classify this binary operation and find the identity element and inverses if there are any.

### Assignment 3: (Due Wednesday, February 6)

- Section 5: 2, 6, 8, 12, 13, 14, 22. (In 13 and 14 you must also give an argument to prove closure.)
- Section 6: 2, 6, 10.

- Show that a group G with identity e and such that x*x = e for all
x in G must be Abelian.
*Hint:*Consider (a*b)*(a*b).

### Assignment 4: (Due Wednesday, February 13)

- Section 6: 7
- Section 7: 2, 9, 14, 23.

- Let (a
_{1},a_{2},a_{3},...,a_{k}) be a k-cycle in S_{n}, and let β ∈ S_{n}. Compute β (a_{1}, a_{2}, a_{3}, ..., a_{k}) β^{-1}. (*Hint:*The result is a k-cycle.)

### Assignment 5: (Due Wednesday, February 20)

- Section 7: 11, 12 (Each worth 2 points)
- Section 9: 21 (2 points)
- Section 8: Determine the group D of symmetries of the regular
hexagon. Label the corners (clockwise and consecutive) 1, 2, 3, 4, 5
and 6, and write permutations on cycle form. Let T = {1,4}. Determine
D
_{T}and D_{(T)}. (4 points)

- Section 7: 22
- Let G be a group with operation * and let H be a finite subset of
G that satisfies (i) H is not empty, and (ii) a*b ∈ H for all a,b
∈ H. Show that H is a subgroup of G. (
*Hint:*Use the subgroup theorem. What is left to establish is a^{-1}∈ H for all a ∈ H. )

### Assignment 6: (Due Wednesday, March 12)

- Section 10: 6, 12, 15, 16, 17 (Each worth 2 points)

- Section 10: 26, 29 (On 29 use 10.15 and 10.16)

### Assignment 7: (Due Wednesday, March 19)

- Section 11: 4, 14
- Section 12: 6, 16, 19
- Section 13: 8, 13, 14
- Section 14: 4, 25 (will be covered on Monday, 17 March)

- Section 12: 21
- Section 14: 38

### Assignment 8: (Due Wednesday, March 26)

- Section 14: 20, 26, 28, 29, 31
- Section 15: 2, 4, 11, 12, 19

- Section 14: 34
- Section 15: 26

### Assignment 9: (Due Wednesday, April 2)

- Section 16: 2, 4, 12, 14, 18

- Section 16: 19, 20

### Assignment 10: (Due Friday, April 18)

- Section 17: 10, 12
- Section 18: 2, 5, 9

- Section 17: 27, 30

### Assignment 11: (Due Wednesday, April 23)

- Section 19: 2, 4, 14, 15, 19
- Section 21: 5, 8, 9, 18, 28

### Assignment 12: (Due Monday, April 28)

This assignment has problems from Sections 21, 22 and 23. Each problem is worth 3 points. Note that 22.10 also offers an additional bonus point.**21.28:**Hint: first find all the subgroups.**21.34:**Also: Use your answer to determine the quotient group (ℤ x ℤ)/<(1,-1)> (up to isomorphism).

[Note: Some browsers (e.g. Explorer) may not display the quotient group correctly. Here is an image version: ( x ) / <(1,-1)> ]**22.1:**Also: give the Cayley table for the quotient groups, and determine what they are (up to isomorphism).**22.10:**Also: The factor group is Abelian. Use the fundamental theorem of finite Abelian groups to determine a representative for each possible isomorphism class for the order of the quotient group. Bonus (1pt): Which of these two representatives is isomorphic to the quotient group?**23.12:**Hint: Mimic Example 23.2. There is also a "clever" solution.

## Course Schedule

The schedule is subject to change. The current version can be found as a PDF file here.Fri Jan 11 17:35:53 EST 2008