## MATH 420/520, Spring 2010

### 9:00-9:50, M W F, DuSable 348

INSTRUCTOR: Professor John Beachy, Watson 355, (815) 753-6753
Email: beachy@math.niu.edu   |   My personal homepage   |   My faculty homepage

OFFICE HOURS: M W F 10:00-10:50, or by appointment.

COURSE DESCRIPTION: Algebra I (3). Introduction to group theory. Properties of the integers, functions, and equivalence relations. A concrete approach to cyclic groups and permutation groups; isomorphisms and the theorems of Lagrange and Cayley.

TEXT: Abstract Algebra, 3rd Ed, by John Beachy and William Blair     |     Study Guide for the 3rd edition (in pdf format)

SYLLABUS: Chapter One, Integers; Chapter Two, Functions; Chapter Three, Groups (3.1 - 3.6)   |   Printed version

COURSE OBJECTIVES: The student is expected to acquire an understanding of the elementary theory of groups, together with the necessary number theoretic prerequisites. There will be some discussion of the computational aspects of these topics, but the main thrust of the course will be theoretical. The student will be expected not only to follow the proofs presented in class and in the text, but also to learn to construct new proofs. Proofs must be logically correct and care must be taken to write them precisely and in grammatically correct English.

COURSE PREREQUISITE: MATH 240, Linear Algebra. We will use matrices in some important examples, but the main reason for the requirement is to attempt to guarantee a certain level of "mathematical maturity."

GRADING: Final grades will be based on 600 points: 3 hour tests (300), homework (100), and final exam (200).
The homework problems are extremely important. In many ways the course is like an English composition course, since it requires you to write out very carefully the reasons for each step in your solutions of problems.

HOUR EXAMS: Exam 1     Exam 2   Solutions     Exam 3     Final Exam

FINAL EXAM: The comprehensive final exam is scheduled for Wednesday May 5, from 8-9:50 AM.

ASSIGNMENTS:
Note: The problems with higher numbers are from the Study Guide.
Homework 1, due 1/20: 1.1 #4(d), 6(d), 7, 9, 13, 16, 21, 39, 40
Homework 2, due 1/29: 1.2 #6, 10, 14, 23, 37, 40, 45, 46
Homework 3, due 2/3: 1.3 #10, 13, 24, 27, 48, 49, 50, 54, 55
Homework 4, due 2/8: 1.4 #10, 12, 24, 28 (recommended: #32, 34, 35, 38, 39)
Homework 5, due 2/24: 2.1 #11, 15, 19, 20, 37, 39, 41, 43, 45     Solutions (to 37,39,41,43,45)
Homework 6, due 3/1: 2.2 #1, 3, 5, 7, 10
Homework 7, due 3/15: 2.3 #11, 13-15, 25-29     Solutions (to 25-29)
Homework 8, due 3/29: 3.1 #10, 11, 23, 24, 36c,d,e,g,h, 37, 39, 40, 44, 45     Solutions (to 36,37,39,40,44,45)
Homework 9, due 4/5: 3.2 #8,9,14,19,54,55,56,60,62
Homework 10, due 4/12: 3.3 #9,11,13     3.4 #10,11,15
Homework 11, due 4/26, in class

SCHEDULE OF LECTURES AND EXAMS (TENTATIVE):

```Monday      Wednesday          Friday           M Tu  W Th  F

1.1           1.1              1.1        JAN 11 12 13 14 15

HOLIDAY        1.2              1.2            18 19 20 21 22

1.2           1.3              1.3            25 26 27 28 29

1.3           1.4              1.4        FEB  1  2  3  4  5

1.4           2.1             EXAM I           8  9 10 11 12

2.1           2.1              2.2            15 16 17 18 19

2.2           2.2              2.3            22 23 24 25 26

2.3           2.3              3.1        MAR  1  2  3  4  5

SPRING BREAK                           8  9 10 11 12

3.1           3.1            EXAM II          15 16 17 18 19

3.2           3.2              3.2            22 23 24 25 26

3.3           3.3              3.3            29 30 31  1  2

3.4           3.4              3.4        APR  5  6  7  8  9

3.4          Review          EXAM III         12 13 14 15 16

3.5           3.5              3.5            19 20 21 22 23

3.6           3.6          READING DAY        26 27 28 29 30

FINAL EXAM: Wednesday May 5, 8-9:50 AM     MAY  3  4  5  6  7
```

CAAR STATEMENT: Students who request accommodation due to a physical or learning disability must contact their instructor at the beginning of the semester. The instructor has the right to see documentation of the student's condition from the CAAR office.

HONORS: The honors section meets an additional hour each week. We will do some extra problems from the text, and then study symmetry groups. Reference: Goodman, Algebra, abstract and concrete, stressing symmetry, Sections 1.1-1.4, 1.12.

Meeting time and place: 9:30-10:20 Thursday mornings in DU 464

ASSIGNMENTS:
Assignment 1: from the text, 1.2 #16,18,25
Assignment 2: from the text, Appendix 4, #4,5,7,9 (pp 443-444)
Assignment 3: from the text, 1.3 #25,61 and 1.4 #29,55,56
Assignment 4: from the text, 2.1 #16,17,18
Assignment 5: from the text, 2.1 #7, 3.6 #12
Assignment 6: find references for a 4 to 5 page paper on Rubik's cube,
which will be due the last week of the semester. Some possibilities are listed below.

References on Group Theory as related to Rubik's Cube
The Complete Cube Book, by Roger Schlafly (51 pages)
Group Theory via Rubik's Cube, by Tom Davis (57 pages)
Group Theory and the Rubik's Cube, by Janet Chen (39 pages)
Mathematics of the Rubik's Cube, by W. D. Joyner (281 pages)
A History of Rubik's Cube