Department of Mathematical Sciences, Northern Illinois University
INSTRUCTOR:
Professor John Beachy,
Watson 355, (815) 7536753
Email: beachy@math.niu.edu

My personal homepage

My faculty homepage
OFFICE HOURS: M W F 10:0010:50, 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 89:50 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, 1315, 2529
Solutions (to 2529)
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, 89:50 AM MAY 3 4 5 6 7
REVIEW
Exam 1, Fall 2006
Exam 1, Fall 2006, solutions
Exam 1, Summer 2009
Exam 1, Summer 2009, solutions
Exam 2, Fall 2006
Exam 2, Fall 2006, solutions
Final, Fall 2006

Solutions

Final, Spring 2003
Some previous final exams

Review of groups

Examples of groups
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 1.11.4, 1.12.
Meeting time and place: 9:3010:20 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 443444)
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