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Syllabus Text Worksheets:
Syllabus for Math 3175 Spring 2016 Prof. A. Iarrobino
Text: Contemporary Abstract Algebra, 8th Ed. J. Gallian, Brooks-Cole
ISBN 13:978-1-133-5999708. There is an e-edition available.
Worksheets: Proofs, Group actions on a set, Sylow theorems, etc.: will be posted or handed out.
Reference: J.L.Alperin and Rowen B. Bell: Groups and Representations, Springer GTM #162.
Online : T.W. Judson, Abstract Algebra http://abstract.ups.edu/index.html (sec. 12-15)
Prerequisite: Linear algebra MTH 2331, and Calculus 3 MTH 2321.
Helpful: Math 1365, Introduction to Mathematical Reasoning.
Time: MWTh 9:15-10:20 AM. Room: 415 Shillman, Registrar CRN 30226
Instructor: Prof. Anthony Iarrobino, 526 NI, x 5524, e-mail [email protected]
E-mail is usually a very quick way to contact me, please address “Prof. Iarrobino”.
Office Hours: Mon 11-12,3:15-4:30. Wed. 11-12, 1:30-2:30; Thurs: 11-2, or by appointment. (I am
frequently in until later Monday and Thursday, but please schedule to be sure).
Spring 2016: I will use Blackboard or e-mail for occasional announcements, and brief summaries of
classwork, as well as HW assignments. I will also post worksheets and sample exams, info on essay at
Blackboard’s Class Materials.
Grading: HW and Quizzes (30%) (About 5 quizzes, usually on alternate weeks). Goal C essay
(10%). The one-hour midterm exam is 20%. A final exam (day TBA) is required of all, and will
count 40%. Optional work such as a project or presentation: may count for up to 20% of your grade
before FE, if arranged with instructor by March 2.
HW: I will assign homework, primarily from the syllabus below and worksheets. On alternate
Thursdays at the beginning of class, I plan to collect HW assigned Wed-Thurs of previous week, and
Monday of current week (if there should be a TA assigned, HW will be collected weekly). At the final
exam, you may pass in your collected HW or notebook to be checked for an Extra Credit HW grade,
part of the before-final course grade.
I encourage you to study with others, and to discuss HW with others, but HW passed in should be
your own work. I will discuss this expectation more in class.
Journal: An optional weekly journal either physical or online will count as EC (discuss with me).
Class work, etc. in small groups: Part of most classes will be devoted to solving problems in group
theory related to HW. We may work in small groups. Also a group of students may choose to do an
optional presentation or project.
Topics: The course introduces the basic ideas and applications of group theory, including symmetry
groups, abelian, cyclic, and permutation groups. Also subgroups, normal subgroups, group
homomorphisms, quotient groups, direct products, group actions on a set, and an introduction to the
Sylow theorems. The theory will be illustrated by examples from geometry, linear algebra and
combinatorics. We will cover chapters 1-10, 29 of text, also topics from Chapters 11,24-27 as time
permits. We will emphasize group action on a set as a unifying theme (handouts).
Goals: A. Students will understand the basic ideas and some applications of groups.
Students will be able to explain groups and factor groups and their relation to symmetry. Students will
recognize mathematical objects that are groups, and be able to classify them as abelian, cyclic, direct
products, etc. Students will understand homomorphism of and quotients of groups, and be able to
determine when a group has a normal subgroup, or a quotient.
B. Students will be able to reason mathematically, to write simple proofs, and are able to judge
when an attempted proof in group theory is correct/complete or is not.
C. Students will have a chance to reflect on doing mathematics, solving problems and
our role and progress as mathematicians
Math 3175 Spring 2016 Prof. A. Iarrobino – Syllabus
p. 2
Homework exercises from text (8th edition). Instructor will make assignments. WS=worksheet
Chapter Topic
Pages
Problems
0
Preliminaries: GCD, LCM, proofs
23
1,2,4-13,20-21,58,59
WS
On proofs
WS
#4A,B,C
1
Examples of symmetry groups
37
5,13,17,19,20,24
2
Definition and examples of groups
54
1-8,11-14
Elementary properties of groups
54
22-25,28,31,33-35,45
3
Finite groups, subgroups
68
1-3,22,26,32,-34,45,46,49
“ “ of GL(2,R) general linear group
72
52-55,61,62,67,69,76,79-80.
WS
Group actions on a set. Orbit, stabilizer
WS
WS
4
Properties of cyclic groups
87
1-10,14,21,26,28
Classification of subgroups of cyclic groups 89
36-39,56-58,61,69-70
Cyclic groups and supplementary exercises
95
1-3,6-7,9,12,20.18,24-25,27,35
5
Permutation groups, cycle decomposition
118
1-10,17,18,27-29
Permutation groups, stabilizer, application
120
31,32,35,37,43,61,81-82
6
Isomorphisms
138
1-10,14,34,35
Isomorphisms: Aut(G), Aut( Z n )
140
37,39,42,45,53,55,56,63
WS
Thurston article: read on ArXiv
WS
Goal C essay.
7
Cosets and Lagrange’s Theorem
156
1-9,15,17,18,20,27,28
Groups of order 2p; rotation groups
158
44,45,46,47,59,60,62,64
8
External direct products
174
1-12,14,18,20,22,24,26,28,29,
35,52,53,54,59,63,67,68,81
Supplementary Exercises
181
5,6,12,15,16,27,28,37,46.47.
49,53,59
9
Normal subgroups, define factor group G/N 200
1,2,3,5,6,7,8, 12
Factor groups G/N
201
14,16,17,18,27,28,29,30,32,34
Orders of elements in G and G/N
202
37,38,42,47,48,51,55,56,67,68,
Center Z(G), and G/Z(G).
70
10
Homomorphisms, kernel, and image
219
5,7-27,48,50,51,56,57,58,66
11
Fundamental Theorem of Abelian groups
234
1,3,4,5,7,13,15,16,19,25.
Spr11FinalPractice Sect 6,
F12ECHW #1,Spr12FE#1
8-11
Supplementary Exercises
238
1,2,4,8,9,10,15,31,32,39,40.
29
Symmetry and Counting
502
1-15
Burnside Theorem WS (proof, applications)
WS
24*
Class equation, Sylow theorems
421
5-8,15-18,21-26,32-39,48,55,
optional Sylow and Group structure
60,61,65,69,70
Applications of Sylow to determine groups WS
Groups actions on a set and proof of Sylow WS
27*
Symmetry groups (also, use as reference)
466
1,2,4,7,13,15
28*
Frieze groups and crystallographic groups
487
1-9,11-15
Readings: I will assign some readings relevant to Goal C, in particular
W. Thurston, “On Proof and Progress in Mathematics” Bull Amer Math Soc 30 #2, April 1994, 161-177.
Students are invited to suggest others.
Attendance and class participation. This is an intensive course, and attendance is needed to participate
fully! Strong participation and ECHW will add to your before FE course grade.
Class: Normally each class will have time for questions, class work on problems, and presentation/discussion
of topics.
Math 3175 Spring 2016 Prof. A. Iarrobino – Syllabus
p. 3
I request that electronic media (cell phones, smart phones, laptops, etc.) not be on during class so that focus
can be on the class discussion. I will make an exception for those who have purchased an e-edition of the text
and wish to use it in class.
Academic Honesty: It is fine to work together to do homework (studies have shown this can be particularly
helpful in learning math), provided such assistance is acknowledged specifically in any work passed in, and
that you understand what you pass in. Collaboration on quizzes and exams is not allowed, unless I make a
specific exemption for a quiz, announced in advance. In any presentation or project, resources used and
sources of assistance must be acknowledged in a professional way.
Student Code of Conduct: see http://www.northeastern.edu/osccr/academic-integrity-policy/ or find at
Undergraduate Student Handbook.
A commitment to the principles of academic integrity is essential to the mission of Northeastern
University. The promotion of independent and original scholarship ensures that students derive the
most from their educational experience and their pursuit of knowledge. Academic dishonesty violates
the most fundamental values of an intellectual community and undermines the achievements of the
entire University.”
The website/handbook goes on to detail examples. In Math 3175, academic dishonesty on a quiz or exam, or
assignment leads to a zero on the quiz or exam or assignment that cannot be made up, as well as a letter
detailing the incident to the Office of Student Conduct and Conflict Resolution. The minimum penalty for a
finding of academic dishonesty by the student Judicial Hearing Board includes one year suspended probation.
Incomplete grade: requires a written understanding (contract) between the Instructor and student with details
about what material will be made up and how. Incompletes are normally appropriate only for a student who is
doing well, but who becomes ill, or has a family emergency late in the semester.
Concerns: In case of concerns about the course that cannot be resolved by speaking with the instructor,
please contact the Mathematics Department Teaching Director, Professor David Massey
529 Nightingale Hall, Phone 617-373-5527 e-mail d.massey(at) neu.edu
Tutoring: http://www.math.neu.edu/undergraduate-program/mathematics-tutoring-services
-available Room 540B Nightingale, sign up on my.neu account (tutoring link).
I will identify some tutors at the Math Dept tutoring center with experience in group theory.
For College of Sciences peer tutoring see www.northeastern.edu/csastutoring/
TRACE participation at the end of the course is expected. If there is class Trace participation above 85% the
lowest quiz will be dropped.
I will also ask from time to time for feedback on how the class is going (3 minute feedback).
Online resource: I will post a list of online resources. One resource is the Gallian Contemporarry Abtract
algebra web page. It has suggestions on using the text, writing proofs, computer program links for working
with groups and the text,. even songs (extra credit for suitable new songs performed)
http://www.d.umn.edu/~jgallian/
Review: Gallian’s True-False questions http://www.d.umn.edu/~jgallian/TF/
MacTutor site: History of group theory (J. J O’Connor and E.F. Robinson).
http://www.gap-system.org/~history/HistTopics/Development_group_theory.html
GAP: software for working with groups (Prof. Gene Cooperman of CS at NU is one of the many authors).
FE: Day, time, TBA. All students will be expected to take the Final Exam as scheduled; your travel
plans at the end of the semester must take the FE into account.
Fly UP