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.