University of Toronto Department of Mathematics
MAT301H1-Y Groups and Symmetries, Summer 2021
Syllabus
Welcome to MAT301 Groups and Symmetries! Please read the full syllabus as it contains lots of important information about the course. It also includes resources and advice that will help you learn more effectively, and policies that you will be expected to know and follow. The time zone used for all scheduling purposes in this course is Eastern Standard Time (EST).
Instructors:
Teaching assistants:
Instructional team
Petr Kosenko
Matthew Sunohara
Lectures Office hours Email
LEC0101 (Wed. 9–10, Fri. 9–11) Tue. 11–13 (tentative) petr.kosenko@mail.utoronto.ca
LEC0201 (Tue. 13–14, Thur. 13–15) Wed. 12–13 (tentative) matthew.sunohara@mail.utoronto.ca
Keirn Munro
Tutorials Email
TUT0301, TUT0302 (Wed. 11–12)
keirn.munro@mail.utoronto.ca
Alexander Slamen
TUT0501 (Thu. 15–16)
alexander.slamen@mail.utoronto.ca
Aaron Tronsgard
Tutorials Email
TUT0101 (Tue. 10–11)
tronsgar@math.toronto.edu
Luke Volk
Sina Zabanfahm
Tutorials Email
TUT0401, TUT0402 (Thu. 9–10)
luke.volk@mail.utoronto.ca
N/A
s.zabanfahm@mail.utoronto.ca
Course overview
Course description: Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.
Prerequisites: MAT224H1/MAT247H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1. (These Prerequisites will be waived for students who have MAT257Y1.) For FASE students, MAT185H, MAT194H, MAT195H.
Exclusions: MAT347Y1
Learning objectives: Upon successful completion of the course, you should be able to:
• define and understand the basic concepts of group theory and basic examples of finite groups, including cyclic groups Cn, dihedral groups Dn, symmetric groups Sn, alternating groups An, and dicyclic groups Dicn ;
• state and understand the basic results of group theory; 1
Adrian She
TUT0201 (Tue. 14–15)
ashe@math.toronto.edu
• write proofs of simple results in group theory;
• recall some of the key ideas and tools that are used to prove the more difficult theorems of group theory
covered in the course;
• apply results from group theory to analyse a given finite group; and
• generate examples and counterexamples for simple group theoretic properties;
• recognise when an object or problem—mathematical or otherwise—has a nontrivial group of symme- tries, determine the group of symmetries, and apply results from group theory to the study of the object or problem.
Course website: The course website is on Quercus (q.utoronto.ca).
Delivery mode: As a result of the Covid-19 pandemic, the lectures, tutorials, and office hours will be held at scheduled times online using Zoom. The Zoom links will be emailed to you or posted on the course website.
Course textbook: Joseph A. Gallian, Contemporary Abstract Algebra, 8th, 9th, or 10th edition. Supplementary resources: Here are some supplementary resources that you may find useful.
• Some resources by Joseph A. Gallian:
– Flash cards for your textbook: text version, software version
– Advice on learning group theory: https://www.d.umn.edu/~jgallian/advice.html
• The School of Mathematics and Statistics at the University of St Andrews has a great resource for the history of mathematics called the MacTutor History of Mathematics Archive: https://mathshistory. st-andrews.ac.uk/
“MacTutor is a free online resource containing biographies of nearly 3000 mathematicians and over 2000 pages of essays and supporting materials.” It has pages on the development of group theory, the history of the abstract group concept, and every mathematician that will be mentioned in the course.
• A more comprehensive and advanced algebra textbook is: Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote. The first part of the book is on group theory and you might find it to be a useful reference.
Technology
Technology requirements: In order to participate in this course, you will need:
• a computer satisfying the University’s minimum technical requirements; and
• reliable internet access. It is recommended that you have a high speed broadband connection with a
minimum download speed of 5 Mbps.
If you are facing financial hardship and are unable to meet these requirements, you are encouraged to con- tact your college or divisional registrar (https://future.utoronto.ca/current-students/registrars/) to apply for an emergency bursary so that you can obtain the required items.
Here are some technology recommendations to help you get the most out of the course.
• It is highly recommended that you have a webcam and especially a microphone for participation in lecture, tutorial, and office hours.
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• Using only the required technology, it is possible for you to typeset your assignments and tests. You can do so using LibreOffice or LATEX. The latter is recommended as it produces better documents, is more flexible, and is a useful tool to be able to use. If you choose not to typeset your assignments and tests, you will need a method of capturing your hand-written work and uploading it to your computer. You will be responsible for making sure that your work is legible. You can use a scanner or a camera such as one on a phone. If you use your phone, we recommend that you use a free scanner app.
Participation in lectures, tutorials, and office hours: Please share your video as often as possible. Being able to see if students look confused can help the instructors and teaching assistants teach more effectively by adjusting the pace and adding clarification. By sharing your video, you also help to make the atmosphere more lively.
To communicate with your instructor or TA, it is ideal if you use a microphone. If you are unable to use a microphone, you should communicate by using the chat window.
Online communication policy:
• You are responsible for regularly checking your email and the course website on Quercus for commu- nication about the course.
• All emails to the instructors and TAs should contain the course code at the beginning of the subject line and come from a university email address. This will help prevent us from missing any of your emails.
• Only send emails to one person: either your course instructor, your TA, or the person responsible for grading an exercise for which you are requesting a regrade. Do not CC people on your emails.
• Only use email to communicate about matters that solely pertain to you (such as regrade requests, absences, accommodation, etc.). All other questions related to the course should be asked in at the beginning or end of lecture, in tutorials, in office hours, or on Piazza. This way, many students benefit from the answer to the question and you your questions may be answered without you having to ask them. The section on course components contains more information about tutorials, office hours, and Piazza.
Recording policy and copyright: Students may not create audio or video recordings of lectures, tutorials, or office hours, with the exception of those students requiring an accommodation for a disability or other circumstances. Students requiring an accommodation should speak to the instructor prior to beginning to record lectures.
Students creating unauthorised recording of lectures violate an instructor’s intellectual property rights and the Canadian Copyright Act. Students violating this agreement will be subject to disciplinary actions under the Code of Student Conduct.
Course components
Lectures: You will have three lecture hours per week. A tentative schedule of the lectures can be found at the end of the syllabus.
The lectures will cover some material that is not in your textbook and also some material in a different order than your textbook. Your textbook will complement the lectures, but not replace them. The lectures will be spent motivating and introducing concepts and then proving theorems about them. The main goal of the lectures is to help you understand the concepts and proofs covered in the course, where they come from, and how you might have come up with them on your own. It is highly recommended that you take notes during your lectures.
Tutorials: You will have one tutorial session per week. Tutorials will start on the second week of courses (May 10–14).
A list of exercise will distributed prior to each tutorial. You will be expected to work on the exercises on your own before the tutorials. In the tutorials, your teaching assistant may review some topics, warn you of common mistakes and misconceptions, provide guidance on the tutorial exercises, and answer questions you
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might have about course content. You should continue to work on the tutorial exercises after your tutorial and complete as many of them as possible as they will prepare you for your assignments.
Office hours: Both instructors will have one office hour per week. Please come to our office hours to ask questions about anything related to the course!
Piazza: As discussed in the online communication policy, all questions related to the course and not solely pertaining to one student may be asked in lecture, tutorial, office hours, or on Piazza, but not via email. Please join the class Piazza at: https://piazza.com/utoronto.ca/summer2021/mat301y1
Discord: There is a class Discord, which you can use to: study with your peers online, find someone to study or collaborate with, ask your peers for notes on a missed lecture or tutorial, or share resources. An invite link will be shared on Quercus.
Assessment
The written assessments in this course are an integral part of the learning process. You should not think of the written assessments as merely tests that you must do well on in order to receive a good grade, but as challenges that you must prepare for, struggle with, and overcome in order to learn. All written assessments will be administered using Crowdmark (https://crowdmark.com/).
Your written work will be graded according to their strengths in three qualities: clarity, correctness, and completeness. Note that completeness is impossible without correctness, and correctness is impossible without clarity. Clarity is the foundation of good proofs and solutions, so it is extremely important.
Completeness is a notion that depends on a hypothetical reader for whom you are writing. You should write your proofs and solutions so that a fellow student in the course would consider them complete. That is, a fellow student should not think anything that any detail is missing besides details that they could immediately supply. Consequently, in your proofs and solutions you may not use results from group theory that have not yet been covered in the class.
Participation: There are many studies demonstrating a statistically significant strong positive correla- tion between attendance and performance in courses. You will receive a participation grade based on your lecture and tutorial attendance. Your participation grade will be the number of lectures and tutorials that you attended divided by the number of lectures and tutorials that you were able to attend (excluding lectures and tutorials missed as a result of a declared absence).
Assignments: There will be six assignments. You will have 10 full days to compete each assignment. All assignments will be distributed automatically and due on the days provided in the table below. The time at which each assignment is distributed and due is 23:59 (11:59 PM) EST.
Collaboration on assignments: Collaborating on an assignment should be understood to mean working on or discussing the exercises or concepts involved with one or more of your academic peers. Asking someone for solutions does not count as collaborating with them and will not help you learn. Collaborating on your assignments with your academic peers is highly recommended. You can use the course Discord (see below) to find someone in the course to collaborate with. You may not collaborate on writing your assignments. Your assignments must be written independently.
Assignment #
Topics from week #
Distributed on
Due on
1
1
Friday, May 7
Monday, May 17
2
2, 3
Monday, May 17
Thursday, May 27
3
4, 5
Monday, May 31
Thursday, June 10
4
6, 7
Monday, July 5
Thursday, July 15
5
8, 9
Monday, July 19
Thursday, July 29
6
10, 11
Monday, August 2
Thursday, August 12
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Seeking help: If you have struggled with an exercise from one of your assignments or tutorials and are stuck on it, you should ask for help from your instructor, teaching assistant, or one of your academic peers. You can ask for help at the beginning or end of a lecture, in a tutorial, in office hours, or on Piazza. Do not ask for help before working on the exercise and struggling with it; that is how you learn. Do not ask for or accept a solution; ask for a hint to get unstuck.
It is probably more efficient to ask for help from your instructor, teaching assistant, or an academic peer, but you can also seek help in a textbook or online. If you are looking online, be careful about the reliability of the source. When using a written source to help you get unstuck on an exercise, treat it as if you are having a discussion with the writing and asking it for a hint. Read the minimum amount necessary to get yourself unstuck and then continue working on your own.
Your solutions must be written on your own and in your own words. Always make sure that you really understand your solution. Do not allow yourself to write things that you do not understand. Reflect on why you were stuck on the problem. What idea were you missing or misunderstanding? Think about what thought processes could have lead you to the solution on your own and try to apply them in your future work.
Late assignment policy: A late assignment will have 10% of its grade deducted for each day it is late. This will be handled automatically by Crowdmark.
While we do not like to deduct marks for late work, there must be a late penalty for the pragmatic reason that we cannot expect the teaching assistants to grade work that is submitted at any time after the due date. The late penalty is chosen so that the assignments can be returned to you as soon as 10 days after the due date. The teaching assistants will aim to finish grading the assignments quickly enough so they can be returned to you before the next assignment is due (within roughly 14 days).
Term tests: There will be two timed term tests. Both term tests will be distributed automatically by Crowdmark. While writing either term test, you may only consult your notes, your solutions to assignment or tutorial exercises, and the course textbook. No other resources are allowed. You may not discuss either term test with anyone until after the time it is due, excluding questions about the test direct to the course instructors.
Term test 1: There are two time slots in which you can write Term Test 1. • Time Slot 1: June 17, from 7–10 PM Eastern Time
• Time Slot 2: June 18, from 7–10 AM Eastern Time
You will receive two automatically generated emails from Crowdmark, one at 7 PM on June 17 for Time Slot 1 and one at 7 AM on June 18 for Time Slot 2.
If you click the link in the email for Time Slot 1 and begin Term Test 1, then you must write Term Test 1 in Time Slot 1 and submit your solutions by 10 PM Eastern Time on June 17. If you begin Term Test 1 during Time Slot 1, then any work submitted for Term Test 1 during Time Slot 2 will be ignored. Thus, if you wish to write Term Test 1 in Time Slot 2, ignore the email for Time Slot 1.
If you do not begin Term Test 1 using the link in the email for Time Slot 1, then you can write Term Test 1 in Time Slot 2 by using the link in the email for Time Slot 2. If you write Term Test 1 in Time Slot 2, then you must submit your solutions by 10 AM Eastern Time on June 18.
Term test 2: Term Test 2 is tentatively scheduled for August 5.
Final assessment: The final assessment will be scheduled by the registrar, and will occur in August. We will communicate information about the final assessment to you on Quercus and in lecture, once it is scheduled.
Academic integrity: “The International Center for Academic Integrity defines academic integrity as a commitment, even in the face of adversity, to six fundamental values: honesty, trust, fairness, respect, responsibility, and courage.” These values are vital for learning, researching, and teaching. Not only must you embody these values, but so must your teachers and the researchers who generate knowledge.
The following quotation is the preamble of the Code of Behaviour on Academic Matters. 5
“The concern of the Code of Behaviour on Academic Matters is with the responsibilities of all parties to the integrity of the teaching and learning relationship. Honesty and fairness must inform this relationship, whose basis remains one of mutual respect for the aims of education and for those ethical principles which must characterize the pursuit and transmission of knowledge in the University.
What distinguishes the University from other centres of research is the central place which the relationship between teaching and learning holds. It is by virtue of this relationship that the University fulfills an essential part of its traditional mandate from society, and, indeed, from history: to be an expression of, and by so doing to encourage, a habit of mind which is discriminating at the same time as it remains curious, which is at once equitable and audacious, valuing openness, honesty and courtesy before any private interests.
This mandate is more than a mere pious hope. It represents a condition necessary for free enquiry, which is the University’s life blood. Its fulfillment depends upon the well being of that relationship whose parties define one another’s roles as teacher and student, based upon differences in expertise, knowledge and experience, though bonded by respect, by a common passion for truth and by mutual responsibility to those principles and ideals that continue to characterize the University.
This Code is concerned, then, with the responsibilities of faculty members and students, not as they belong to administrative or professional or social groups, but as they co-operate in all phases of the teaching and learning relationship.
Such co-operation is threatened when teacher or student forsakes respect for the other–and for others involved in learning–in favour of self-interest, when truth becomes a hostage of expediency. On behalf of teacher and student and in fulfillment of its own principles and ideals, the University has a responsibility to ensure that academic achievement is not obscured or undermined by cheating or misrepresentation, that the evaluative process meets the highest standards of fairness and honesty, and that malevolent or even mischievous disruption is not allowed to threaten the educational process.
These are areas in which teacher and student necessarily share a common interest as well as common responsibil- ities.”
All suspected cases of academic dishonesty will be investigated following procedures outlined in the Code of Behaviour on Academic Matters. If you have questions or concerns about what constitutes appro- priate academic behaviour or appropriate research and citation methods, please reach out to your Course Instructor. Note that you are expected to seek out additional information on academic integrity from me or from other institutional resources (for example, the University of Toronto website on Academic Integrity: http://academicintegrity.utoronto.ca/).
Grading scheme: Your final grade for the course will be a weighted sum of your participation grade P, the average of your assignment grades A, your term test grades T1 and T2, and your final assessment grade F. Your final grade will be:
0.05P + 0.25A + 0.20T1 + 0.20T2 + 0.30F. Note that the weight of each assignment is 0.25/6 = 4 1 %.
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Regrade policy: Regrade requests must be emailed, following the online communication policy, to the person who graded the work.
Policy on missed assessments: Verification of Illness forms (“doctor’s notes”) are temporarily not required. Students who are absent from academic work for any reason (medical or non-medical) and at any time (not just when you have missed a course deadline), should report their absence through the Absence Declaration tool, which the University has created in response to the effects of the pandemic.
For more information see the FAQ. Note: “Your instructor will not be automatically alerted when you declare an absence. Therefore, it is your responsibility to let your instructor know that you have used the Absence Declaration so that you can discuss any needed consideration, where appropriate. Some instructors may ask their department to confirm absences reported by students to ensure that they have been entered into the system on the dates indicated by a student.” For help on how to use the Absence Declaration tool, visit: https://help.acorn.utoronto.ca/blog/ufaqs/declare-an-absence/
If you submit in a late assignment because of a declared absence, the late policy will be waived. If you miss one (or both) term test(s) because of a declared absence, your grading scheme will be re-weighted by increasing the weight of the final assessment grade to include the weight of the missed term test(s). If you miss the final assessment because of a declared absence, it will be deferred to a future offering of the course.
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Tips for success.
• As discussed above, clarity in your writing is essential for good proofs and solutions. Your writing must be unambiguous, detailed, grammatically correct, and correctly punctuated. If your writing is lacking in any of these qualities, it might fail to convey what you intend and your proof or solution might be rendered incomplete, incorrect, or meaningless. The University has several resources to help students improve their writing. For more information, visit: https://writing.utoronto.ca
• A simple tip to that can help to improve the quality of many students’ proofs and solutions is to use words instead of logical symbols such as: ∴ , ∵ , ¬ , ∨, ∧, =⇒ , ∀, and ∃. Doing so might prevent you from making mistakes in the logic of your proofs, and it will surely make your proofs more readable and have better style.
• Attend and actively participate in lectures and tutorials. Ask questions and offer answers when ques- tions are posed to you. Attend office hours frequently to ask questions and listen to the questions of others.
• Find people to study with and discuss course material with other students. (The class Discord might help with this.) Ask and answer questions on Piazza. You will benefit from learning the mistakes and questions of other students, which can reveal gaps in your understanding. You will learn a lot by explaining concepts to people, and doing so will reveal to yourself things that you do not understand well, which will help you refine your understanding.
• Learn from many, complementary resources. You have access to many free textbooks and sources (including online ones) via the University of Toronto Libraries. If an explanation in one source is confusing, a different explanation in another source might make things click for you.
• Always reflect on proofs and solutions. After you read or write a proof or solution, review the key steps and ideas. Reflect on what thought processes could lead to these ideas. Review how the hypotheses are used. Can the hypotheses be weakened? Can the hypotheses be weakened in order to obtain a weaker result? Can you find an alternative proof or solution. Read the feedback given to you on your graded work.
• Make sure you become and/or remain interested in and excited about the course material. It is hard to do well otherwise. Every topic in any field of study has aspects that you will find interesting or exciting; you just need to find them. Look for connections between the course material and other your interests.
Accessibility, inclusion, and support
Accessibility accommodations: The University provides academic accommodations for students with disabilities in accordance with the terms of the Ontario Human Rights Code. This occurs through a collaborative process that acknowledges a collective obligation to develop an accessible learning environment that both meets the needs of students and preserves the essential academic requirements of the University’s courses and programs.
Students with diverse learning styles and needs are welcome in this course. In particular, if you have a disability/health consideration that may require accommodations, please feel free to approach me and/or Accessibility Services at (416) 978 8060. The sooner you let us know your needs the quicker we can assist you in achieving your learning goals in this course. For more information, visit: http://studentlife. utoronto.ca/as
Diversity and inclusion: The University of Toronto is committed to equity, human rights and respect for diversity. All members of the learning environment in this course should strive to create an atmosphere of mutual respect where all members of our community can express themselves, engage with each other, and respect one another’s differences. U of T does not condone discrimination or harassment against any persons or communities.
Other academic and personal support:
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• Academic Success Centre: https://studentlife.utoronto.ca/department/academic-success/ • College/Faculty Registrars: https://future.utoronto.ca/current-students/registrars/
• Feeling distressed? Please seek support. We care about your well-being.
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Tentative lecture schedule
Week
Dates
Topics
Reading
1
May 3–7
review of modular arithmetic and equivalence re- lations; groups; Caley tables; basic concepts and properties; examples (including Z/nZ, (Z/nZ)×, some infinite groups, some matrix groups, and di- hedral groups)
Ch. 0, 1, 2
2
May 10–14
the order of an element; subgroups; subgroup tests; the cyclic subgroup generated by an element; cyclic groups; the order of gk; subgroups of cyclic groups; the centre and centralisers
Ch. 3, 4
3
May 17–21
permutation groups; cycles; the Cycle Decomposi- tion Theorem; order of a permutation
Ch. 5
4
May 24–28
transpositions; the Parity Theorem; inversions; the sign of a permutation; alternating groups
Ch. 5
5
May 31–June 4
isomorphisms and automorphisms; homomor- phisms; kernel and image of a homomorphism; Cay- ley’s representation theorem; homomorphisms from cyclic groups
Ch. 6&partofCh. 10
6
June 7–11
cosets; Lagrange’s theorem and applications; groups of order 2p
Ch. 7
• No classes from June 14–July 2.
• June 17–28: Final assessments for F courses and term tests for Y courses.
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July 5–9
surjective homomorphisms; normal subgroups; quo- tient (or factor) groups; First Isomorphism The- orem; simple groups; Cauchy’s Theorem for finite abelian groups
Ch. 9, 10
8
July 12–16
9
July 19–23
10
July 26–30
11
August 2–6
12
August 9–13
• Classes end on August 16.
• From August 18–30: final assessments for S and Y courses
Key dates: For key dates set by the Faculty of Arts and Science, including deadlines and holidays, visit: https://www.artsci.utoronto.ca/current/dates-deadlines
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