| Week |
Topic
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Assignments
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| 1 |
Introduction to Groups
I realized that to me, Number Theory, Geometry, and the Theory of Algebraic Solutions were only shadows cast in different directions by some central solid essence. I tried to reconstruct that central object and realized it was Abstract Algebra.
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Read Chapters 0 & 1,
Perusall Syllabus Assignment, Pre-Semester survey https://forms.gle/cVThcvziKWWWyBJb6,
Perusall Humanizing Mathematics Assignment (due Friday 9/3)
Homework Assignment 1 (due Tuesday 9/7)
- I encourage you to work in groups of 2 or 3, turning in one write up for your group and discussing not only the turned in problems, but suggested problems together
- Read Chapter 2.
- LaTeX and turn-in the following 5 problems (problem numbers refer to exercises in Gallian's Abstract Algebra 8th edition).
- Ch. 0/ #12, Ch. 1/ #24, Ch. 2/ #19 (you may do #16 instead if you haven't had Linear Algebra), #37, and #47
- Read and think about (but do not turn-in) the following problems:
- Ch.1/ 5, 6, 8, 9, 17, 19, 22 and Ch. 2/13, 16, 17, 25, 31, 49, 59
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| 2 |
Finite Groups, Subgroups, & Cyclic Groups
I realized that to me, the symmetries of an equilateral triangle, the symmetries of a tri-hexaflexagon, and the permutations of 3 elements were only shadows cast in different directions by some central solid essence. I tried to reconstruct that central object and derived the concept of group isomorphism.
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Perusall Assignments: (due Sunday 9/12)
- Group Theory and the Rubik's Cube (22 minutes across 3 videos)
- Quanta article on Polynomial Building Blocks
Homework 2 (due Tuesday 9/14) : LaTeX and turn-in the following 5 problems as well as the Top 5 individual assignment.
- Ch.2/ #37
- Ch.3/ #31, 32, 68, and 70
- Top 5 list (in preparation for Group Exam #1 next week). This part of the assignment needs to be done individually, even if working in a group for the rest of the assignment:
- What are the five most important ideas/definitions/theorems we have seen so far in this course?
- Which of these five do you think you understand best and why?
- Which do you think you understand the least and why?
- Propose a problem for the upcoming group exam and explain why you think it would be a good problem.
Read and think about (but do not turn-in) the following problems:
- Ch.3/ 4, 14, 36, 38, 40, 43, 44, 53, 56, 59, 60, 65, 66, 70, 74, 77, and 79
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| 3 |
Ch. 4 Cyclic Groups
I realized that to me, all cyclic groups of order n were only shadows cast in different directions by some central solid essence. I tried to reconstruct that central object and realized that all cyclic groups of order n are isomorphic to $Z_n$.
Group Exam 1 Thursday 9/16
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Group Exam (Thursday 9/16)
Perusall Assignments: Monster group Numberphile Videos (due Sunday 9/19)
Homework 3 (due Tuesday 9/21) : LaTeX and turn-in the following 5 problems
- Ch.3/ #4 and 77
- Ch.4/ #39, 40, and 82
- Homework 1 rewrites (optional)
Read and think about (but do not turn-in) the following problems: Ch.4/ 14, 19, 23, 37, 64, 69, 74, and 76
Group research journals collected Thursday 9/23
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| 4 |
Ch. 5 Permutations
I realized that to me, all groups were only shadows cast in different directions by some central solid essence. I tried to reconstruct that central object and realized that every group is isomorphic to a group of permutations.
Group Research Journal due Friday 9/24 (by 2pm)
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Tuesday 9/21: Look over Ch 4 before class (cyclic groups)
Thursday 9/23: Ch. 5 Permutations
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Homework 3 (due Thursday 9/23 midnight - upload to WISE > Assignments)
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Key to Homework 3 will be sent out Thursday night. Use this key to correct your work. You can submit two questions for me to answer after your correct your work (maybe you are not sure if your approach works, or you wonder if you needed to include that part you didn't include but I do on the key, or vice versa, etc). These questions need to be submitted by Saturday 9/25. Details on how to submit your questions are forthcoming.
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Group research journals turn in by 2pm Friday 9/24
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Week 4 Survey (https://forms.gle/zt8r1TBN3fDjm7ea9). Please complete by Sunday night 9/2
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Homework 4 (due Tuesday 9/28) : LaTeX and turn-in the following 5 problems
- Ch.4/ #19, 20, 49, and 58
- Supplementary Exercises Ch.s 1-4/ #36
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Read and think about (but do not turn-in) the following problems: Ch.4/ 24, 45, 46, 84, Supplementary Exercises / 1-4, 11, 15, 16, 19, 28, 36-37
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| 5 |
Ch. 6 Isomorphism
"Cayley's Theorem tells us that abstract groups are not different from permutation groups. Rather, it is the viewpoint that is different. It is this difference of viewpoint that has stimulated the tremendous progress in group theory and many other branches of mathematics in the twentieth century." - J. Gallian
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Tuesday 9/28: Look over Ch 5 before class (permutation groups)
Thursday 9/30: Look over Ch. 6 before class (Isomorphism)
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Week 4 Survey (https://forms.gle/zt8r1TBN3fDjm7ea9). Please complete by Wednesday night 9/29 at the latest
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Homework 4 (due Tuesday 9/28) : Turn-in a hard copy in class or to my office
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Perusall Assignments: Change Ringing and Group Theory (due Sunday 10/3)
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Homework 5 (due Tuesday 10/5) : LaTeX and turn-in the following 5 problems
- Ch.5/ #32, 35, 43, 61, and 81
- Top 5 list (in preparation for Group Exam #2 next week). This part of the assignment needs to be done individually, even if working in a group for the rest of the assignment:
- What are the five most important ideas/definitions/theorems from chapters 4, 5, and 6?
- Which of these five do you think you understand best and why?
- Which do you think you understand the least and why?
- Propose a problem for the upcoming group exam and explain why you think it would be a good problem.
- Read and think about (but do not turn-in) the following problems: Ch.5/ 11, 18, 17, 23, 27, 31, 34, 37, 39, 45, 61, 65, 68
Group Celebration of Knowledge #2 will be Thursday 10/7. Study guide will be posted by Tuesday 10/5
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| 6 |
Ch. 6 Isomorphism
Group Celebration #2 Thursday 10/7
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Tuesday 10/5: Look over Ch 7 before class (Cosets & Lagrange's Theorem)
Thursday 10/7: Group Celebration of Knowledge #2 (see study guide below)
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Homework 5 (due Tuesday 10/5) : Turn-in a hard copy in class or to my office
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Perusall Assignment: Group Theory Overview (due Thursday 10/7) . Note the change in due date.
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Group Celebration of Knowledge #2 (Thursday 10/7)
- Concepts covered:
- Cyclic Groups: Definition of cyclic group, definition of cyclic subgroups, relationships between |a| and |a^k|, what a^n=e implies about |a|, relationships between element orders and the order of a cyclic group, generators of <a>, the Fundamental Theorem of cyclic groups.
- Permutations: Definition of a permutation, various ways of presenting permutations, Ruffini's theorem, Cayley's theorem, S_n, A_n
- Isomorphisms: Definition of an isomorphism, definition of an automorphism, how to prove something is an isomorphism, Aut(G), Inn(G), properties of isomorphimsm
- Still need to know definitions of and how to work with: Groups, subgroups, centers, and centralizers
- Suggested problems to look over
- Ch.4/ 21, 24
- Sup 1-4/ 1, 2, 16
- Ch. 5/ 23, 38, 39
- Ch.6/ 15, 27, 61
- Proof that isomorphism is an equivalence relation on the set of all groups
- Proofs of any of the properties of isomorphism from the handout given in class on Thursday (9/30)
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Homework 6 (due Tuesday 10/12) : LaTeX and turn-in the following 5 problems
- Ch.6/ #7, 10, 27, 15 (Aut(G) is a group), and 45
- Read and think about (but do not turn-in) the following problems: Ch.6/2, 3, 9, 12, 15, 27, 28, 30, 33-35, 37-39, 42-46, 56, 61, 63
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| 7 |
Ch. 7 Cosets & Lagrange's Theorem, Ch. 8 Group Products
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Tuesday 10/12: Look over Ch 7 (Cosets, Lagrange's Theorem, orbits and stabilizers) before class
Thursday 10/14: Look over Ch 8 (external direct products) before class
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Homework 6 (due Tuesday 10/12): Turn-in a hard copy in class or to my office
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Perusall Assignments: The Genius Box & Conway Reflects (due Thursday 10/14)
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Homework 7 (due Tuesday 10/19 by midnight) : LaTeX and turn-in the following 6 problems to WISE > Assignments. I will post solutions Thursday 10/21. Use this key to correct your work. You can submit two questions for me to answer after your correct your work.
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Ch.7/ #6, 12, 42, 45, 48, 61
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Read and think about (but do not turn-in) the following problems: Ch.7/ #1, 4, 7-9, 13, 17, 18, 20, 22, 24, 27, 30, 33, 43, 58-62
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| 8 |
Ch.s 8 & 9: Group Products
Group Research Journal due Friday 10/22 (by 2pm)
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Tuesday 10/19: Look over Ch 8 (External Direct Products) before class
Thursday 10/21: Look over Ch 9 (Normal Subgroups & Internal Direct Products) before class
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Homework 7 (due Thursday 10/21 by 5pm) : LaTeX and turn-in the following 6 problems to WISE > Assignments. I will post solutions Thursday night. Use this key to correct your work. If you have questions after looking over the key, please submit them through WISE Assignments or on Discord by Saturday night (10/23)
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Perusall Assignments: Euler's Formula from a Group Theoretic Perspective (due Thursday 10/21)
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Group Research Journal: 2nd Check-in (due by 2pm Friday 10/22 to my office or via WISE Assignments)
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Week 8 Survey (please complete by Monday 10/25) https://forms.gle/DhZAqsifXsr6pShx9
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Homework 8 (due Tuesday 10/26) : LaTeX and turn-in the following 6 problems
- Ch.8/ #39, 56, 70
- Sup 5-8/ #6
- Ch. 9/ #2, 7
- Read and think about (but do not turn-in) the following problems: Ch.8/ #1, 2, 11, 16, 17, 23, 26, 41, 42, 47, 55, 61, 63, 69, 71, 73, 75-79 and Ch. 9/#1, 6, 9,
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| 9 |
Ch. 9: Factor Groups
Individual Celebration of Knowledge 10/28
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Tuesday 10/26: Look over Ch 9 (Normal Subgroups & Factor Groups) before class
Thursday 10/28: Individual Celebration of Knowledge (See the Celebration Study Guide one WISE > Resources > Handouts)
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Homework 8 (due Tuesday 10/26) : LaTeX and turn-in hard copy in class or to my office.
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Take-home problems from Individual Celebration of Knowledge (due Tuesday 11/2)
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Homework 9 (due Tuesday 11/2) : LaTeX and turn-in the following 3 problems
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Ch.9/ # 11, 55, 68
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Read and think about (but do not turn-in) the following problems: Ch.9/#12, 17, 19, 25, 29, 31, 33, 38, 39, 43, 44, 49, 51, 56, 57, 61, 63, 64, 67, 69, 72, 74, 75
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| 10 |
Ch. 10: Homomorphism
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Tuesday 11/2: Look over Ch 10 Homomorphism
Thursday 11/4: Look over Ch 10 Homomorphism
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Homework 9 (due Tuesday 11/2) : LaTeX and turn in hard copy in class or to my office.
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Take-home problems from Individual Celebration of Knowledge (due Tuesday 11/2)
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Perusall Assignment: Quaternions! (due Thursday 11/4)
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Homework 10 (due Tuesday 11/9) : LaTeX and turn in hard copy in class or to my office.
- Ch.10/ #7, 41, 45 or 50 - your choice, 51, 58
- Read and think about (but do not turn-in) the following problems: Ch.10/ #8, 9, 11, 15, 17, 19, 61
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| 11 |
Ch. 11: Fundamental Theorem of Finite Abelian Groups
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Tuesday 11/9: Look over Ch 11 Fundamental Theorem of Finite Abelian Groups
Thursday 11/11: Look over Ch 12 Rings
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Homework 10 (due Tuesday 11/9) : LaTeX and turn in hard copy in class or to my office.
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Perusall Assignment: Map of Mathematics (due Thursday 11/9)
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Homework 11 (due Tuesday 11/16) : LaTeX the following problems and turn in hard copy in class or to my office.
- Write down the elements of $$<[5],[3]> \in U(12)\oplus\mathbb{Z}_9$$
- Find an example of an external direct product of cyclic groups that is not cyclic
- Fill in Parts I and II of the Fundamental Theorem of Finite Abelian Groups.
- Ch.11/ 26, 31, 38 (Extra Credit: 14, 21)
- Read and think about (but do not turn-in) the following problems: Ch.11/ 3, 5, 7, 9, 11, 12, 13, 15, 19, 23, 27, 37
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| 12 |
Ch. 12: Rings & Group Celebration #3 11/18
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Tuesday 11/16: Look over Ch 12 Rings
Thursday 11/18: Group Celebration of Knowledge #3 (See Study Guide Below)
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Homework 11 (due Tuesday 11/16) : LaTeX the following problems and turn in hard copy in class or to my office
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Group Celebration #3 (Thursday 11/18) Prepare one page of notes, one-sided. Theorems and Definitions are allowed, as well as general outlines of proof approaches, but no fully worked out problems or proofs.
- Concepts covered:
- Normal subgroups and Factor groups.
- Internal & External Direct Products
- Group Homomorphisms & Properties of Homomorphism
- The Fundamental Theorem of Abelian Finite Groups & Isomorphism Classes
- Still need to know definitions of and how to work with: Groups, subgroups, centers, and centralizers, cyclic groups, permutations, isomorphism, element order, group order, Lagrange's Theorem, properties of isomorphism, etc.
- Suggested problems to look over
- Ch.9/ 9, 27, 29, 31, 33, 38, 43, 51, 61, 63
- Ch. 10/ 12, 17, 19, 23, 42
- Ch. 11/ 7, 14, 23
- Supp 9-11/ 5, 7, 8, 11, 19. 26, 27
- Proofs of any of the properties of homomorphism from the handout given in class
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| 13 |
Fall Break
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| 14 |
Ch. 13 & 14: Rings, Fields, & Ideals
Group Research Journal due Friday 12/3
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Tuesday 11/30: Look over Ch 13 Rings, Fields, Integral Domains
Thursday 12/2: Look over Ch 14 Ideals & Factors Rings
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Week #12 Survey (https://forms.gle/6NTAK9Yv1jeTC42v9) please fill out by Tuesday 11/30
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Homework 12 (due Tuesday 11/30):
- Read Ch. 12 on Perusall and comment with either a question or your favorite thing about rings
- LaTeX the following problems and turn in hard copy in class or to my office
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Perusall Assignment: Math in Seventeen Syllables (due Thursday 12/2)
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Group Journal - Final Collection (due Friday 12/3) : Please turn-in to my office or upload on WISE
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| 15 |
Ch. 14 & 15: Factor Rings & Ring Homomorphism
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Tuesday 12/7: Look over Ch 14 Ideals and Factor Rings, Final Brainstorm
Thursday 12/9: Look over Ch 15 Ring Homomorphism, Group Journal Swap-meet
- Homework 13 (due Thursday 12/9) : LaTeX the following problems and turn in hard copy in class or to my office.
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- Ch.13/ 7, 35, 63
- Ch.14/ 3, 29, 39
- Read and think about (but do not turn-in) the following problems: Ch. 14/ 6, 7, 9-13, 16, 29, 34, 41, 51, 57, 60
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Poem Assignment: Compose 1-3 Abstract Algebra themed haikus, lyrics, epic poems, or catchy songs (due Thursday 12/9)
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| 16 |
Final Celebration of Knowledge, Monday 12/13, 2-5pm |
