Calendar
There is a proposed calendar on the syllabus, but here I will record what we actually get through in class.
- 8/29/19 – We went through the Review of Discrete and Proofs that is linked to below and then looked briefly at some examples. We saw how we could make more complicated structures out of simpler ones and discussed groups.
- 9/5/19 – We covered introductory material on groups from chapters 1 and 2.
- 9/12/19 – We covered material from chapter 4, sorry for getting bogged down in minutia. We will cover chapter 3 next week and push off the first exam to 9/26/19. The assignment due dates are changed as well (I updated them below).
- 9/19/2019 – We covered chapter 3 on subgroups. We looked at a lot of examples, and looked at the one step, two step, and finite group subgroup tests. The next class is your exam. Make sure to look at Practice Problems and at the Big Theorems listed for each unit. The undergraduate students should be able to apply theorems, and the graduate students should, in addition, be able to prove or outline a proof of the theorems.
- 9/26/2019 – You took your first exam. Next week we will look at a couple of questions where knowing, and understanding, your definitions was the difference between a problem being simple or impossible.
- 10/3/2019 – We covered material from chapter 5, next week we will do chapter 6, or what we need from it, and start chapter 7. The exercises for chapter 5 are below. Try to get a 2 or 3 of them done.
- 10/10/2019 – We covered a lot of material from chapter 6, including Cayley’s Theorem. Then we started chapter 7 and covered enough material so that we have the ground work to cover Lagrange’s Theorem in the next class.
- 10/17/2019 – We covered Lagrange’s Theorem and some of its consequences. We also started looking at normal subgroups in detail. We ended by laying out the pieces we need to show that cosets of a normal subgroup form a group themselves. For now we are skipping over the chapter 8 material and will come back to it when we need it.
- 10/25/19 – Tonight we discussed equivalence relations, equivalence classes, and quotient structures. The emphasis was on the idea that a quotient structure is an equivalence relation which respects the algebraic operations defined on the set. Then we discussed a particular example of this, Factor Groups (a.k.a Quotient Groups). We saw that if a subgroup is normal then the cosets acted in a consistent manner as group elements, i.e. if \(G\) is a group and \(N\triangleleft G\), then the set of left cosets of \(N\), denoted \[G/N=\{gN | g\in G\},\]form a group with \[gN\cdot hN = gh N,\ \forall g,h\in G.\] We then looked at a couple of ways to view quotient groups and the transformation form a group to a quotient group. For next weeks exam you should look at chapters 5 and 7 in their entirety, chapter 6 up to and including Cayley’s Theorem, and Chapter 9 through the section on Factor Groups. The lists of definitions and theorems that you need to study are updated below, ignore practice problems on internal and external direct products.
- 10/31/2019 – Announcement, Announcement, Announcement … I am flipping the next two classes so that we have time to answer questions about the material before the exam. I won’t spend the whole class on it, but we will spend a little time commenting on the content. With the time not spent discussing material from Part II, we started on Part III. We discussed homomorphisms as opposed to isomorphisms, so functions that are just required to satisfy the condition \[\phi(a*_G b)=\phi(a)*_{\overline{G}}\phi(b).\] Make sure to look at theorems 6.2, 6.3, 10.1, and 10.2.
- 11/7/2019 – Exam II
- 11/14/2019 – Handed back the exam. Reviewed material on homomorphisms, discussed kernels, and looked at the First Isomorphism Theorem.
- 11/21/2019 – We covered preliminary material from chapter 11 and looked at one or two examples of what the Fundamental Theorem of Finite Abelian groups tells us. We then launched into a proof of the theorem which we will finish after break. This is easily the longest theorem we have done this semester and really draws on a lot of previous work so that it acts as a cap stone to the course.
- 12/5/2019 – We finished the Fundamental Theorem of Finite Abelian Groups, you should be able to outline the overall structure of the proof and explain what it tells us; I don;t expect you to be able to write out the entire proof. Make sure you familiarize yourself with the definitions, theorems, and practice problems listed for Part III, particularly those from chapters 6, 10, and 11.
Links and Handouts:
Part I – Basics
Unit 1: Introduction, Definitions, and Examples
- Chapters: 0 – 2
- Definitions:
- Equivalence Relation and Class, Modular Equivalence, Relatively Prime, Functions, One-to-One, Onto
- Group, Abelian Group, Non-Abelian Group, Dihedral Group, Units Modulo n
- Big Theorems:
- Elementary Properties of Groups (Theorems 2.1-2.4)
- Practice Problems:
- Chapter 0 (p.23): 1-23 odd
- Chapter 1 (p.37): 1-15 odd
- Chapter 2 (p.54): 1-15, 21,33,45 odd
- Skills Assignments: Do six of the following (at least 1 from each chapter), Due 9/12
- Chapter 0 (p.23): 4, 16, 22
- Chapter 1 (p.37): 2, 10, 24
- Chapter 2 (p.54): 6, 4, 8, 32
Unit 2: Finite and Cyclic Groups
- Chapters: 3 & 4
- Definitions:
- Subgroup, Order of a Group, Order of an Element, Center of a Group, Centralizer of an Element
- Cyclic Group, Generator(s)
- Big Theorems:
- Subgroup Tests (Theorems 3.1-3.3)
- Special Subgroups (Theorems 3.4-3.6)
- Orders of Elements and Generators (Theorems 4.1, 4.2, and corollaries)
- Fundamental Theorem of Cyclic Groups (Theorem 4.3)
- Practice Problems:
- Chapter 3 (p.68): 1-9, 17, 23, 47 odd
- Chapter 4 (p.85): 1-15, 21, 33, 41, 51, 53, 61, 65, 73 odd
- Skills Assignments: Do six of the following (at least 1 from each chapter), Due 9/26
- Chapter 3 (p.68): 2,4,6,22,28
- Chapter 4 (p.85): 2,8,10,12,38,74
Part I Problem Solving Exercises:
Do six of the following (at least 1 from each chapter), Due 10/3
- Chapter 0 (p.23): 14, 30, 36
- Chapter 2 (p.54): 34, 42, 46
- Chapter 3 (p.68): 16, 34, 54
- Chapter 4 (p.85): 14 (hint apply Theorem 4.3), 40
Part II – Subgroups
Unit 3: Permutation Groups and Cayley’s Theorem
- Chapters: 5 & 6
- Definitions:
- Permutation Group (or Symmetric Group), Cycles, Permutations, Even and Odd Permutations
- Isomorphism, Aut(G), Inn(G)
- Big Theorems:
- Properties of Permutation (Theorems 5.1-5.3)
- Even and Odd Permutations (Theorems 5.5)
- Cayley’s Theorem (Theorem 6.1)
- Practice Problems:
- Chapter 5 (p.112): 1-11 odd
- Chapter 6 (p.132): 1,3,9,17 odd
- Skills Assignments: Do 4 of the following problems, at least one from each chapter.
- Chapter 5 (p.112): 8, 10, 16, 19 & 20 together, 26, 40
- Chapter 6 (p.132): 6,8,12,42
Unit 4: Cosets, Subgroups, and Products
- Chapters: 7-9
- Definitions:
- Coset
External Direct Product- Normal Subgroup, Factor Group(or Quotient Group),
Internal Direct Product
- Big Theorems:
- Lagrange’s Theorem (Theorem 7.1)
- Groups of Order 2p (Theorem 7.3)
External Direct Products of Units (Theorem 8.3)- Factor Groups (Theorem 9.2)
- Cauchy’s Theorem for Abelian Groups (Theorem 9.5)
External and Internal Products (Theorem 9.6)Classification of Groups of order \(p^2\) (Theorem 9.7)
- Practice Problems:
- Chapter 7 (p.150): 1-7, 11-21, 25, 29
Chapter 8 (p.167): 1,3,7,9, …- Chapter 9 (p.187): 1,3,5,9,11,13,23,47
- Skills Assignments:Do 4 of the following problems, at least one from each chapter.
- Chapter 7 (p.150): 4,6,10,12,16,20,22,30,34
Chapter 8 (p.167): 4, 10&11, 14, 18, 19&20- Chapter 9 (p.187): 2,6,12,14,18,28,38
Part II Problem Solving Exercises
Do one problem from each chapter.
- Chapter 5 (p.112): 14, 18, 38, 46 (Hint look at the orders of elements)
- Chapter 6: (p.132): 20, 24, 28
- Chapter 7 (p.150): 14, 18, 24, 26, 36, 40
Chapter 8 (p.167): 2, 12, 26- Chapter 9 (p.187): 22&24, 26, 32&37, 42
Part III – Structure
Unit 5: Morphisms
- Chapters: 6 & 10
- Definitions:
- Isomorphism, Aut(G), Inn(G)
- Homomorphism, Kernel
- Big Theorems:
- Properties of Isomorphisms (Theorem 6.2 & 6.3)
- Properties of Homomorphisms (Theorem 10.1 & 10.2)
- First Isomorphism Theorem (Theorem 10.3)
- Practice Problems:
- Chapter 6 (p.132): 7,11,13,21,31,39,49
- Chapter 10 (p.205): 1,3,5,9,11,13,15,17,19,31,33,35,45,47
- Skills Assignments: Do 4 of the following problems, at least one from each chapter.
- Chapter 6 (p.132): 44, ??
- Chapter 10 (p.205): 7,8,10,14,18,21,24,40,51 (for the odd problems be sure to give details)
Unit 6: Group Structures
- Chapters: 11,
24, & 26 - Definitions:
- Conjugacy Class, p-Group,
Sylow p-Subgroup, Generators, Relation, Equivalence Classes
- Previous Theorems:
- Cayley’s Theorem (Theorem 6.1)
- Lagrange’s Theorem (Theorem 7.1)
- External and Internal Products (Theorem 9.6)
- Cauchy’s Theorem for Abelian Groups (Theorem 9.5)
- Classification of Groups of order \(p^2\) (Theorem 9.7)
- Big Theorems:
- Fundamental Theorem of Finite Abelian Groups (Theorem 11.1)
Sylow’s First Theorem (Theorem 24.3)Cauchy’s Theorem (Corollary to Theorem 24.3)Universal Mapping Theorem (Theorem 26.2)
- Practice Problems:
- Chapter 11 (p. 220): 1-9,15-23
Chapter 24:Chapter 26:
- Skills Assignments: Do 2 of the following problems.
- Chapter 11 (p.220): 6,8,10,12,16,26
Chapter 24:Chapter 26:
Problem Solving Exercises for Part III
Do one problem from each chapter.
- Chapter 6 (p.132): 18, 26, 32, 38
- Chapter 10 (p.205): 6,20,22
- Chapter 11 (p.220): 18,22,29 & 30 (together), 32
Chapter 24:Chapter 26: