Final Exam Week Office Hours:

• Tuesday 12-7-2021: 2:30-4:30pm
• Thursday 12-9-2021: 1:00-3:00pm
• Monday 12-13-2021: 12:00-2:00pm
• Tuesday 12-14-2021: 3:00-5:00pm

Calendar

There is a proposed calendar on the syllabus, but here I will record what we actually get through in class.

• 8-31-2021: Tonight we looked at the syllabus and how the class is going to run. Then we spent time going over this slide show on some concepts from Number Theory and Proofs. The emphasis was on building common vocabulary and base example before beginning truly new material. But we also tried to understand the importance of looking at how proofs are constructed from axioms, definitions and previous results. This material relates to chapter 0, next week we launch into chapters 1 and 2.
• 9-7-2021: We spent most of our time tonight looking at the Dihedral Group $$D_4$$ with a focus on the properties that are common to all groups. We then looked at the definition of a group and some basic group properties. This covers most of the material from chapters 1 and 2. You should scan through chapter 1 to get the author’s perspective on dihedral groups and look at the other examples of group in chapter 2. We will discuss other examples of groups in class as they become relevant. beyond the basics in the text we looked at ways of visualizing groups, you can play with this further using the online resource Group Explorer (https://nathancarter.github.io/group-explorer/index.html). Finally, if you have not already done so you should start looking at exercises and practice problems from chapters 0 through 2.
• 9-14-2021: Tonight we looked at chapter 3 from the text. We started class by talking about modular equivalence $a\equiv b\pmod{n} \Leftrightarrow n|(b-a)$ and arithmetic so we could discuss the groups $$\mathbb{Z}_m$$. We saw that while $$\mathbb{Z}_8$$ and $$\mathbb{Z}_2\oplus\mathbb{Z}_4$$ are different groups, $$\mathbb{Z}_{12}$$ and $$\mathbb{Z}_3\oplus\mathbb{Z}_4$$ are the same. Following this we covered the meanings of order of a group and order of and element. For the last half of class we discussed subgroups and three different subgroup tests. We finished by showing that the center of a group, $$Z(G)=\left\{x\in G|\forall a\in G: xa=ax\right\}$$, is a subgroup by using the two step subgroup test. For next time you should try to think about how we would do the same for the centralizer of an element, $$C(a)=\left\{x\in G| ax=xa\right\}$$.
• 9-21-2021: Today we finished up Part 1 of the course by discussing the group of units modulo n, $$\mathbb{Z}^*_n=U(n)$$ and going over the material on cyclic groups in chapter 4. We also spent some time discussing Euler’s $$\phi$$-Function which counts the number of positive integers less than or equal to a given number which are relatively prime to that number. Your first exam is next week on Tuesday the 28th.
• 9-28-21: Exam 1
• 10-5-2021: Tonight we started unit 2 with a discussion of permutations. In this class we introduced the symmetric group, $$S_n$$, the alternating group, $$A_n$$, cycles and cycle notation, even and odd permutations, and transpositions. In the last portion of class we covered the following tools which can be used to understand the theorems in chapter 5 (i.e. read through the theorems and their proofs):
• If $$\alpha$$ is a permutation and $$\alpha^k(a)=\alpha^j(b)$$, then there exists $$m$$ such that $$\alpha^m(a)=b$$.
• If $$\alpha$$ is an $$n$$-cycle with $$n\geq 3$$, then $$\alpha$$ can be written as a product of a transposition and an $$(n-1)$$-cycle, e.g. $(a_1a_2a_3\cdots a_{k-1}a_k)=(a_1a_k)(a_1a_2a_3\cdots a_{k-1})$
• Every transposition $$(xy)$$ can be expanded to a product of three transpositions like so $(xy)=(xz)(yz)(xz)\ \text{or}\ (yz)(xz)(yz).$
• In a product of two transpositions, any single element permuted by the product can be shifted to the left most transposition so that it doesn’t appear in the right hand transposition, e.g. $(ab)(ac)=(ac)(bc).$
• 10-12-21: Tonight you got back your first assignment. We discussed the importance of clear communication and for being sure to do your own work and not submitting answers found on the internet as your own. The deadline for the Unit 1 Problem solving exercises was extended to next week, Tuesday October 19th, so that you can be sure to submit work that is yours alone and that demonstrates good communication skills. Next we reviewed material from last week on the symmetric group $$S_n$$ and worked toward an understanding of Cayley’s Theorem which states that every group is isomorphic to a group of permutations. Along the way we discussed one-to-one and onto functions, bijections, operation preserving functions, and isomorphisms.
• 10-19-21: Tonight we looked at chapter 7. We covered Lagrange’s Theorem which tells us that given a group and a subgroup, the group can be written as a union of disjoint cosets of the subgroup and if the group has finite order then the order of the subgroup divides the order of the group. Then we looked at some consequences of and subsequent theorems related to Lagrange. These theorems can all be used to help us understand what types of elements a group must have. Next week we will be looking at chapter 8 and some of chapter 9.
• 10-26-2021: Tonight we covered material on external direct products, $$G\oplus H$$, and normal subgroups, $$gH=Hg$$. We also introduced internal direct products and outlined the significant theorems connecting them to external direct products; we will start class next time with these.
• 11-2-2021: Tonight we finished off material on external, $$G=H\oplus K$$, and internal, $$G=HK$$, direct products. In particular we saw that if $$G=HK$$, then $$G\cong H\oplus K$$. This allows us to better understand the structure of certain groups, i.e. it must be the case that $\mathbb{Z}_{210}\cong \mathbb{Z}_{2}\oplus \mathbb{Z}_{3}\oplus \mathbb{Z}_{5}\oplus \mathbb{Z}_{7},$ and so the later group is cyclic. This idea will be extended further when we get to the Fundamental Theorem of Finite Abelian Groups. In the second half of class we saw that if a subgroup is normal, $$H\triangleleft G$$, then we can extend the binary operation from $$G$$ to one defined on the cosets of $$H$$ so that $G/H=\left\{gH\middle|g\in G\right\}$ forms a group called a quotient group. In particular concepts of modular arithmetic can now be understood to be a special case of what we can call a quotient structure made up of a set with some binary operations, and an equivalence relation which respects the binary operations. We ended class by looking at how we can use quotient groups to prove Cauchy’s Theorems for Abelian Groups. Don’t forget that next week is exam 2, if you haven’t already be sure to look carefully at the review material below.
• 11-9-2021: Exam 2
• 11-16-2021: We covered the material from chapter 10 culminating in the First Isomorphism Theorem. We also drew a parallel between the way homomorphisms split a group into elements in the kernel and equivalence classes of elements not in the kernel and the way linear transformation split up a vector space into a null space and row space. (FYI vector spaces are actually a special case of an algebraic structure called a module.)
• 11-23-2021: Tonight we looked at the Fundamental Theorem of Finite Abelian Groups which allows us to write every abelian group as an external (or internal) direct product of cyclic groups, e.g. if $$|G|=60$$ then either $G\cong \mathbb{Z}_4\oplus\mathbb{Z}_3\oplus\mathbb{Z}_5$ or $G\cong\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_3\oplus\mathbb{Z}_5.$
• 11-30-2021: We covered Cauchy’s Theorem and Sylow’s Theorem. We will cover the rest of chapter 23 next week.

Typesetting Out-of-Class Work (Template)

1. Create an account at Overleaf.com using your WCSU email account
2. Watch the introductory video here VIDEO LINK, the document created in the video is here: https://www.overleaf.com/read/khbktbvvvkjd

If you are having problems typing up a document and would like feedback you can share a link to it by following the directions here: https://www.overleaf.com/learn/how-to/Sharing_a_project; send me the link to view but not edit. Here is a quick reference sheet with symbols and such: Quick Reference

Part 1: Basics

Skills Exercises: Do six of the following, at least 1 from each chapter – Due 10/05

• Chapter 0 (p.21): 4, 18
• Chapter 1 (p.33): 2, 10, 22
• Chapter 2 (p.52): 4, 6, 10
• Chapter 3 (p.69): 2,4,8,32
• Chapter 4 (p.89): 2,8,10,12,88

Problem Solving Exercises: Do four problems, one from each chapter – Due 10/12 10/19

• Chapter 0 (p.21): 16, 30, 36
• Chapter 2 (p.52): 36, 40
• Chapter 3 (p.69): 38, 58
• Chapter 4 (p.89): 14 (hint apply Theorem 4.3), 46

Test Preparation Materials: (Test on 9/28)

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, Subgroup, Order of a Group, Order of an Element, Center of a Group, Centralizer of an Element, Cyclic Group, Generator(s)

Theorems:

• Elementary Properties of Groups (Theorems 2.1-2.4)
• 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 0 (p.21): 1-23 odd
• Chapter 1 (p.33): 1-15 odd
• Chapter 2 (p.52): 1-15, 21,33 odd
• Chapter 3 (p.69): 1-9, 21, 27, 51 odd
• Chapter 4 (p.89): 1-15, 29, 49, 59, 61, 69, 85 odd

Part 2: Subgroups

Skills Exercises: Do six of the following, at least 1 from each chapter – Due 11/16

• Chapter 5 (p.118): 8, 12, 15 & 16 together, 30, 48
• Chapter 6 (p.141): 6,8,12,44
• Chapter 7 (p.163): 8,10,14,18,20,28
• Chapter 8 (p.184): 4, 10&11, 14, 18, 19&20
• Chapter 9 (p.209): 2,6,12,14,28,38

Problem Solving Exercises: Do four problems, no more than one from each chapter – Due 11/23

• Chapter 5 (p.118): 18, 44, 52 (Hint look at the orders of elements)
• Chapter 6: (p.141): 22, 40
• Chapter 7 (p.163): 12, 16, 38, 42
• Chapter 8 (p.184): 2, 12
• Chapter 9 (p.209): 22&24, 26, 32&37, 42

Test Preparation Materials: (Test on 11/9)

Definitions:

Permutation Group (or Symmetric Group), Cycles, Permutations, Even and Odd Permutations, One-to-One and Onto Functions, Isomorphism, Aut(G), Inn(G), Coset, External Direct Product, Normal Subgroup, Factor Group(or Quotient Group), Internal Direct Product

Theorems:

• Properties of Permutation (Theorems 5.1-5.3)
• Even and Odd Permutations (Theorems 5.5)
• Cayley’s Theorem (Theorem 6.1)
• Lagrange’s Theorem (Theorem 7.1)
• Cardinalities of Products of Subgroups $$\left(|HK|=|H||K|/|H\cap K|\right)$$ (Theorem 7.2)
• 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 5 (p.118): 1-11 odd
• Chapter 6 (p.141): 1,3,9,17 odd
• Chapter 7 (p.163): 1,3, 11-21, 23, 27
• Chapter 8 (p.184): 1,3,7,9
• Chapter 9 (p.209): 1,3,5,9,11,13,23,47

Part 3: Group Structure

Skills Exercises: Do six of the following from at least three chapters – Due on 12/14

• Chapter 6 (p.141): 46, ??
• Chapter 10 (p.232): 7,8,10,14,18,21,24,40,59 (for the odd problems be sure to give details)
• Chapter 11 (p.249): 6,8,10,12,16,26
• Chapter 23: (p.440): 4,8,18,20(applying Cor p.434)
• Chapter 25: TBA

Problem Solving Exercises: Do four problems from at least three chapters – Due on 12/14

• Chapter 6 (p.141): 18, 28, 34, 40
• Chapter 10 (p.232): 6,20,22
• Chapter 11 (p.249): 18,22,31 & 32 (together), 34
• Chapter 23 (p.440): 10 (Use #9 and Lagrange’s Theorem), 14, 16 (Hint: $$a^2=ab^{-1}ba$$),22,26 (Hint: how many distinct 5 cycles or 3 cycles can you count)
• Chapter 25: TBA

Test Preparation Materials: (Test on 12/14)

Definitions:

Isomorphism, Aut(G), Inn(G), Homomorphism, Kernel, Conjugacy Class, p-Group, Sylow p-Subgroup, Generators, Relation, Equivalence Classes

Theorems:

• Properties of Isomorphisms (Theorem 6.2 & 6.3)
• Properties of Homomorphisms (Theorem 10.1 & 10.2)
• First Isomorphism Theorem (Theorem 10.3)
• 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)
• Cauchy’s Theorem
• Sylow’s First Theorem

Practice Problems:

• Chapter 6 (p.141): 7,11,13,23,33,41,51
• Chapter 10 (p.232): 1,3,5,9,11,13,15,17,19,31,33,35
• Chapter 11 (p. 249): 1-9,15-23
• Chapter 23 (p.440): 1-8,…
• Chapter 25:TBA