## 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.

## 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

• 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, 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)
• 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 p2 (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: TBA
• 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: TBA
• 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)

#### 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: TBA
• Chapter 25:TBA