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*When introduced to a new idea, always ask why you should care.\
Do not expect an answer right away, but demand one eventually.\
--- Ravi Vakil *
If you like this book and want to support me, please consider buying me a coffee!\
[{width="32ex"}](https://ko-fi.com/evanchen)\
<https://ko-fi.com/... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Title Embellishments | 01_title-embellishments.md | 0 | 436 | |
The origin of the name "Napkin" comes from the following quote of mine.
> I'll be eating a quick lunch with some friends of mine who are still in high school. They'll ask me what I've been up to the last few weeks, and I'll tell them that I've been learning category theory. They'll ask me what category theory is about... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Preface | 02_preface.md | 0 | 682 | |
The *Infinitely Large Napkin* is a light but mostly self-contained introduction to a large amount of higher math.
I should say at once that this book is not intended as a replacement for dedicated books or courses; the amount of depth is not comparable. On the flip side, the benefit of this "light" approach is that it... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Preface | About this book | 02_preface.md | 1 | 689 |
The project is hosted on GitHub at <https://github.com/vEnhance/napkin>. Pull requests are quite welcome! I am also happy to receive suggestions and corrections by email.
The project is hosted on GitHub at <https://github.com/vEnhance/napkin>. Pull requests are quite welcome! I am also happy to receive suggestions and... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Preface | Source code | 02_preface.md | 2 | 1,460 |
The preface would become too long if I talked about some of my pedagogical decisions chapter by chapter, so contains those comments instead.
In particular, I often name specific references, and the end of that appendix has more references. So this is a good place to look if you want further reading.
The preface would... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Preface | More pedagogical comments and references | 02_preface.md | 3 | 813 |
I am indebted to countless people for this work. Here is a partial (surely incomplete) list.
- Thanks to all my teachers and professors for teaching me much of the material covered in these notes, as well as the authors of all the references I have cited here. A special call-out to , , , , , , , , , , which were espec... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Preface | Acknowledgements | 02_preface.md | 4 | 1,112 |
As explained in the preface, the main prerequisite is some amount of mathematical maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on.
I also assume the reader is familiar with basic terminology about sets and functions (e.g. "what is a bijection?"). If n... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Prerequisites | 03_advice.md | 0 | 1,205 |
In this book, there are three hierarchies:
- An inline **question** is intended to be offensively easy, mostly a chance to help you internalize definitions. If you find yourself unable to answer one or two of them, it probably means I explained it badly and you should complain to me. But if you can't answer many, you ... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Questions, exercises, and problems | 03_advice.md | 1 | 764 |
At the risk of being blunt,
Read this book with pencil and paper.
Here's why:
{width="50%"}\
Image from [@img:read_with_pencil]
You are not God. You cannot keep everything in your head.[^2] If you've printed out a hard copy, then write in the margins. If you're trying to save pap... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Paper | 03_advice.md | 2 | 872 |
This part describes some of the less familiar notations and definitions and settles for once and for all some annoying issues ("is zero a natural number?"). Most of these are "remarks for experts": if something doesn't make sense, you probably don't have to worry about it for now.
A full glossary of notation used can ... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Conventions and notations | 03_advice.md | 3 | 306 |
This is brief, intended as a reminder for experts. Consult for full details.
An **equivalence relation** on a set $X$ is a relation $\sim$ which is symmetric, reflexive, and transitive. An equivalence relation partitions $X$ into several **equivalence classes**. We will denote this by $X / {\sim}$. An element of such ... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Sets and equivalence relations | 03_advice.md | 4 | 608 |
This is brief, intended as a reminder for experts. Consult for full details.
Let $X \taking f Y$ be a function:
- By $f\pre(T)$ I mean the **pre-image** $$f\pre(T) := \left\{ x \in X \mid f(x) \in T \right\}.$$ This is in contrast to the $f^{-1}(T)$ used in the rest of the world; I only use $f^{-1}$ for an inverse *f... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Functions | 03_advice.md | 5 | 621 |
Additionally, a permutation on a finite set may be denoted in *cycle notation*, as described in say <https://en.wikipedia.org/wiki/Permutation#Cycle_notation>. For example the notation $(1 \; 2 \; 3 \; 4)(5 \; 6 \; 7)$ refers to the permutation with $1 \mapsto 2$, $2 \mapsto 3$, $3 \mapsto 4$, $4 \mapsto 1$, $5 \mapsto... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Cycle notation for permutations | 03_advice.md | 6 | 655 |
We accept the Axiom of Choice, and use it freely.
We accept the Axiom of Choice, and use it freely.
The appendix contains a list of resources I like, and explanations of pedagogical choices that I made for each chapter. I encourage you to check it out.
In particular, this is where you should go for further reading! ... | An Infinitely Large Napkin | napkin | general | advanced | Front Matter | Advice for the reader | Choice | 03_advice.md | 7 | 400 |
This chapter contains a pitch for each part, to help you decide what you want to read and to elaborate more on how they are interconnected.
For convenience, here is again the dependency plot that appeared in the frontmatter.
,45:Ch ,-[Abs Alg](#part:absalg) ,45:Ch ,-[Topology](#part:basictop) ,45:Ch -,[Lin Alg](#part... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | 04_salespitch.md | 0 | 749 | |
- [**.**]\
I made a design decision that the first part should have a little bit of both algebra and topology: so this first chapter begins by defining a **group**, while the second chapter begins by defining a **metric space**. The intention is so that newcomers get to see two different examples of "sets with addition... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | The basics | 04_salespitch.md | 1 | 1,149 |
- [**.**]\
In high school, linear algebra is often really unsatisfying. You are given these arrays of numbers, and they're manipulated in some ways that don't really make sense. For example, the determinant is defined as this funny-looking sum with a bunch of products that seems to come out of thin air. Where does it c... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Abstract algebra | 04_salespitch.md | 2 | 892 |
- [**.**]\
In high school, linear algebra is often really unsatisfying. You are given these arrays of numbers, and they're manipulated in some ways that don't really make sense. For example, the determinant is defined as this funny-looking sum with a bunch of products that seems to come out of thin air. Where does it c... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Abstract algebra | 04_salespitch.md | 3 | 892 |
- [**.**]\
In this part, we'll use our built-up knowledge of metric and topological spaces to give short, rigorous definitions and theorems typical of high school calculus. That is, we'll really define and prove most everything you've seen about **limits**, **series**, **derivatives**, and **integrals**.
Although this... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Real and complex analysis | 04_salespitch.md | 4 | 1,091 |
- [**.**]\
In this part, we'll use our built-up knowledge of metric and topological spaces to give short, rigorous definitions and theorems typical of high school calculus. That is, we'll really define and prove most everything you've seen about **limits**, **series**, **derivatives**, and **integrals**.
Although this... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Real and complex analysis | 04_salespitch.md | 5 | 1,091 |
- [**.**]\
Why is $3+\sqrt5$ the conjugate of $3-\sqrt5$? How come the norm $\norm{a+b\sqrt5} = a^2-5b^2$ used in Pell's equations just happens to be multiplicative? Why is it we can do factoring into primes in $\mathbb{Z}[i]$ but not in $\mathbb{Z}[\sqrt{-5}]$? All these questions and more will be answered in this par... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Algebraic number theory | 04_salespitch.md | 6 | 677 |
- [**.**]\
What's the difference between an annulus and disk? Well, one of them has a "hole" in it, but if we are just given intrinsic topological spaces it's hard to make this notion precise. The **fundamental group** $\pi_1(X)$ and more general **homotopy group** will make this precise --- we'll find a way to define ... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Algebraic topology | 04_salespitch.md | 7 | 1,432 |
- [**.**]\
We begin with a classical study of classical **complex varieties**: the study of intersections of polynomial equations over $\mathbb{C}$. This will naturally lead us into the geometry of rings, giving ways to draw pictures of ideals, and motivating **Hilbert's nullstellensatz**. The **Zariski topology** will... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Algebraic geometry | 04_salespitch.md | 8 | 745 |
- [**.**]\
Why is **Russell's paradox** such a big deal and how is it resolved? What is this **Zorn's lemma** that everyone keeps talking about? In this part we'll learn the answers to these questions by giving a real description of the **Zermelo-Frankel** axioms, and the **axiom of choice**, delving into the details o... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Sales pitches | Set theory | 04_salespitch.md | 9 | 672 |
A group is one of the most basic structures in higher mathematics. In this chapter I will tell you only the bare minimum: what a group is, and when two groups are the same.
A group is one of the most basic structures in higher mathematics. In this chapter I will tell you only the bare minimum: what a group is, and whe... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | 05_grp-intro.md | 0 | 98 | |
A group consists of two pieces of data: a set $G$, and an associative binary operation $\star$ with some properties. Before I write down the definition of a group, let me give two examples.
The pair $(\mathbb{Z}, +)$ is a group: $\mathbb{Z} = \left\{ \dots,-2,-1,0,1,2,\dots \right\}$ is the set and the associative ope... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Definition and examples of groups | 05_grp-intro.md | 1 | 1,488 |
Let $S_n$ be the set of permutations of $\left\{ 1,\dots,n \right\}$. By viewing these permutations as functions from $\left\{ 1,\dots,n \right\}$ to itself, we can consider *compositions* of permutations. Then the pair $(S_n, \circ)$ (here $\circ$ is function composition) is also a group, because
- There is an identi... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Definition and examples of groups | 05_grp-intro.md | 2 | 853 |
A group consists of two pieces of data: a set $G$, and an associative binary operation $\star$ with some properties. Before I write down the definition of a group, let me give two examples.
The pair $(\mathbb{Z}, +)$ is a group: $\mathbb{Z} = \left\{ \dots,-2,-1,0,1,2,\dots \right\}$ is the set and the associative ope... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Definition and examples of groups | 05_grp-intro.md | 3 | 1,488 |
Let $S_n$ be the set of permutations of $\left\{ 1,\dots,n \right\}$. By viewing these permutations as functions from $\left\{ 1,\dots,n \right\}$ to itself, we can consider *compositions* of permutations. Then the pair $(S_n, \circ)$ (here $\circ$ is function composition) is also a group, because
- There is an identi... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Definition and examples of groups | 05_grp-intro.md | 4 | 853 |
From now on, we'll often refer to a group $(G, \star)$ by just $G$. Moreover, we'll abbreviate $a \star b$ to just $ab$. Also, because the operation $\star$ is associative, we will omit unnecessary parentheses: $(ab)c = a(bc) = abc$.
From now on, for any $g \in G$ and $n \in \mathbb{N}$ we abbreviate $$g^n
=
\underbra... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Properties of groups | 05_grp-intro.md | 5 | 956 |
First, let me talk about what it means for groups to be isomorphic. Consider the two groups
- $\mathbb{Z} = (\left\{ \dots,-2,-1,0,1,2,\dots \right\}, +)$.
- $10\mathbb{Z} = (\left\{ \dots, -20, -10, 0, 10, 20, \dots \right\}, +)$.
These groups are "different", but only superficially so -- you might even say they onl... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Isomorphisms | 05_grp-intro.md | 6 | 972 |
First, let me talk about what it means for groups to be isomorphic. Consider the two groups
- $\mathbb{Z} = (\left\{ \dots,-2,-1,0,1,2,\dots \right\}, +)$.
- $10\mathbb{Z} = (\left\{ \dots, -20, -10, 0, 10, 20, \dots \right\}, +)$.
These groups are "different", but only superficially so -- you might even say they onl... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Isomorphisms | 05_grp-intro.md | 7 | 972 |
As is typical in math, we use the word "order" for way too many things. In groups, there are two notions of order.
The **order of a group** $G$ is the number of elements of $G$. We denote this by $\left\lvert G \right\rvert$. Note that the order may not be finite, as in $\mathbb{Z}$. We say $G$ is a **finite group** j... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Orders of groups, and Lagrange's theorem | 05_grp-intro.md | 8 | 1,330 |
Earlier we saw that $\GL_n(\mathbb{R})$, the $n \times n$ matrices with nonzero determinant, formed a group under matrix multiplication. But we also saw that a subset of $\GL_n(\mathbb{R})$, namely $\SL_n(\mathbb{R})$, also formed a group with the same operation. For that reason we say that $\SL_n(\mathbb{R})$ is a sub... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Subgroups | 05_grp-intro.md | 9 | 1,383 |
Just for fun, here is a list of all groups of order less than or equal to ten (up to isomorphism, of course).
1. The only group of order $1$ is the trivial group.
2. The only group of order $2$ is $\mathbb{Z}/2\mathbb{Z}$.
3. The only group of order $3$ is $\mathbb{Z}/3\mathbb{Z}$.
4. The only groups of order $4$ a... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Groups of small orders | 05_grp-intro.md | 10 | 1,070 |
A common question is: why these axioms? For example, why associative but not commutative? This answer will likely not make sense until later, but here are some comments that may help.
One general heuristic is: Whenever you define a new type of general object, there's always a balancing act going on. On the one hand, y... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Unimportant long digression | 05_grp-intro.md | 11 | 1,508 |
Look at the group $G$ of $2 \times 2$ matrices mod $p$ with determinant $\pm 1$ (whose entries are the integers mod $p$). Let $g = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and then use $g^{\left\lvert G \right\rvert} = 1_G$.
[^1]: In other words, permutation groups can be arbitrarily weird. I remember being highl... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Unimportant long digression | 05_grp-intro.md | 12 | 118 |
A common question is: why these axioms? For example, why associative but not commutative? This answer will likely not make sense until later, but here are some comments that may help.
One general heuristic is: Whenever you define a new type of general object, there's always a balancing act going on. On the one hand, y... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Unimportant long digression | 05_grp-intro.md | 13 | 1,508 |
Look at the group $G$ of $2 \times 2$ matrices mod $p$ with determinant $\pm 1$ (whose entries are the integers mod $p$). Let $g = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$ and then use $g^{\left\lvert G \right\rvert} = 1_G$.
[^1]: In other words, permutation groups can be arbitrarily weird. I remember being highl... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Groups | Unimportant long digression | 05_grp-intro.md | 14 | 118 |
At the time of writing, I'm convinced that metric topology is the morally correct way to motivate point-set topology as well as to generalize normal calculus.[^1] So here is my best attempt.
The concept of a metric space is very "concrete", and lends itself easily to visualization. Hence throughout this chapter you sh... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | 06_metric-top.md | 0 | 1,341 | |
A **metric space** is a pair $(M, d)$ consisting of a set of points $M$ and a **metric** $d \colon M \times M \to \mathbb R_{\ge 0}$. The distance function must obey:
- For any $x,y \in M$, we have $d(x,y) = d(y,x)$; i.e. $d$ is symmetric.
- The function $d$ must be **positive definite** which means that $d(x,y) \ge 0... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Definition and examples of metric spaces | 06_metric-top.md | 1 | 1,088 |
Since we can talk about the distance between two points, we can talk about what it means for a sequence of points to converge. This is the same as the typical epsilon-delta definition, with absolute values replaced by the distance function.
Let $(x_n)_{n \ge 1}$ be a sequence of points in a metric space $M$. We say th... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Convergence in metric spaces | 06_metric-top.md | 2 | 1,061 |
In calculus you were also told (or have at least heard) of what it means for a function to be continuous. Probably something like
> A function $f \colon \mathbb{R} \to \mathbb{R}$ is continuous at a point $p \in \mathbb{R}$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $\left\lvert x-p \right\rv... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Continuous maps | 06_metric-top.md | 3 | 816 |
When do we consider two groups to be the same? Answer: if there's a structure-preserving map between them which is also a bijection. For metric spaces, we do exactly the same thing, but replace "structure-preserving" with "continuous".
Let $M$ and $N$ be metric spaces. A function $f \colon M \to N$ is a **homeomorphis... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Homeomorphisms | 06_metric-top.md | 4 | 783 |
When do we consider two groups to be the same? Answer: if there's a structure-preserving map between them which is also a bijection. For metric spaces, we do exactly the same thing, but replace "structure-preserving" with "continuous".
Let $M$ and $N$ be metric spaces. A function $f \colon M \to N$ is a **homeomorphis... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Homeomorphisms | 06_metric-top.md | 5 | 783 |
Here is an extended example which will occur later on. Let $M = (M, d_M)$ and $N = (N, d_N)$ be metric spaces (say, $M = N = \mathbb{R}$). Our goal is to define a metric space on $M \times N$.
Let $p_i = (x_i,y_i) \in M \times N$ for $i=1,2$. Consider the following metrics on the set of points $M \times N$: $$\begin{a... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Extended example/definition: product metric | 06_metric-top.md | 6 | 853 |
Continuity is really about what happens "locally": how a function behaves "close to a certain point $p$". One way to capture this notion of "closeness" is to use metrics as we've done above. In this way we can define an $r$-neighborhood of a point.
Let $M$ be a metric space. For each real number $r > 0$ and point $p \... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Open sets | 06_metric-top.md | 7 | 1,328 |
It would be criminal for me to talk about open sets without talking about closed sets. The name "closed" comes from the definition in a metric space.
Let $M$ be a metric space. A subset $S \subseteq M$ is **closed** in $M$ if the following property holds: let $x_1$, $x_2$, ... be a sequence of points in $S$ and suppos... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Closed sets | 06_metric-top.md | 8 | 1,457 |
Assume for contradiction it is completely discontinuous; by scaling set $f(0) = 0$, $f(1) = 1$ and focus just on $f \colon [0,1] \to [0,1]$. Since it's discontinuous everywhere, for every $x \in [0,1]$ there's an $\varepsilon_x > 0$ such that the continuity condition fails. Since the function is strictly increasing, th... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Closed sets | 06_metric-top.md | 9 | 999 |
It would be criminal for me to talk about open sets without talking about closed sets. The name "closed" comes from the definition in a metric space.
Let $M$ be a metric space. A subset $S \subseteq M$ is **closed** in $M$ if the following property holds: let $x_1$, $x_2$, ... be a sequence of points in $S$ and suppos... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Closed sets | 06_metric-top.md | 10 | 1,457 |
Assume for contradiction it is completely discontinuous; by scaling set $f(0) = 0$, $f(1) = 1$ and focus just on $f \colon [0,1] \to [0,1]$. Since it's discontinuous everywhere, for every $x \in [0,1]$ there's an $\varepsilon_x > 0$ such that the continuity condition fails. Since the function is strictly increasing, th... | An Infinitely Large Napkin | napkin | general | advanced | Starting Out | Metric spaces | Closed sets | 06_metric-top.md | 11 | 999 |
Homomorphisms and quotient groups
ch:homomorphisms_quotient
Generators and group presentations
= < r,s r^n=s^2=1>$
Let $G$ be a group.
Recall that for some element $x G$,
we could consider the subgroup
\[ \ , x^-2, x^-1, 1, x, x^2, \ \]
of $G$.
Here's a more pictorial version of what we did:
put $x$ in a box, seal it ... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 0 | 1,559 | |
Now let me define a very important property of a homomorphism.
The kernel of a homomorphism $ G H$ is defined by
\[
\ g G : (g) = 1_H \.
\]
It is a subgroup of $G$
(in particular, $1_G $ for obvious reasons).
Verify that $$ is in fact a subgroup of $G$.
We also have the following important fact, which we also encour... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 1 | 1,568 | |
Now that we've done this, we can give an intrinsic
definition for the quotient group we alluded to earlier.
A subgroup $N$ of $G$ is called normal if it is the
kernel of some homomorphism.
We write this as $N G$.
Let $N G$.
Then the quotient group, denoted $G/N$
(and read ``$G$ mod $N$''),
is the group defined as fol... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 2 | 1,563 | |
Consider again the product group $G H$.
Earlier we identified a subgroup
\[ G' = \ (g, 1_H) g G \ G. \]
You can easily see that $G' G H$.
(Easy calculation.)
Moreover, you can check that
\[ (G H) / (G') H. \]
Indeed, we have $(g, h) _G' (1_G, h)$ for all $g G$ and $h H$.
It is not necessarily true that $(G/H) H G$.
F... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 3 | 1,537 | |
Since $(12123, 63) = 9$, we find $a^9 = 1$, hence finally $c^2 = 1$.
So the presentation above simplifies to
\[ G = < a,c a^9=c^2=1, \; ac = ca^-1 > \]
which is the presentation of the dihedral group of order $18$.
% a = r, r^9 = 1
% b = r^3
% c = s, s^2 = 2
This completes the proof.
[Homophony group]
The homophony g... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 4 | 485 | |
Homomorphisms and quotient groups
ch:homomorphisms_quotient
Generators and group presentations
= < r,s r^n=s^2=1>$
Let $G$ be a group.
Recall that for some element $x G$,
we could consider the subgroup
\[ \ , x^-2, x^-1, 1, x, x^2, \ \]
of $G$.
Here's a more pictorial version of what we did:
put $x$ in a box, seal it ... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 5 | 1,559 | |
Now let me define a very important property of a homomorphism.
The kernel of a homomorphism $ G H$ is defined by
\[
\ g G : (g) = 1_H \.
\]
It is a subgroup of $G$
(in particular, $1_G $ for obvious reasons).
Verify that $$ is in fact a subgroup of $G$.
We also have the following important fact, which we also encour... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 6 | 1,568 | |
Now that we've done this, we can give an intrinsic
definition for the quotient group we alluded to earlier.
A subgroup $N$ of $G$ is called normal if it is the
kernel of some homomorphism.
We write this as $N G$.
Let $N G$.
Then the quotient group, denoted $G/N$
(and read ``$G$ mod $N$''),
is the group defined as fol... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 7 | 1,563 | |
Consider again the product group $G H$.
Earlier we identified a subgroup
\[ G' = \ (g, 1_H) g G \ G. \]
You can easily see that $G' G H$.
(Easy calculation.)
Moreover, you can check that
\[ (G H) / (G') H. \]
Indeed, we have $(g, h) _G' (1_G, h)$ for all $g G$ and $h H$.
It is not necessarily true that $(G/H) H G$.
F... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 8 | 1,537 | |
Since $(12123, 63) = 9$, we find $a^9 = 1$, hence finally $c^2 = 1$.
So the presentation above simplifies to
\[ G = < a,c a^9=c^2=1, \; ac = ca^-1 > \]
which is the presentation of the dihedral group of order $18$.
% a = r, r^9 = 1
% b = r^3
% c = s, s^2 = 2
This completes the proof.
[Homophony group]
The homophony g... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 9 | 485 | |
In this chapter we'll introduce the notion of a **commutative ring** $R$. It is a larger structure than a group: it will have two operations addition and multiplication, rather than just one. We will then immediately define a **ring homomorphism** $R \to S$ between pairs of rings.
This time, instead of having normal s... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Some motivational metaphors about rings vs groups | 08_ring-intro.md | 0 | 580 |
I wrote most of these examples with a number theoretic eye in mind; thus if you liked elementary number theory, a lot of your intuition will carry over. Basically, we'll try to generalize properties of the ring $\mathbb{Z}$ to any abelian structure in which we can also multiply. That's why, for example, you can talk ab... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | (Optional) Pedagogical notes on motivation | 08_ring-intro.md | 1 | 715 |
Well, I guess I'll define a ring[^1].
A **ring** is a triple $(R, +, \times)$, the two operations usually called addition and multiplication, such that
(i) $(R,+)$ is an abelian group, with identity $0_R$, or just $0$.
(ii) $\times$ is an associative, binary operation on $R$ with some identity, written $1_R$ or just... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Definition and examples of rings | 08_ring-intro.md | 2 | 1,087 |
Well, I guess I'll define a ring[^1].
A **ring** is a triple $(R, +, \times)$, the two operations usually called addition and multiplication, such that
(i) $(R,+)$ is an abelian group, with identity $0_R$, or just $0$.
(ii) $\times$ is an associative, binary operation on $R$ with some identity, written $1_R$ or just... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Definition and examples of rings | 08_ring-intro.md | 3 | 1,087 |
Although we won't need to know what a field is until next chapter, they're so convenient for examples I will go ahead and introduce them now.
As you might already know, if the multiplication is invertible, then we call the ring a field. To be explicit, let me write the relevant definitions.
A **unit** of a ring $R$ i... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Fields | 08_ring-intro.md | 4 | 764 |
This section is going to go briskly -- it's the obvious generalization of all the stuff we did with quotient groups.[^2]
First, we define a homomorphism and isomorphism.
Let $R = (R, +_R, \times_R)$ and $S = (S, +_S, \times_S)$ be rings. A **ring homomorphism** is a map $\phi \colon R \to S$ such that
(i) $\phi(x +_... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Homomorphisms | 08_ring-intro.md | 5 | 984 |
Now, just like we were able to mod out by groups, we'd also like to define quotient rings. So once again,
The **kernel** of a ring homomorphism $\phi \colon R \to S$, denoted $\ker \phi$, is the set of $r \in R$ such that $\phi(r) = 0$.
In group theory, we were able to characterize the "normal" subgroups by a few obv... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Ideals | 08_ring-intro.md | 6 | 1,309 |
Let's give you some practice with ideals.
An important piece of intuition is that once an ideal contains a unit, it contains $1$, and thus must contain the entire ring. That's why the notion of "proper ideal" is useful language. To expand on that:
Let $R$ be a ring and $I \subseteq R$ an ideal. Then $I$ is proper (i.... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Generating ideals | 08_ring-intro.md | 7 | 1,083 |
What happens if we put multiple generators in an ideal, like $(10,15) \subseteq \mathbb{Z}$? Well, we have by definition that $(10,15)$ is given as a set by $$(10,15) := \left\{ 10x + 15y \mid x,y \in \mathbb{Z} \right\}.$$ If you're good at number theory you'll instantly recognize this as $5\mathbb{Z} = (5)$. Surprise... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Principal ideal domains | 08_ring-intro.md | 8 | 1,145 |
If it's too much to ask that an ideal is generated by *one* element, perhaps we can at least ask that our ideals are generated by *finitely many* elements. Unfortunately, in certain weird rings this is also not the case.
Consider the ring $R = \mathbb{Z}[x_1, x_2, x_3, \dots]$ which has *infinitely* many free variable... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Noetherian rings | 08_ring-intro.md | 9 | 1,552 |
If it's too much to ask that an ideal is generated by *one* element, perhaps we can at least ask that our ideals are generated by *finitely many* elements. Unfortunately, in certain weird rings this is also not the case.
Consider the ring $R = \mathbb{Z}[x_1, x_2, x_3, \dots]$ which has *infinitely* many free variable... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Rings and ideals | Noetherian rings | 08_ring-intro.md | 10 | 1,552 |
We continue our exploration of rings by considering some nice-ness properties that rings or ideals can satisfy, which will be valuable later on. As before, number theory is interlaced as motivation. I guess I can tell you at the outset what the completed table is going to look like, so you know what to expect.
Ring no... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | 09_ring-flavors.md | 0 | 417 | |
We already saw this definition last chapter: a field $K$ is a nontrivial ring for which every nonzero element is a unit.
In particular, there are only two ideals in a field: the ideal $(0)$, which is maximal, and the entire field $K$.
We already saw this definition last chapter: a field $K$ is a nontrivial ring for w... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Fields | 09_ring-flavors.md | 1 | 1,477 |
We know that every integer can be factored (up to sign) as a unique product of primes; for example $15 = 3 \cdot 5$ and $-10 = -2 \cdot 5$. You might remember the proof involves the so-called Bézout's lemma, which essentially says that $(a,b) = (\gcd(a,b))$; in other words we've carefully used the fact that $\mathbb{Z}... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Prime ideals | 09_ring-flavors.md | 2 | 813 |
We know that every integer can be factored (up to sign) as a unique product of primes; for example $15 = 3 \cdot 5$ and $-10 = -2 \cdot 5$. You might remember the proof involves the so-called Bézout's lemma, which essentially says that $(a,b) = (\gcd(a,b))$; in other words we've carefully used the fact that $\mathbb{Z}... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Prime ideals | 09_ring-flavors.md | 3 | 813 |
Here's another flavor of an ideal.
A proper ideal $I$ of a ring $R$ is **maximal** if it is not contained in any other proper ideal.
(a) The ideal $I = (7)$ of $\mathbb{Z}$ is maximal, because if an ideal $J$ contains $7$ and an element $n$ not in $I$ it must contain $\gcd(7,n) = 1$, and hence $J = \mathbb{Z}$.
(b) ... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Maximal ideals | 09_ring-flavors.md | 4 | 532 |
As long as we are here, we take the time to introduce a useful construction that turns any integral domain into a field.
Given an integral domain $R$, we define its **field of fractions** or **fraction field** $\Frac(R)$ as follows: it consists of elements $a / b$, where $a,b \in R$ and $b \neq 0$. We set $a / b \sim ... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Field of fractions | 09_ring-flavors.md | 5 | 966 |
Here is one stray definition that will be important for those with a number-theoretic inclination. Over the positive integers, we have a fundamental theorem of arithmetic, stating that every integer is uniquely the product of prime numbers.
We can even make an analogous statement in $\mathbb{Z}$ or $\mathbb{Z}[i]$, if... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Unique factorization domains (UFD's) | 09_ring-flavors.md | 6 | 1,422 |
This chapter will not be used later on, but it is of historical interest.
Recall that a PID is a ring where you can take the $\gcd$ of any family of elements.
We all know that the most popular algorithm to compute the $\gcd$ of two elements in $\mathbb{Z}$ is the Euclidean algorithm:
- Start with two integers $a$ an... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Extra: Euclidean domains | 09_ring-flavors.md | 7 | 1,535 |
That having said, sometimes the natural norm of a Euclidean domain need not be Euclidean. $\mathbb{Z}[\frac{1 + \sqrt{69}}{2}]$ is the first example.
Similarly, in $\mathbb{Q}[x]$ we can let the norm be the degree of a polynomial --- the polynomial division with remainder algorithm will take care of computing the $\gc... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Extra: Euclidean domains | 09_ring-flavors.md | 8 | 1,379 |
*Proof.* All numbers are of the form, $\frac{a}{2} + \frac{b\sqrt{-19}}{2}$ where $a$ and $b$ have the same parity. The absolute value of a complex number, defined as $\frac{a^2}{4} + \frac{19b^2}{4}$, is multiplicative and is greater than $1$ for all numbers in $\mathbb Z [\frac{1+\sqrt{-19}}{2}]$ except for $-1, 0, 1... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Extra: Euclidean domains | 09_ring-flavors.md | 9 | 1,563 |
(b) More generally, show that every prime ideal in an Artinian ring is maximal.
[^1]: Some authors abbreviate this to "domain", notably Artin.
[^2]: See for the explanation why this norm is the natural one.
[^3]: Note that $\psi$ cannot be the zero map for us, since we require $\psi(1_K) = 1_R$. You sometimes find d... | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Flavors of rings | Extra: Euclidean domains | 09_ring-flavors.md | 10 | 102 |
At the end of the last chapter on metric spaces, we introduced two adjectives "open" and "closed". These are important because they'll grow up to be the *definition* for a general topological space, once we graduate from metric spaces.
To move forward, we provide a couple niceness adjectives that applies to *entire me... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | 10_metric-prop.md | 0 | 485 | |
Here is one notion of how to prevent a metric space from being a bit too large.
A metric space $M$ is **bounded** if there is a constant $D$ such that $d(p,q) \le D$ for all $p,q \in M$.
You can change the order of the quantifiers:
A metric space $M$ is bounded if and only if for every point $p \in M$, there is a ra... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Boundedness | 10_metric-prop.md | 1 | 1,092 |
So far we can only talk about sequences converging if they have a limit. But consider the sequence $$x_1 = 1, \; x_2 = 1.4, \; x_3 = 1.41, \; x_4 = 1.414, \dots.$$ It converges to $\sqrt 2$ in $\mathbb{R}$, of course. But it fails to converge in $\mathbb{Q}$; there is no *rational* number this sequence converges to. An... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Completeness | 10_metric-prop.md | 2 | 1,279 |
There is something suspicious about both these notions: neither are preserved under homeomorphism!
Let $M = (0,1)$ and $N = \mathbb{R}$. As we saw much earlier $M$ and $N$ are homeomorphic. However:
- $(0,1)$ is totally bounded, but not complete.
- $\mathbb{R}$ is complete, but not bounded.
This is the first hint of... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Let the buyer beware | 10_metric-prop.md | 3 | 414 |
As we've already been doing implicitly in examples, we'll now say:
Every subset $S \subseteq M$ is a metric space in its own right, by reusing the distance function on $M$. We say that $S$ is a **subspace** of $M$.
For example, we saw that the circle $S^1$ is just a subspace of $\mathbb{R}^2$.
It thus becomes import... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Subspaces, and (inb4) a confusing linguistic point | 10_metric-prop.md | 4 | 1,497 |
This illustrates that $M \cong N$ despite the fact that $M$ is both complete and bounded but $N$ is neither complete nor bounded. On the other hand, we will later see that complete and totally bounded implies *compact*, which is a very strong property preserved under homeomorphism.
Let $M$ be a metric space. Construct... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Subspaces, and (inb4) a confusing linguistic point | 10_metric-prop.md | 5 | 290 |
As we've already been doing implicitly in examples, we'll now say:
Every subset $S \subseteq M$ is a metric space in its own right, by reusing the distance function on $M$. We say that $S$ is a **subspace** of $M$.
For example, we saw that the circle $S^1$ is just a subspace of $\mathbb{R}^2$.
It thus becomes import... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Subspaces, and (inb4) a confusing linguistic point | 10_metric-prop.md | 6 | 1,497 |
This illustrates that $M \cong N$ despite the fact that $M$ is both complete and bounded but $N$ is neither complete nor bounded. On the other hand, we will later see that complete and totally bounded implies *compact*, which is a very strong property preserved under homeomorphism.
Let $M$ be a metric space. Construct... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Properties of metric spaces | Subspaces, and (inb4) a confusing linguistic point | 10_metric-prop.md | 7 | 290 |
Topological spaces
ch:top_more
In ch:metric_space we introduced the notion
of space by describing metrics on them.
This gives you a lot of examples, and nice intuition,
and tells you how you should draw pictures of open and closed sets.
However, moving forward, it will be useful to begin
thinking about topological sp... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 0 | 1,391 | |
%
% A sequence $(x_n)$ of points in a topological space $X$ is said to converge to $x X$ if for every open neighborhood of $x$,
% eventually all terms of the sequence lie in that open neighborhood.
%
%
% Unfortunately, for general topological spaces we no longer have the nice property
% that any function which preserve... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 1 | 1,531 | |
Show that a space $X$ has a nontrivial clopen set (one other than $$ and $X$)
if and only if $X$ can be written as a disjoint union of two nonempty open sets.
We say $X$ is disconnected if there are nontrivial clopen sets,
and connected otherwise.
To see why this should be a reasonable definition,
it might help to sol... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 2 | 1,520 | |
What this definition is doing is taking $$ and ``continuously deforming'' it to $$, while keeping the endpoints fixed.
Note that for each particular $s$, $F_s$ is itself a function.
So $s$ represents time as we deform $$ to $$:
it goes from $0$ to $1$, starting at $$ and ending at $$.
size(9cm);
bigbox("$ C$");
pair A... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 3 | 1,514 | |
%
% A topological space $X$ is called locally path-connected
% if for every point $x X$ and open neighborhood $U$ of $x$,
% some open neighborhood $V$ of $x$ contained in $U$ is path-connected.
% Prove that $X$ is path-connected if and only if it is connected
% and locally path-connected.
% prob:local_path_connected
%
... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 4 | 231 | |
Topological spaces
ch:top_more
In ch:metric_space we introduced the notion
of space by describing metrics on them.
This gives you a lot of examples, and nice intuition,
and tells you how you should draw pictures of open and closed sets.
However, moving forward, it will be useful to begin
thinking about topological sp... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 5 | 1,391 | |
%
% A sequence $(x_n)$ of points in a topological space $X$ is said to converge to $x X$ if for every open neighborhood of $x$,
% eventually all terms of the sequence lie in that open neighborhood.
%
%
% Unfortunately, for general topological spaces we no longer have the nice property
% that any function which preserve... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 6 | 1,531 | |
Show that a space $X$ has a nontrivial clopen set (one other than $$ and $X$)
if and only if $X$ can be written as a disjoint union of two nonempty open sets.
We say $X$ is disconnected if there are nontrivial clopen sets,
and connected otherwise.
To see why this should be a reasonable definition,
it might help to sol... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 7 | 1,520 | |
What this definition is doing is taking $$ and ``continuously deforming'' it to $$, while keeping the endpoints fixed.
Note that for each particular $s$, $F_s$ is itself a function.
So $s$ represents time as we deform $$ to $$:
it goes from $0$ to $1$, starting at $$ and ending at $$.
size(9cm);
bigbox("$ C$");
pair A... | An Infinitely Large Napkin | napkin | general | advanced | Basic Topology | Topological spaces | 11_top-more.md | 8 | 1,514 |
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