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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/evanchen/>
*For Brian and Lisa, who finally got me to write it*.\
Evan Chen.\
Text licensed under [CC-by-SA-4.0](https://creativecommons.org/licenses/by-sa/4.0/). Source files licensed under [GNU GPL v3](https://choosealicense.com/licenses/gpl-3.0/).\
This is (still!) an **incomplete draft**. Please send corrections, comments, pictures of kittens, etc. to , or pull-request at <https://github.com/vEnhance/napkin>.\
Last updated 2026-02-25.
*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/evanchen/>
*For Brian and Lisa, who finally got me to write it*.\
Evan Chen.\
Text licensed under [CC-by-SA-4.0](https://creativecommons.org/licenses/by-sa/4.0/). Source files licensed under [GNU GPL v3](https://choosealicense.com/licenses/gpl-3.0/).\
This is (still!) an **incomplete draft**. Please send corrections, comments, pictures of kittens, etc. to , or pull-request at <https://github.com/vEnhance/napkin>.\
Last updated 2026-02-25. | 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. I tell them it's about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I'll try to give them the standard example $\catname{Grp}$, but then I'll realize that they don't know what a homomorphism is. So then I'll start trying to explain what a homomorphism is, but then I'll remember that they haven't learned what a group is. So then I'll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they've already forgotten why I was talking about groups in the first place. And then it's 1PM, people need to go places, and I can't help but think:\
> *"Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all".*
This book was initially my attempt at those forty hours, but has grown considerably since then.
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. I tell them it's about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I'll try to give them the standard example $\catname{Grp}$, but then I'll realize that they don't know what a homomorphism is. So then I'll start trying to explain what a homomorphism is, but then I'll remember that they haven't learned what a group is. So then I'll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they've already forgotten why I was talking about groups in the first place. And then it's 1PM, people need to go places, and I can't help but think:\
> *"Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all".*
This book was initially my attempt at those forty hours, but has grown considerably since then. | 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 becomes accessible to a larger audience, since the goal is merely to give the reader a feeling for any particular topic rather than to emulate a full semester of lectures.
I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, advanced high schoolers who like math but not math contests, and unusually intelligent kittens fluent in English.
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 becomes accessible to a larger audience, since the goal is merely to give the reader a feeling for any particular topic rather than to emulate a full semester of lectures.
I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, advanced high schoolers who like math but not math contests, and unusually intelligent kittens fluent in English. | 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 corrections by email.
As far as I can tell, higher math for high-school students comes in two flavors:
- Someone tells you about the hairy ball theorem in the form "you can't comb the hair on a spherical cat" then doesn't tell you anything about why it should be true, what it means to actually "comb the hair", or any of the underlying theory, leaving you with just some vague notion in your head.
- You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.
Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here.
Unlike university, it is *not* the purpose of this book to train you to solve exercises or write proofs,[^1] or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem *should* be true rather than writing down its proof. This is a recurrent theme of this book:
Natural explanations supersede proofs.
My hope is that after reading any particular chapter in Napkin, one might get the following out of it:
- Knowing what the precise definitions are of the main characters,
- Being acquainted with the few really major examples,
- Knowing the precise statements of famous theorems, and having a sense of why they *should* be true.
Understanding "why" something is true can have many forms. This is sometimes accomplished with a complete rigorous proof; in other cases, it is given by the idea of the proof; in still other cases, it is just a few key examples with extensive cheerleading.
Obviously this is nowhere near enough if you want to e.g. do research in a field; but if you are just a curious outsider, I hope that it's more satisfying than the elevator pitch or Wikipedia articles. Even if you do want to learn a topic with serious depth, I hope that it can be a good zoomed-out overview before you really dive in, because in many senses the choice of material is "what I wish someone had told me before I started".
As far as I can tell, higher math for high-school students comes in two flavors:
- Someone tells you about the hairy ball theorem in the form "you can't comb the hair on a spherical cat" then doesn't tell you anything about why it should be true, what it means to actually "comb the hair", or any of the underlying theory, leaving you with just some vague notion in your head.
- You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.
Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here.
Unlike university, it is *not* the purpose of this book to train you to solve exercises or write proofs,[^1] or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem *should* be true rather than writing down its proof. This is a recurrent theme of this book:
Natural explanations supersede proofs.
My hope is that after reading any particular chapter in Napkin, one might get the following out of it:
- Knowing what the precise definitions are of the main characters,
- Being acquainted with the few really major examples,
- Knowing the precise statements of famous theorems, and having a sense of why they *should* be true.
Understanding "why" something is true can have many forms. This is sometimes accomplished with a complete rigorous proof; in other cases, it is given by the idea of the proof; in still other cases, it is just a few key examples with extensive cheerleading.
Obviously this is nowhere near enough if you want to e.g. do research in a field; but if you are just a curious outsider, I hope that it's more satisfying than the elevator pitch or Wikipedia articles. Even if you do want to learn a topic with serious depth, I hope that it can be a good zoomed-out overview before you really dive in, because in many senses the choice of material is "what I wish someone had told me before I started". | 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 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.
I began writing this book in December 2014, after having finished my first semester of undergraduate at Harvard. It became my main focus for about 18 months after that, as I became immersed in higher math. I essentially took only math classes (gleefully ignoring all my other graduation requirements), and merged as much of it as I could (as well as lots of other math I learned on my own time) into the Napkin.
Towards August 2016, though, I finally lost steam. The first public drafts went online then, and I decided to step back. Having burnt out slightly, I then took a break from higher math, and spent the remaining two undergraduate years[^2] working extensively as a coach for the American math olympiad team, and trying to spend as much time with my friends as I could before they graduated and went their own ways.
During those two years, readers sent me many kind words of gratitude, many reports of errors, and many suggestions for additions. So in November 2018, some weeks into my first semester as a math PhD student, I decided I should finish what I had started. Some months later, here is what I have.
I began writing this book in December 2014, after having finished my first semester of undergraduate at Harvard. It became my main focus for about 18 months after that, as I became immersed in higher math. I essentially took only math classes (gleefully ignoring all my other graduation requirements), and merged as much of it as I could (as well as lots of other math I learned on my own time) into the Napkin.
Towards August 2016, though, I finally lost steam. The first public drafts went online then, and I decided to step back. Having burnt out slightly, I then took a break from higher math, and spent the remaining two undergraduate years[^2] working extensively as a coach for the American math olympiad team, and trying to spend as much time with my friends as I could before they graduated and went their own ways.
During those two years, readers sent me many kind words of gratitude, many reports of errors, and many suggestions for additions. So in November 2018, some weeks into my first semester as a math PhD student, I decided I should finish what I had started. Some months later, here is what I have. | 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 especially influential.
- Thanks also to dozens of friends and strangers who read through preview copies of my draft, and pointed out errors and gave other suggestions. Special mention to Andrej Vuković and Alexander Chua for together catching over a thousand errors. Thanks also to Brian Gu and Tom Tseng for many corrections. (If you find mistakes or have suggestions yourself, I would love to hear them!) Thanks also to Royce Yao and `user202729` for their contributions of guest chapters to the document.
- Thanks to Jenny Chu and Lanie Deng for the cover artwork.
- I'd also like to express my gratitude for many, many kind words I received during the development of this project. These generous comments led me to keep working on this, and were largely responsible for my decision in November 2018 to begin updating the Napkin again.
Finally, a huge thanks to the math olympiad community, from which the Napkin (and me) has its roots. All the enthusiasm, encouragement, and thank-you notes I have received over the years led me to begin writing this in the first place. I otherwise would never have the arrogance to dream a project like this was at all possible. And of course I would be nowhere near where I am today were it not for the life-changing journey I took in chasing my dreams to the IMO. Forever TWN2!
[^1]: Which is not to say problem-solving isn't valuable; I myself am a math olympiad coach at heart. It's just not the point of this book.
[^2]: Alternatively: " ... and spent the next two years forgetting everything I had painstakingly learned". Which made me grateful for all the past notes in the Napkin!
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 especially influential.
- Thanks also to dozens of friends and strangers who read through preview copies of my draft, and pointed out errors and gave other suggestions. Special mention to Andrej Vuković and Alexander Chua for together catching over a thousand errors. Thanks also to Brian Gu and Tom Tseng for many corrections. (If you find mistakes or have suggestions yourself, I would love to hear them!) Thanks also to Royce Yao and `user202729` for their contributions of guest chapters to the document.
- Thanks to Jenny Chu and Lanie Deng for the cover artwork.
- I'd also like to express my gratitude for many, many kind words I received during the development of this project. These generous comments led me to keep working on this, and were largely responsible for my decision in November 2018 to begin updating the Napkin again.
Finally, a huge thanks to the math olympiad community, from which the Napkin (and me) has its roots. All the enthusiasm, encouragement, and thank-you notes I have received over the years led me to begin writing this in the first place. I otherwise would never have the arrogance to dream a project like this was at all possible. And of course I would be nowhere near where I am today were it not for the life-changing journey I took in chasing my dreams to the IMO. Forever TWN2!
[^1]: Which is not to say problem-solving isn't valuable; I myself am a math olympiad coach at heart. It's just not the point of this book.
[^2]: Alternatively: " ... and spent the next two years forgetting everything I had painstakingly learned". Which made me grateful for all the past notes in the Napkin! | 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 not, one should consult .
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 not, one should consult .
There is no need to read this book in linear order: it covers all sorts of areas in mathematics, and there are many paths you can take. In , I give a short overview for each part explaining what you might expect to see in that part.
For now, here is a brief chart showing how the chapters depend on each other; again see for details. Dependencies are indicated by arrows; dotted lines are optional dependencies. **I suggest that you simply pick a chapter you find interesting, and then find the shortest path.** With that in mind, I hope the length of the entire PDF is not intimidating.
(The text in the following diagram should be clickable and links to the relevant part.)
,45:Ch ,-[Abs Alg](#part:absalg) ,45:Ch ,-[Topology](#part:basictop) ,45:Ch -,[Lin Alg](#part:linalg) ,35:Ch [Grp Act](#part:groups) ,24:Ch [Grp Classif](#ch:sylow) ,35:Ch -[Rep Th](#part:repth) ,43:Ch -[Quantum](#part:quantum) ,38:Ch -[Calc](#part:calc) ,30:Ch -[Cmplx Ana](#part:cmplxana) ,20:Ch -[Measure/Pr](#part:measure) ,28:Ch -[Diff Geo](#part:diffgeo) ,10:Ch -[Alg NT 1](#part:algnt1) ,0:Ch -[Alg NT 2](#part:algnt2) ,10:Ch -[Alg Top 1](#part:algtop1) ,28:Ch -[Cat Th](#part:cats) ,0:Ch -[Alg Top 2](#part:algtop2) ,10:Ch -[Alg Geo 1](#part:ag1) ,0:Ch -[Alg Geo 2-3](#part:ag2) ,45:Ch -[Set Theory](#part:st1)
,45,30,35;0: ,45,33,45;0: ,45,30,35;0: ,45,45,43;0: ,45,20,28;0: ,45,20,28;0: ,45,20,28;80: ,45,20,28;-40: ,28,30,35;0: ,45,5,35;0: ,35,5,24;0: ,45,48,28;30: ,45,64,38;0: ,38,48,28;50: ,45,48,28;10: ,45,64,38;0: ,38,64,30;0: ,45,64,30;-10: ,10,64,30;0: ,45,55,20;0: ,38,55,20;50: ,10,6,0;0: ,45,6,10;0: ,45,6,10;-40: ,28,6,10;0: ,28,6,0;-18: ,10,23,0;0: ,45,23,10;0: ,35,23,10;20: ,10,20,28;20: ,28,23,0;-190: ,10,40,0;0: ,45,40,10;-40: ,45,40,10;50: ,35,40,0;10:
There is no need to read this book in linear order: it covers all sorts of areas in mathematics, and there are many paths you can take. In , I give a short overview for each part explaining what you might expect to see in that part.
For now, here is a brief chart showing how the chapters depend on each other; again see for details. Dependencies are indicated by arrows; dotted lines are optional dependencies. **I suggest that you simply pick a chapter you find interesting, and then find the shortest path.** With that in mind, I hope the length of the entire PDF is not intimidating.
(The text in the following diagram should be clickable and links to the relevant part.)
,45:Ch ,-[Abs Alg](#part:absalg) ,45:Ch ,-[Topology](#part:basictop) ,45:Ch -,[Lin Alg](#part:linalg) ,35:Ch [Grp Act](#part:groups) ,24:Ch [Grp Classif](#ch:sylow) ,35:Ch -[Rep Th](#part:repth) ,43:Ch -[Quantum](#part:quantum) ,38:Ch -[Calc](#part:calc) ,30:Ch -[Cmplx Ana](#part:cmplxana) ,20:Ch -[Measure/Pr](#part:measure) ,28:Ch -[Diff Geo](#part:diffgeo) ,10:Ch -[Alg NT 1](#part:algnt1) ,0:Ch -[Alg NT 2](#part:algnt2) ,10:Ch -[Alg Top 1](#part:algtop1) ,28:Ch -[Cat Th](#part:cats) ,0:Ch -[Alg Top 2](#part:algtop2) ,10:Ch -[Alg Geo 1](#part:ag1) ,0:Ch -[Alg Geo 2-3](#part:ag2) ,45:Ch -[Set Theory](#part:st1)
,45,30,35;0: ,45,33,45;0: ,45,30,35;0: ,45,45,43;0: ,45,20,28;0: ,45,20,28;0: ,45,20,28;80: ,45,20,28;-40: ,28,30,35;0: ,45,5,35;0: ,35,5,24;0: ,45,48,28;30: ,45,64,38;0: ,38,48,28;50: ,45,48,28;10: ,45,64,38;0: ,38,64,30;0: ,45,64,30;-10: ,10,64,30;0: ,45,55,20;0: ,38,55,20;50: ,10,6,0;0: ,45,6,10;0: ,45,6,10;-40: ,28,6,10;0: ,28,6,0;-18: ,10,23,0;0: ,45,23,10;0: ,35,23,10;20: ,10,20,28;20: ,28,23,0;-190: ,10,40,0;0: ,45,40,10;-40: ,45,40,10;50: ,35,40,0;10: | 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 likely missed something important: read back.
- An inline **exercise** is more meaty than a question, but shouldn't have any "tricky" steps. Often I leave proofs of theorems and propositions as exercises if they are instructive and at least somewhat interesting.
- Each chapter features several trickier **problems** at the end. Some are reasonable, but others are legitimately difficult olympiad-style problems. Harder problems are marked with up to three chili peppers (), like this paragraph.
In addition to difficulty annotations, the problems are also marked by how important they are to the big picture.
- **Normal problems**, which are hopefully fun but non-central.
- **Daggered problems**, which are (usually interesting) results that one should know, but won't be used directly later.
- **Starred problems**, which are results which will be used later on in the book.[^1]
Several hints and solutions can be found in .
{width="14cm"}\
Image from [@img:exercise]
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 likely missed something important: read back.
- An inline **exercise** is more meaty than a question, but shouldn't have any "tricky" steps. Often I leave proofs of theorems and propositions as exercises if they are instructive and at least somewhat interesting.
- Each chapter features several trickier **problems** at the end. Some are reasonable, but others are legitimately difficult olympiad-style problems. Harder problems are marked with up to three chili peppers (), like this paragraph.
In addition to difficulty annotations, the problems are also marked by how important they are to the big picture.
- **Normal problems**, which are hopefully fun but non-central.
- **Daggered problems**, which are (usually interesting) results that one should know, but won't be used directly later.
- **Starred problems**, which are results which will be used later on in the book.[^1]
Several hints and solutions can be found in .
{width="14cm"}\
Image from [@img:exercise] | 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 paper, grab a notebook or something along with the ride. Somehow, some way, make sure you can write. Thanks.
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 paper, grab a notebook or something along with the ride. Somehow, some way, make sure you can write. Thanks.
I am pathologically obsessed with examples. In this book, I place all examples in large boxes to draw emphasis to them, which leads to some pages of the book simply consisting of sequences of boxes one after another. I hope the reader doesn't mind.
I also often highlight a "prototypical example" for some sections, and reserve the color red for such a note. The philosophy is that any time the reader sees a definition or a theorem about such an object, they should test it against the prototypical example. If the example is a good prototype, it should be immediately clear why this definition is intuitive, or why the theorem should be true, or why the theorem is interesting, et cetera.
Let me tell you a secret. Whenever I wrote a definition or a theorem in this book, I would have to recall the exact statement from my (quite poor) memory. So instead I often consider the prototypical example, and then only after that do I remember what the definition or the theorem is. Incidentally, this is also how I learned all the definitions in the first place. I hope you'll find it useful as well.
I am pathologically obsessed with examples. In this book, I place all examples in large boxes to draw emphasis to them, which leads to some pages of the book simply consisting of sequences of boxes one after another. I hope the reader doesn't mind.
I also often highlight a "prototypical example" for some sections, and reserve the color red for such a note. The philosophy is that any time the reader sees a definition or a theorem about such an object, they should test it against the prototypical example. If the example is a good prototype, it should be immediately clear why this definition is intuitive, or why the theorem should be true, or why the theorem is interesting, et cetera.
Let me tell you a secret. Whenever I wrote a definition or a theorem in this book, I would have to recall the exact statement from my (quite poor) memory. So instead I often consider the prototypical example, and then only after that do I remember what the definition or the theorem is. Incidentally, this is also how I learned all the definitions in the first place. I hope you'll find it useful as well. | 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 be found in the appendix.
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 be found in the appendix.
The set $\mathbb{N}$ is the set of *positive* integers, not including $0$. In the set theory chapters, we use $\omega = \{0, 1, \dots\}$ instead, for consistency with the rest of the book.
The set $\mathbb{N}$ is the set of *positive* integers, not including $0$. In the set theory chapters, we use $\omega = \{0, 1, \dots\}$ instead, for consistency with the rest of the book. | 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 equivalence class is a **representative** of that equivalence class.
I always use $\cong$ for an "isomorphism"-style relation (formally: a relation which is an isomorphism in a reasonable category). The only time $\simeq$ is used in the Napkin is for homotopic paths.
I generally use $\subseteq$ and $\subsetneq$ since these are non-ambiguous, unlike $\subset$. I only use $\subset$ on rare occasions in which equality obviously does not hold yet pointing it out would be distracting. For example, I write $\mathbb{Q} \subset \mathbb{R}$ since "$\mathbb{Q} \subsetneq \mathbb{R}$" is distracting.
I prefer $S \setminus T$ to $S - T$.
The power set of $S$ (i.e., the set of subsets of $S$), is denoted either by $2^S$ or $\mathcal P(S)$.
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 equivalence class is a **representative** of that equivalence class.
I always use $\cong$ for an "isomorphism"-style relation (formally: a relation which is an isomorphism in a reasonable category). The only time $\simeq$ is used in the Napkin is for homotopic paths.
I generally use $\subseteq$ and $\subsetneq$ since these are non-ambiguous, unlike $\subset$. I only use $\subset$ on rare occasions in which equality obviously does not hold yet pointing it out would be distracting. For example, I write $\mathbb{Q} \subset \mathbb{R}$ since "$\mathbb{Q} \subsetneq \mathbb{R}$" is distracting.
I prefer $S \setminus T$ to $S - T$.
The power set of $S$ (i.e., the set of subsets of $S$), is denoted either by $2^S$ or $\mathcal P(S)$. | 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 *function*.
By abuse of notation, we may abbreviate $f\pre(\{y\})$ to $f\pre(y)$. We call $f\pre(y)$ a **fiber**.
- By $f\im(S)$ I mean the **image** $$f\im(S) := \left\{ f(x) \mid x \in S \right\}.$$ Almost everyone else in the world uses $f(S)$ (though $f[S]$ sees some use, and $f''(S)$ is often used in logic) but this is abuse of notation, and I prefer $f\im(S)$ for emphasis. This image notation is *not* standard.
- If $S \subseteq X$, then the **restriction** of $f$ to $S$ is denoted $f \restrict{S}$, i.e. it is the function $f \restrict{S} \colon S \to Y$.
- Sometimes functions $f \colon X \to Y$ are *injective* or *surjective*; I may emphasize this sometimes by writing $f \colon X \hookrightarrow Y$ or $f \colon X \twoheadrightarrow Y$, respectively.
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 *function*.
By abuse of notation, we may abbreviate $f\pre(\{y\})$ to $f\pre(y)$. We call $f\pre(y)$ a **fiber**.
- By $f\im(S)$ I mean the **image** $$f\im(S) := \left\{ f(x) \mid x \in S \right\}.$$ Almost everyone else in the world uses $f(S)$ (though $f[S]$ sees some use, and $f''(S)$ is often used in logic) but this is abuse of notation, and I prefer $f\im(S)$ for emphasis. This image notation is *not* standard.
- If $S \subseteq X$, then the **restriction** of $f$ to $S$ is denoted $f \restrict{S}$, i.e. it is the function $f \restrict{S} \colon S \to Y$.
- Sometimes functions $f \colon X \to Y$ are *injective* or *surjective*; I may emphasize this sometimes by writing $f \colon X \hookrightarrow Y$ or $f \colon X \twoheadrightarrow Y$, respectively. | 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 $1 \mapsto 2$, $2 \mapsto 3$, $3 \mapsto 4$, $4 \mapsto 1$, $5 \mapsto 6$, $6 \mapsto 7$, $7 \mapsto 5$. Usage of this notation will usually be obvious from context.
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 $1 \mapsto 2$, $2 \mapsto 3$, $3 \mapsto 4$, $4 \mapsto 1$, $5 \mapsto 6$, $6 \mapsto 7$, $7 \mapsto 5$. Usage of this notation will usually be obvious from context.
All rings have a multiplicative identity $1$ unless otherwise specified. We allow $0=1$ in general rings but not in integral domains.
**All rings are commutative unless otherwise specified.** There is an elaborate scheme for naming rings which are not commutative, used only in the chapter on cohomology rings:
Graded Not Graded
----------------------------------- ----------------------------- -------------
$1$ not required graded pseudo-ring pseudo-ring
Anticommutative, $1$ not required anticommutative pseudo-ring N/A
Has $1$ graded ring N/A
Anticommutative with $1$ anticommutative ring N/A
Commutative with $1$ commutative graded ring ring
On the other hand, an *algebra* always has $1$, but it need not be commutative.
All rings have a multiplicative identity $1$ unless otherwise specified. We allow $0=1$ in general rings but not in integral domains.
**All rings are commutative unless otherwise specified.** There is an elaborate scheme for naming rings which are not commutative, used only in the chapter on cohomology rings:
Graded Not Graded
----------------------------------- ----------------------------- -------------
$1$ not required graded pseudo-ring pseudo-ring
Anticommutative, $1$ not required anticommutative pseudo-ring N/A
Has $1$ graded ring N/A
Anticommutative with $1$ anticommutative ring N/A
Commutative with $1$ commutative graded ring ring
On the other hand, an *algebra* always has $1$, but it need not be commutative. | 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! There are some topics that should be covered in the Napkin, but are not, due to my own ignorance or laziness. The references provided in this appendix should hopefully help partially atone for my omissions.
[^1]: This is to avoid the classic "we are done by PSet 4, Problem 8" that happens in college sometimes, as if I remembered what that was.
[^2]: See also <https://blog.evanchen.cc/2015/03/14/writing/> and the source above.
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! There are some topics that should be covered in the Napkin, but are not, due to my own ignorance or laziness. The references provided in this appendix should hopefully help partially atone for my omissions.
[^1]: This is to avoid the classic "we are done by PSet 4, Problem 8" that happens in college sometimes, as if I remembered what that was.
[^2]: See also <https://blog.evanchen.cc/2015/03/14/writing/> and the source above. | 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:linalg) ,35:Ch [Grp Act](#part:groups) ,24:Ch [Grp Classif](#ch:sylow) ,35:Ch -[Rep Th](#part:repth) ,43:Ch -[Quantum](#part:quantum) ,38:Ch -[Calc](#part:calc) ,30:Ch -[Cmplx Ana](#part:cmplxana) ,20:Ch -[Measure/Pr](#part:measure) ,28:Ch -[Diff Geo](#part:diffgeo) ,10:Ch -[Alg NT 1](#part:algnt1) ,0:Ch -[Alg NT 2](#part:algnt2) ,10:Ch -[Alg Top 1](#part:algtop1) ,28:Ch -[Cat Th](#part:cats) ,0:Ch -[Alg Top 2](#part:algtop2) ,10:Ch -[Alg Geo 1](#part:ag1) ,0:Ch -[Alg Geo 2-3](#part:ag2) ,45:Ch -[Set Theory](#part:st1)
,45,30,35;0: ,45,33,45;0: ,45,30,35;0: ,45,45,43;0: ,45,20,28;0: ,45,20,28;0: ,45,20,28;80: ,45,20,28;-40: ,28,30,35;0: ,45,5,35;0: ,35,5,24;0: ,45,48,28;30: ,45,64,38;0: ,38,48,28;50: ,45,48,28;10: ,45,64,38;0: ,38,64,30;0: ,45,64,30;-10: ,10,64,30;0: ,45,55,20;0: ,38,55,20;50: ,10,6,0;0: ,45,6,10;0: ,45,6,10;-40: ,28,6,10;0: ,28,6,0;-18: ,10,23,0;0: ,45,23,10;0: ,35,23,10;20: ,10,20,28;20: ,28,23,0;-190: ,10,40,0;0: ,45,40,10;-40: ,45,40,10;50: ,35,40,0;10:
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:linalg) ,35:Ch [Grp Act](#part:groups) ,24:Ch [Grp Classif](#ch:sylow) ,35:Ch -[Rep Th](#part:repth) ,43:Ch -[Quantum](#part:quantum) ,38:Ch -[Calc](#part:calc) ,30:Ch -[Cmplx Ana](#part:cmplxana) ,20:Ch -[Measure/Pr](#part:measure) ,28:Ch -[Diff Geo](#part:diffgeo) ,10:Ch -[Alg NT 1](#part:algnt1) ,0:Ch -[Alg NT 2](#part:algnt2) ,10:Ch -[Alg Top 1](#part:algtop1) ,28:Ch -[Cat Th](#part:cats) ,0:Ch -[Alg Top 2](#part:algtop2) ,10:Ch -[Alg Geo 1](#part:ag1) ,0:Ch -[Alg Geo 2-3](#part:ag2) ,45:Ch -[Set Theory](#part:st1)
,45,30,35;0: ,45,33,45;0: ,45,30,35;0: ,45,45,43;0: ,45,20,28;0: ,45,20,28;0: ,45,20,28;80: ,45,20,28;-40: ,28,30,35;0: ,45,5,35;0: ,35,5,24;0: ,45,48,28;30: ,45,64,38;0: ,38,48,28;50: ,45,48,28;10: ,45,64,38;0: ,38,64,30;0: ,45,64,30;-10: ,10,64,30;0: ,45,55,20;0: ,38,55,20;50: ,10,6,0;0: ,45,6,10;0: ,45,6,10;-40: ,28,6,10;0: ,28,6,0;-18: ,10,23,0;0: ,45,23,10;0: ,35,23,10;20: ,10,20,28;20: ,28,23,0;-190: ,10,40,0;0: ,45,40,10;-40: ,45,40,10;50: ,35,40,0;10: | 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 additional structure" in somewhat different contexts, and to have a minimal amount of literacy as these sorts of definitions appear over and over.[^1]
- [**.**]\
The algebraically inclined can then delve into further types of algebraic structures: some more details of **groups**, and then **rings** and **fields** --- which will let you generalize $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$. So you'll learn to become familiar with all sorts of other nouns that appear in algebra, unlocking a whole host of objects that one couldn't talk about before.
We'll also come to **ideals**, which generalize the GCD in $\mathbb{Z}$ that you might know of. For example, you know in $\mathbb{Z}$ that any integer can be written in the form $3a+5b$ for $a,b \in \mathbb{Z}$, since $\gcd(3,5)=1$. We'll see that this statement is really a statement of ideals: "$(3,5)=1$ in $\mathbb{Z}$", and thus we'll understand in what situations it can be generalized, e.g. to polynomials.
- [**.**]\
The more analytically inclined can instead move into topology, learning more about spaces. We'll find out that "metric spaces" are actually too specific, and that it's better to work with **topological spaces**, which are based on the so-called **open sets**. You'll then get to see the buddings of some geometrical ideals, ending with the really great notion of **compactness**, a powerful notion that makes real analysis tick.
One example of an application of compactness to tempt you now: a continuous function $f \colon [0,1] \to \mathbb{R}$ always achieves a *maximum* value. (In contrast, $f \colon (0,1) \to \mathbb{R}$ by $x \mapsto 1/x$ does not.) We'll see the reason is that $[0,1]$ is compact.
- [**.**]\
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 additional structure" in somewhat different contexts, and to have a minimal amount of literacy as these sorts of definitions appear over and over.[^1]
- [**.**]\
The algebraically inclined can then delve into further types of algebraic structures: some more details of **groups**, and then **rings** and **fields** --- which will let you generalize $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$. So you'll learn to become familiar with all sorts of other nouns that appear in algebra, unlocking a whole host of objects that one couldn't talk about before.
We'll also come to **ideals**, which generalize the GCD in $\mathbb{Z}$ that you might know of. For example, you know in $\mathbb{Z}$ that any integer can be written in the form $3a+5b$ for $a,b \in \mathbb{Z}$, since $\gcd(3,5)=1$. We'll see that this statement is really a statement of ideals: "$(3,5)=1$ in $\mathbb{Z}$", and thus we'll understand in what situations it can be generalized, e.g. to polynomials.
- [**.**]\
The more analytically inclined can instead move into topology, learning more about spaces. We'll find out that "metric spaces" are actually too specific, and that it's better to work with **topological spaces**, which are based on the so-called **open sets**. You'll then get to see the buddings of some geometrical ideals, ending with the really great notion of **compactness**, a powerful notion that makes real analysis tick.
One example of an application of compactness to tempt you now: a continuous function $f \colon [0,1] \to \mathbb{R}$ always achieves a *maximum* value. (In contrast, $f \colon (0,1) \to \mathbb{R}$ by $x \mapsto 1/x$ does not.) We'll see the reason is that $[0,1]$ is compact. | 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 come from? Why does $\det(AB) = \det A \det B$ with such a bizarre formula?
Well, it turns out that you *can* explain all of these things! The trick is to not think of linear algebra as the study of matrices, but instead as the study of *linear maps*. In earlier chapters we saw that we got great generalizations by speaking of "sets with enriched structure" and "maps between them". This time, our sets are **vector spaces** and our maps are **linear maps**. We'll find out that a matrix is actually just a way of writing down a linear map as an array of numbers, but using the "intrinsic" definitions we'll de-mystify all the strange formulas from high school and show you where they all come from.
In particular, we'll see *easy* proofs that column rank equals row rank, determinant is multiplicative, trace is the sum of the diagonal entries. We'll see how the dot product works, and learn all the words starting with "eigen-". We'll even have a bonus chapter for Fourier analysis showing that you can also explain all the big buzz-words by just being comfortable with vector spaces.
- [**.**]\
Some of you might be interested in more about groups, and this chapter will give you a way to play further. It starts with an exploration of **group actions**, then goes into a bit on **Sylow theorems**, which are the tools that let us try to *classify all groups*.
- [**.**]\
If $G$ is a group, we can try to understand it by implementing it as a *matrix*, i.e. considering embeddings $G \hookrightarrow \GL_n(\mathbb{C})$. These are called **representations** of $G$; it turns out that they can be decomposed into **irreducible** ones. Astonishingly we will find that we can *basically characterize all of them*: the results turn out to be short and completely unexpected.
For example, we will find out that there are finitely many irreducible representations of a given finite group $G$; if we label them $V_1$, $V_2$, ..., $V_r$, then we will find that $r$ is the number of conjugacy classes of $G$, and moreover that $$|G| = (\dim V_1)^2 + \dots + (\dim V_r)^2$$ which comes out of nowhere!
The last chapter of this part will show you some unexpected corollaries. Here is one of them: let $G$ be a finite group and create variables $x_g$ for each $g \in G$. A $|G| \times |G|$ matrix $M$ is defined by setting the $(g,h)$th entry to be the variable $x_{g \cdot h}$. Then this determinant will turn out to *factor*, and the factors will correspond to the $V_i$ we described above: there will be an irreducible factor of degree $\dim V_i$ appearing $\dim V_i$ times. This result, called the **Frobenius determinant**, is said to have given birth to representation theory.
- [**.**]\
If you ever wondered what **Shor's algorithm** is, this chapter will use the built-up linear algebra to tell you! | 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 come from? Why does $\det(AB) = \det A \det B$ with such a bizarre formula?
Well, it turns out that you *can* explain all of these things! The trick is to not think of linear algebra as the study of matrices, but instead as the study of *linear maps*. In earlier chapters we saw that we got great generalizations by speaking of "sets with enriched structure" and "maps between them". This time, our sets are **vector spaces** and our maps are **linear maps**. We'll find out that a matrix is actually just a way of writing down a linear map as an array of numbers, but using the "intrinsic" definitions we'll de-mystify all the strange formulas from high school and show you where they all come from.
In particular, we'll see *easy* proofs that column rank equals row rank, determinant is multiplicative, trace is the sum of the diagonal entries. We'll see how the dot product works, and learn all the words starting with "eigen-". We'll even have a bonus chapter for Fourier analysis showing that you can also explain all the big buzz-words by just being comfortable with vector spaces.
- [**.**]\
Some of you might be interested in more about groups, and this chapter will give you a way to play further. It starts with an exploration of **group actions**, then goes into a bit on **Sylow theorems**, which are the tools that let us try to *classify all groups*.
- [**.**]\
If $G$ is a group, we can try to understand it by implementing it as a *matrix*, i.e. considering embeddings $G \hookrightarrow \GL_n(\mathbb{C})$. These are called **representations** of $G$; it turns out that they can be decomposed into **irreducible** ones. Astonishingly we will find that we can *basically characterize all of them*: the results turn out to be short and completely unexpected.
For example, we will find out that there are finitely many irreducible representations of a given finite group $G$; if we label them $V_1$, $V_2$, ..., $V_r$, then we will find that $r$ is the number of conjugacy classes of $G$, and moreover that $$|G| = (\dim V_1)^2 + \dots + (\dim V_r)^2$$ which comes out of nowhere!
The last chapter of this part will show you some unexpected corollaries. Here is one of them: let $G$ be a finite group and create variables $x_g$ for each $g \in G$. A $|G| \times |G|$ matrix $M$ is defined by setting the $(g,h)$th entry to be the variable $x_{g \cdot h}$. Then this determinant will turn out to *factor*, and the factors will correspond to the $V_i$ we described above: there will be an irreducible factor of degree $\dim V_i$ appearing $\dim V_i$ times. This result, called the **Frobenius determinant**, is said to have given birth to representation theory.
- [**.**]\
If you ever wondered what **Shor's algorithm** is, this chapter will use the built-up linear algebra to tell you! | 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 might seem intimidating, it turns out that actually, by the time we start this chapter, *the hard work has already been done*: the notion of limits, open sets, and compactness will make short work of what was swept under the rug in AP calculus. Most of the proofs will thus actually be quite short. We sit back and watch all the pieces slowly come together as a reward for our careful study of topology beforehand.
That said, if you are willing to suspend belief, you can actually read most of the other parts without knowing the exact details of all the calculus here, so in some sense this part is "optional".
- [**.**]\
It turns out that **holomorphic functions** (complex-differentiable functions) are close to the nicest things ever: they turn out to be given by a Taylor series (i.e. are basically polynomials). This means we'll be able to prove unreasonably nice results about holomorphic functions $\mathbb{C} \to \mathbb{C}$, like
- they are determined by just a few inputs,
- their contour integrals are all zero,
- they can't be bounded unless they are constant,
- ....
We then introduce **meromorphic functions**, which are like quotients of holomorphic functions, and find that we can detect their zeros by simply drawing loops in the plane and integrating over them: the famous **residue theorem** appears. (In the practice problems, you will see this even gives us a way to evaluate real integrals that can't be evaluated otherwise.)
- [**.**]\
Measure theory is the upgraded version of integration. The Riemann integration is for a lot of purposes not really sufficient; for example, if $f$ is the function equals $1$ at rational numbers but $0$ at irrational numbers, we would hope that $\int_0^1 f(x) \; dx = 0$, but the Riemann integral is not capable of handling this function $f$.
The **Lebesgue integral** will handle these mistakes by assigning a *measure* to a generic space $\Omega$, making it into a **measure space**. This will let us develop a richer theory of integration where the above integral *does* work out to zero because the "rational numbers have measure zero". Even the development of the measure will be an achievement, because it means we've developed a rigorous, complete way of talking about what notions like area and volume mean --- on any space, not just $\mathbb{R}^n$! So for example the Lebesgue integral will let us integrate functions over any **measure space**.
- [**.**]\
Using the tools of measure theory, we'll be able to start giving rigorous definitions of **probability**, too. We'll see that a **random variable** is actually a function from a measure space of worlds to $\mathbb{R}$, giving us a rigorous way to talk about its probabilities. We can then start actually stating results like the **law of large numbers** and **central limit theorem** in ways that make them both easy to state and straightforward to prove.
- [**.**]\
Multivariable calculus is often confusing because of all the partial derivatives. But we'll find out that, armed with our good understanding of linear algebra, that we're really looking at a **total derivative**: at every point of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ we can associate a *linear map* $Df$ which captures in one object the notion of partial derivatives. Set up this way, we'll get to see versions of **differential forms** and **Stokes' theorem**, and we finally will know what the notation $dx$ really means. In the end, we'll say a little bit about manifolds in general. | 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 might seem intimidating, it turns out that actually, by the time we start this chapter, *the hard work has already been done*: the notion of limits, open sets, and compactness will make short work of what was swept under the rug in AP calculus. Most of the proofs will thus actually be quite short. We sit back and watch all the pieces slowly come together as a reward for our careful study of topology beforehand.
That said, if you are willing to suspend belief, you can actually read most of the other parts without knowing the exact details of all the calculus here, so in some sense this part is "optional".
- [**.**]\
It turns out that **holomorphic functions** (complex-differentiable functions) are close to the nicest things ever: they turn out to be given by a Taylor series (i.e. are basically polynomials). This means we'll be able to prove unreasonably nice results about holomorphic functions $\mathbb{C} \to \mathbb{C}$, like
- they are determined by just a few inputs,
- their contour integrals are all zero,
- they can't be bounded unless they are constant,
- ....
We then introduce **meromorphic functions**, which are like quotients of holomorphic functions, and find that we can detect their zeros by simply drawing loops in the plane and integrating over them: the famous **residue theorem** appears. (In the practice problems, you will see this even gives us a way to evaluate real integrals that can't be evaluated otherwise.)
- [**.**]\
Measure theory is the upgraded version of integration. The Riemann integration is for a lot of purposes not really sufficient; for example, if $f$ is the function equals $1$ at rational numbers but $0$ at irrational numbers, we would hope that $\int_0^1 f(x) \; dx = 0$, but the Riemann integral is not capable of handling this function $f$.
The **Lebesgue integral** will handle these mistakes by assigning a *measure* to a generic space $\Omega$, making it into a **measure space**. This will let us develop a richer theory of integration where the above integral *does* work out to zero because the "rational numbers have measure zero". Even the development of the measure will be an achievement, because it means we've developed a rigorous, complete way of talking about what notions like area and volume mean --- on any space, not just $\mathbb{R}^n$! So for example the Lebesgue integral will let us integrate functions over any **measure space**.
- [**.**]\
Using the tools of measure theory, we'll be able to start giving rigorous definitions of **probability**, too. We'll see that a **random variable** is actually a function from a measure space of worlds to $\mathbb{R}$, giving us a rigorous way to talk about its probabilities. We can then start actually stating results like the **law of large numbers** and **central limit theorem** in ways that make them both easy to state and straightforward to prove.
- [**.**]\
Multivariable calculus is often confusing because of all the partial derivatives. But we'll find out that, armed with our good understanding of linear algebra, that we're really looking at a **total derivative**: at every point of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ we can associate a *linear map* $Df$ which captures in one object the notion of partial derivatives. Set up this way, we'll get to see versions of **differential forms** and **Stokes' theorem**, and we finally will know what the notation $dx$ really means. In the end, we'll say a little bit about manifolds in general. | 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 part, when we learn about **number fields**, a generalization of $\mathbb{Q}$ and $\mathbb{Z}$ to things like $\mathbb{Q}(\sqrt5)$ and $\mathbb{Z}[\sqrt{5}]$. We'll find out that we have unique factorization into prime ideals, that there is a real *multiplicative norm* in play here, and so on. We'll also see that Pell's equation falls out of this theory.
- [**.**]\
All the big buzz-words come out now: **Galois groups**, the **Frobenius**, and friends. We'll see quadratic reciprocity is just a shadow of the behavior of the Frobenius element, and meet the **Chebotarev density theorem**, which generalizes greatly the Dirichlet theorem on the infinitude of primes which are $a \pmod n$. Towards the end, we'll also state **Artin reciprocity**, one of the great results of **class field theory**, and how it generalizes quadratic reciprocity and cubic reciprocity.
- [**.**]\
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 part, when we learn about **number fields**, a generalization of $\mathbb{Q}$ and $\mathbb{Z}$ to things like $\mathbb{Q}(\sqrt5)$ and $\mathbb{Z}[\sqrt{5}]$. We'll find out that we have unique factorization into prime ideals, that there is a real *multiplicative norm* in play here, and so on. We'll also see that Pell's equation falls out of this theory.
- [**.**]\
All the big buzz-words come out now: **Galois groups**, the **Frobenius**, and friends. We'll see quadratic reciprocity is just a shadow of the behavior of the Frobenius element, and meet the **Chebotarev density theorem**, which generalizes greatly the Dirichlet theorem on the infinitude of primes which are $a \pmod n$. Towards the end, we'll also state **Artin reciprocity**, one of the great results of **class field theory**, and how it generalizes quadratic reciprocity and cubic reciprocity. | 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 abelian group $\pi_1(X)$ for every topological space $X$ which captures the idea there is a hole in the space, by throwing lassos into the space and seeing if we can reel them in.
Amazingly, the fundamental group $\pi_1(X)$ will, under mild conditions, tell you about ways to cover $X$ with a so-called **covering projection**. One picture is that one can wrap a real line $\mathbb{R}$ into a helix shape and then project it down into the circle $S^1$. This will turn out to correspond to the fact that $\pi_1(S^1) = \mathbb{Z}$ which has only one subgroup. More generally the subgroups of $\pi_1(X)$ will be in bijection with ways to cover the space $X$!
- [**.**]\
What do fields, groups, manifolds, metric spaces, measure spaces, modules, representations, rings, topological spaces, vector spaces, all have in common? Answer: they are all "objects with additional structure", with maps between them.
The notion of **category** will appropriately generalize all of them. We'll see that all sorts of constructions and ideas can be abstracted into the framework of a category, in which we *only* think about objects and arrows between them, without probing too hard into the details of what those objects are. This results in drawing many **commutative diagrams**.
For example, any way of taking an object in one category and getting another one (for example $\pi_1$ as above, from the category of spaces into the category of groups) will probably be a **functor**. We'll unify $G \times H$, $X \times Y$, $R \times S$, and anything with the $\times$ symbol into the notion of a product, and then even more generally into a **limit**. Towards the end, we talk about **abelian categories** and talk about the famous **snake lemma**, **five lemma**, and so on.
- [**.**]\
Using the language of category theory, we then resume our adventures in algebraic topology, in which we define the **homology groups** which give a different way of noticing holes in a space, in a way that is longer to define but easier to compute in practice. We'll then reverse the construction to get so-called **cohomology rings** instead, which give us an even finer invariant for telling spaces apart.
- [**.**]\
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 abelian group $\pi_1(X)$ for every topological space $X$ which captures the idea there is a hole in the space, by throwing lassos into the space and seeing if we can reel them in.
Amazingly, the fundamental group $\pi_1(X)$ will, under mild conditions, tell you about ways to cover $X$ with a so-called **covering projection**. One picture is that one can wrap a real line $\mathbb{R}$ into a helix shape and then project it down into the circle $S^1$. This will turn out to correspond to the fact that $\pi_1(S^1) = \mathbb{Z}$ which has only one subgroup. More generally the subgroups of $\pi_1(X)$ will be in bijection with ways to cover the space $X$!
- [**.**]\
What do fields, groups, manifolds, metric spaces, measure spaces, modules, representations, rings, topological spaces, vector spaces, all have in common? Answer: they are all "objects with additional structure", with maps between them.
The notion of **category** will appropriately generalize all of them. We'll see that all sorts of constructions and ideas can be abstracted into the framework of a category, in which we *only* think about objects and arrows between them, without probing too hard into the details of what those objects are. This results in drawing many **commutative diagrams**.
For example, any way of taking an object in one category and getting another one (for example $\pi_1$ as above, from the category of spaces into the category of groups) will probably be a **functor**. We'll unify $G \times H$, $X \times Y$, $R \times S$, and anything with the $\times$ symbol into the notion of a product, and then even more generally into a **limit**. Towards the end, we talk about **abelian categories** and talk about the famous **snake lemma**, **five lemma**, and so on.
- [**.**]\
Using the language of category theory, we then resume our adventures in algebraic topology, in which we define the **homology groups** which give a different way of noticing holes in a space, in a way that is longer to define but easier to compute in practice. We'll then reverse the construction to get so-called **cohomology rings** instead, which give us an even finer invariant for telling spaces apart. | 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 show its face, and then we'll play with **projective varieties** and **quasi-projective varieties**, with a bonus detour into **Bézout's theorem**. All this prepares us for our journey into schemes.
- [**.**]\
We now get serious and delve into Grothendieck's definition of an **affine scheme**: a generalization of our classical varieties that allows us to start with any ring $A$ and construct a space $\Spec A$ on it. We'll equip it with its own Zariski topology and then a sheaf of functions on it, making it into a **locally ringed space**; we will find that the sheaf can be understood effectively in terms of **localization** on it. We'll find that the language of commutative algebra provides elegant generalizations of what's going on geometrically: prime ideals correspond to irreducible closed subsets, radical ideals correspond to closed subsets, maximal ideals correspond to closed points, and so on. We'll draw lots of pictures of spaces and examples to accompany this.
- [**.**]\
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 show its face, and then we'll play with **projective varieties** and **quasi-projective varieties**, with a bonus detour into **Bézout's theorem**. All this prepares us for our journey into schemes.
- [**.**]\
We now get serious and delve into Grothendieck's definition of an **affine scheme**: a generalization of our classical varieties that allows us to start with any ring $A$ and construct a space $\Spec A$ on it. We'll equip it with its own Zariski topology and then a sheaf of functions on it, making it into a **locally ringed space**; we will find that the sheaf can be understood effectively in terms of **localization** on it. We'll find that the language of commutative algebra provides elegant generalizations of what's going on geometrically: prime ideals correspond to irreducible closed subsets, radical ideals correspond to closed subsets, maximal ideals correspond to closed points, and so on. We'll draw lots of pictures of spaces and examples to accompany this. | 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 of how math is built axiomatically at the very bottom foundations. We'll meet the **ordinal numbers** and **cardinal numbers** and learn how to do **transfinite induction** with them.
- [**.**]\
The **continuum hypothesis** states that there are no cardinalities between the size of the natural numbers and the size of the real numbers. It was shown to be *independent* of the axioms --- one cannot prove or disprove it. How could a result like that possibly be proved? Using our understanding of the ZF axioms, we'll develop a bit of **model theory** and then use **forcing** in order to show how to construct entire models of the universe in which the continuum hypothesis is true or false.
[^1]: In particular, I think it's easier to learn what a homeomorphism is after seeing group isomorphism, and what a homomorphism is after seeing continuous map.
- [**.**]\
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 of how math is built axiomatically at the very bottom foundations. We'll meet the **ordinal numbers** and **cardinal numbers** and learn how to do **transfinite induction** with them.
- [**.**]\
The **continuum hypothesis** states that there are no cardinalities between the size of the natural numbers and the size of the real numbers. It was shown to be *independent* of the axioms --- one cannot prove or disprove it. How could a result like that possibly be proved? Using our understanding of the ZF axioms, we'll develop a bit of **model theory** and then use **forcing** in order to show how to construct entire models of the universe in which the continuum hypothesis is true or false.
[^1]: In particular, I think it's easier to learn what a homeomorphism is after seeing group isomorphism, and what a homomorphism is after seeing continuous map. | 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 when two groups are the same. | 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 operation is *addition*. Note that
- The element $0 \in \mathbb{Z}$ is an *identity*: $a+0=0+a = a$ for any $a$.
- Every element $a \in \mathbb{Z}$ has an additive *inverse*: $a + (-a) = (-a) + a = 0$.
We call this group $\mathbb{Z}$.
Let $\mathbb{Q}^\times$ be the set of *nonzero rational numbers*. The pair $(\mathbb{Q}^\times, \cdot)$ is a group: the set is $\mathbb{Q}^\times$ and the associative operation is *multiplication*.
Again we see the same two nice properties.
- The element $1 \in \mathbb{Q}^\times$ is an *identity*: for any rational number, $a \cdot 1 = 1 \cdot a = a$.
- For any rational number $x \in \mathbb{Q}^\times$, we have an inverse $x^{-1}$, such that $$x \cdot x^{-1} = x^{-1} \cdot x = 1.$$
From this you might already have a guess what the definition of a group is.
A **group** is a pair $G = (G, \star)$ consisting of a set of elements $G$, and a binary operation $\star$ on $G$, such that:
- $G$ has an **identity element**, usually denoted $1_G$ or just $1$, with the property that $$1_G \star g = g \star 1_G = g \text{ for all $g \in G$}.$$
- The operation is **associative**, meaning $(a \star b) \star c = a \star (b \star c)$ for any $a,b,c \in G$. Consequently we generally don't write the parentheses.
- Each element $g \in G$ has an **inverse**, that is, an element $h \in G$ such that $$g \star h = h \star g = 1_G.$$
Some authors like to add a "closure" axiom, i.e. to say explicitly that $g \star h \in G$. This is implied already by the fact that $\star$ is a binary operation on $G$, but is worth keeping in mind for the examples below.
It is not required that $\star$ is commutative ($a \star b = b \star a$). So we say that a group is **abelian** if the operation is commutative and **non-abelian** otherwise.
- The pair $(\mathbb{Q}, \cdot)$ is NOT a group. (Here $\mathbb{Q}$ is rational numbers.) While there is an identity element, the element $0 \in \mathbb{Q}$ does not have an inverse.
- The pair $(\mathbb{Z}, \cdot)$ is also NOT a group. (Why?)
- Let $\mathrm{Mat}_{2 \times 2}(\mathbb{R})$ be the set of $2 \times 2$ real matrices. Then $(\mathrm{Mat}_{2 \times 2}(\mathbb{R}), \cdot)$ (where $\cdot$ is matrix multiplication) is NOT a group. Indeed, even though we have an identity matrix $$\begin{bmatrix}
1 & 0 \\ 0 & 1
\end{bmatrix}$$ we still run into the same issue as before: the zero matrix does not have a multiplicative inverse.
(Even if we delete the zero matrix from the set, the resulting structure is still not a group: those of you that know some linear algebra might recall that any matrix with determinant zero cannot have an inverse.)
Let's resume writing down examples. Here are some more **abelian examples** of groups:
Let $S^1$ denote the set of complex numbers $z$ with absolute value one; that is $$S^1 := \left\{ z \in \mathbb{C} \mid \left\lvert z \right\rvert = 1 \right\}.$$ Then $(S^1, \times)$ is a group because
- The complex number $1 \in S^1$ serves as the identity, and
- Each complex number $z \in S^1$ has an inverse $\frac 1z$ which is also in $S^1$, since $\left\lvert z^{-1} \right\rvert = \left\lvert z \right\rvert^{-1} = 1$.
There is one thing I ought to also check: that $z_1 \times z_2$ is actually still in $S^1$. But this follows from the fact that $\left\lvert z_1z_2 \right\rvert = \left\lvert z_1 \right\rvert \left\lvert z_2 \right\rvert = 1$.
Here is an example from number theory: Let $n > 1$ be an integer, and consider the residues (remainders) modulo $n$. These form a group under addition. We call this the **cyclic group of order $n$**, and denote it as $\mathbb{Z}/n\mathbb{Z}$, with elements $\overline 0, \overline 1, \dots$. The identity is $\overline 0$.
Let $p$ be a prime. Consider the *nonzero residues modulo $p$*, which we denote by $(\mathbb{Z}/p\mathbb{Z})^\times$. Then $\left( (\mathbb{Z}/p\mathbb{Z})^\times, \times \right)$ is a group.
Why do we need the fact that $p$ is prime?
(Digression: the notation $\mathbb{Z}/n\mathbb{Z}$ and $(\mathbb{Z}/p\mathbb{Z})^\times$ may seem strange but will make sense when we talk about rings and ideals. Set aside your worry for now.)
Here are some **non-abelian examples**:
Let $n$ be a positive integer. Then $\GL_n(\mathbb{R})$ is defined as the set of $n \times n$ real matrices which have nonzero determinant. It turns out that with this condition, every matrix does indeed have an inverse, so $(\GL_n(\mathbb{R}), \times)$ is a group, called the **general linear group**.
(The fact that $\GL_n(\mathbb{R})$ is closed under $\times$ follows from the linear algebra fact that $\det (AB) = \det A \det B$, proved in later chapters.)
Following the example above, let $\SL_n(\mathbb{R})$ denote the set of $n \times n$ matrices whose determinant is actually $1$. Again, for linear algebra reasons it turns out that $(\SL_n(\mathbb{R}), \times)$ is also a group, called the **special linear group**. | 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 identity permutation, and
- Each permutation has an inverse.
The group $S_n$ is called the **symmetric group** on $n$ elements.
The **dihedral group of order $2n$**, denoted $D_{2n}$, is the group of symmetries of a regular $n$-gon $A_1A_2 \dots A_n$, which includes rotations and reflections. It consists of the $2n$ elements $$\left\{ 1, r, r^2, \dots, r^{n-1}, s, sr, sr^2, \dots, sr^{n-1} \right\}.$$ The element $r$ corresponds to rotating the $n$-gon by $\frac{2\pi}{n}$, while $s$ corresponds to reflecting it across the line $OA_1$ (here $O$ is the center of the polygon). So $rs$ means "reflect then rotate" (like with function composition, we read from right to left).
In particular, $r^n = s^2 = 1$. You can also see that $r^k s = sr^{-k}$.
Here is a picture of some elements of $D_{10}$.
size(12cm); picture aoeu(string a, string b, string c, string d, string e, string x) draw(dir(0)--dir(72)--dir(144)--dir(216)--dir(288)--cycle); MP(a, dir(0), dir(0)); MP(b, dir(72), dir(72)); MP(c, dir(144), dir(144)); MP(d, dir(216), dir(216)); MP(e, dir(288), dir(288)); MP(x, origin, origin); return CC(); picture one = aoeu(\"1\", \"2\", \"3\", \"4\", \"5\", \"1\"); picture r = aoeu(\"5\", \"1\", \"2\", \"3\", \"4\", \"r\"); picture s = aoeu(\"1\", \"5\", \"4\", \"3\", \"2\", \"s\"); picture sr = aoeu(\"5\", \"4\", \"3\", \"2\", \"1\", \"sr\"); picture rs = aoeu(\"2\", \"1\", \"5\", \"4\", \"3\", \"rs\"); add(shift( (0,0) ) \* one); add(shift( (3,0) ) \* r); add(shift( (6,0) ) \* s); add(shift( (9,0) ) \* sr); add(shift( (12,0) ) \* rs);
Trivia: the dihedral group $D_{12}$ is my favorite example of a non-abelian group, and is the first group I try for any exam question of the form "find an example...".
More examples:
Let $(G, \star)$ and $(H, \ast)$ be groups. We can define a **product group** $(G \times H, {\cdot})$, as follows. The elements of the group will be ordered pairs $(g,h) \in G \times H$. Then $$(g_1, h_1) \cdot (g_2, h_2) = (g_1 \star g_2, h_1 \ast h_2) \in G \times H$$ is the group operation.
What are the identity and inverses of the product group?
The **trivial group**, often denoted $0$ or $1$, is the group with only an identity element. I will use the notation $\{1\}$.
Which of these are groups?
(a) Rational numbers with odd denominators (in simplest form), where the operation is addition. (This includes integers, written as $n/1$, and $0 = 0/1$).
(b) The set of rational numbers with denominator at most $2$, where the operation is addition.
(c) The set of rational numbers with denominator at most $2$, where the operation is multiplication.
(d) The set of nonnegative integers, where the operation is addition. | 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 operation is *addition*. Note that
- The element $0 \in \mathbb{Z}$ is an *identity*: $a+0=0+a = a$ for any $a$.
- Every element $a \in \mathbb{Z}$ has an additive *inverse*: $a + (-a) = (-a) + a = 0$.
We call this group $\mathbb{Z}$.
Let $\mathbb{Q}^\times$ be the set of *nonzero rational numbers*. The pair $(\mathbb{Q}^\times, \cdot)$ is a group: the set is $\mathbb{Q}^\times$ and the associative operation is *multiplication*.
Again we see the same two nice properties.
- The element $1 \in \mathbb{Q}^\times$ is an *identity*: for any rational number, $a \cdot 1 = 1 \cdot a = a$.
- For any rational number $x \in \mathbb{Q}^\times$, we have an inverse $x^{-1}$, such that $$x \cdot x^{-1} = x^{-1} \cdot x = 1.$$
From this you might already have a guess what the definition of a group is.
A **group** is a pair $G = (G, \star)$ consisting of a set of elements $G$, and a binary operation $\star$ on $G$, such that:
- $G$ has an **identity element**, usually denoted $1_G$ or just $1$, with the property that $$1_G \star g = g \star 1_G = g \text{ for all $g \in G$}.$$
- The operation is **associative**, meaning $(a \star b) \star c = a \star (b \star c)$ for any $a,b,c \in G$. Consequently we generally don't write the parentheses.
- Each element $g \in G$ has an **inverse**, that is, an element $h \in G$ such that $$g \star h = h \star g = 1_G.$$
Some authors like to add a "closure" axiom, i.e. to say explicitly that $g \star h \in G$. This is implied already by the fact that $\star$ is a binary operation on $G$, but is worth keeping in mind for the examples below.
It is not required that $\star$ is commutative ($a \star b = b \star a$). So we say that a group is **abelian** if the operation is commutative and **non-abelian** otherwise.
- The pair $(\mathbb{Q}, \cdot)$ is NOT a group. (Here $\mathbb{Q}$ is rational numbers.) While there is an identity element, the element $0 \in \mathbb{Q}$ does not have an inverse.
- The pair $(\mathbb{Z}, \cdot)$ is also NOT a group. (Why?)
- Let $\mathrm{Mat}_{2 \times 2}(\mathbb{R})$ be the set of $2 \times 2$ real matrices. Then $(\mathrm{Mat}_{2 \times 2}(\mathbb{R}), \cdot)$ (where $\cdot$ is matrix multiplication) is NOT a group. Indeed, even though we have an identity matrix $$\begin{bmatrix}
1 & 0 \\ 0 & 1
\end{bmatrix}$$ we still run into the same issue as before: the zero matrix does not have a multiplicative inverse.
(Even if we delete the zero matrix from the set, the resulting structure is still not a group: those of you that know some linear algebra might recall that any matrix with determinant zero cannot have an inverse.)
Let's resume writing down examples. Here are some more **abelian examples** of groups:
Let $S^1$ denote the set of complex numbers $z$ with absolute value one; that is $$S^1 := \left\{ z \in \mathbb{C} \mid \left\lvert z \right\rvert = 1 \right\}.$$ Then $(S^1, \times)$ is a group because
- The complex number $1 \in S^1$ serves as the identity, and
- Each complex number $z \in S^1$ has an inverse $\frac 1z$ which is also in $S^1$, since $\left\lvert z^{-1} \right\rvert = \left\lvert z \right\rvert^{-1} = 1$.
There is one thing I ought to also check: that $z_1 \times z_2$ is actually still in $S^1$. But this follows from the fact that $\left\lvert z_1z_2 \right\rvert = \left\lvert z_1 \right\rvert \left\lvert z_2 \right\rvert = 1$.
Here is an example from number theory: Let $n > 1$ be an integer, and consider the residues (remainders) modulo $n$. These form a group under addition. We call this the **cyclic group of order $n$**, and denote it as $\mathbb{Z}/n\mathbb{Z}$, with elements $\overline 0, \overline 1, \dots$. The identity is $\overline 0$.
Let $p$ be a prime. Consider the *nonzero residues modulo $p$*, which we denote by $(\mathbb{Z}/p\mathbb{Z})^\times$. Then $\left( (\mathbb{Z}/p\mathbb{Z})^\times, \times \right)$ is a group.
Why do we need the fact that $p$ is prime?
(Digression: the notation $\mathbb{Z}/n\mathbb{Z}$ and $(\mathbb{Z}/p\mathbb{Z})^\times$ may seem strange but will make sense when we talk about rings and ideals. Set aside your worry for now.)
Here are some **non-abelian examples**:
Let $n$ be a positive integer. Then $\GL_n(\mathbb{R})$ is defined as the set of $n \times n$ real matrices which have nonzero determinant. It turns out that with this condition, every matrix does indeed have an inverse, so $(\GL_n(\mathbb{R}), \times)$ is a group, called the **general linear group**.
(The fact that $\GL_n(\mathbb{R})$ is closed under $\times$ follows from the linear algebra fact that $\det (AB) = \det A \det B$, proved in later chapters.)
Following the example above, let $\SL_n(\mathbb{R})$ denote the set of $n \times n$ matrices whose determinant is actually $1$. Again, for linear algebra reasons it turns out that $(\SL_n(\mathbb{R}), \times)$ is also a group, called the **special linear group**. | 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 identity permutation, and
- Each permutation has an inverse.
The group $S_n$ is called the **symmetric group** on $n$ elements.
The **dihedral group of order $2n$**, denoted $D_{2n}$, is the group of symmetries of a regular $n$-gon $A_1A_2 \dots A_n$, which includes rotations and reflections. It consists of the $2n$ elements $$\left\{ 1, r, r^2, \dots, r^{n-1}, s, sr, sr^2, \dots, sr^{n-1} \right\}.$$ The element $r$ corresponds to rotating the $n$-gon by $\frac{2\pi}{n}$, while $s$ corresponds to reflecting it across the line $OA_1$ (here $O$ is the center of the polygon). So $rs$ means "reflect then rotate" (like with function composition, we read from right to left).
In particular, $r^n = s^2 = 1$. You can also see that $r^k s = sr^{-k}$.
Here is a picture of some elements of $D_{10}$.
size(12cm); picture aoeu(string a, string b, string c, string d, string e, string x) draw(dir(0)--dir(72)--dir(144)--dir(216)--dir(288)--cycle); MP(a, dir(0), dir(0)); MP(b, dir(72), dir(72)); MP(c, dir(144), dir(144)); MP(d, dir(216), dir(216)); MP(e, dir(288), dir(288)); MP(x, origin, origin); return CC(); picture one = aoeu(\"1\", \"2\", \"3\", \"4\", \"5\", \"1\"); picture r = aoeu(\"5\", \"1\", \"2\", \"3\", \"4\", \"r\"); picture s = aoeu(\"1\", \"5\", \"4\", \"3\", \"2\", \"s\"); picture sr = aoeu(\"5\", \"4\", \"3\", \"2\", \"1\", \"sr\"); picture rs = aoeu(\"2\", \"1\", \"5\", \"4\", \"3\", \"rs\"); add(shift( (0,0) ) \* one); add(shift( (3,0) ) \* r); add(shift( (6,0) ) \* s); add(shift( (9,0) ) \* sr); add(shift( (12,0) ) \* rs);
Trivia: the dihedral group $D_{12}$ is my favorite example of a non-abelian group, and is the first group I try for any exam question of the form "find an example...".
More examples:
Let $(G, \star)$ and $(H, \ast)$ be groups. We can define a **product group** $(G \times H, {\cdot})$, as follows. The elements of the group will be ordered pairs $(g,h) \in G \times H$. Then $$(g_1, h_1) \cdot (g_2, h_2) = (g_1 \star g_2, h_1 \ast h_2) \in G \times H$$ is the group operation.
What are the identity and inverses of the product group?
The **trivial group**, often denoted $0$ or $1$, is the group with only an identity element. I will use the notation $\{1\}$.
Which of these are groups?
(a) Rational numbers with odd denominators (in simplest form), where the operation is addition. (This includes integers, written as $n/1$, and $0 = 0/1$).
(b) The set of rational numbers with denominator at most $2$, where the operation is addition.
(c) The set of rational numbers with denominator at most $2$, where the operation is multiplication.
(d) The set of nonnegative integers, where the operation is addition. | 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 $$g^n
=
\underbrace{g \star \dots \star g}_{\text{$n$ times}}.$$ Moreover, we let $g^{-1}$ denote the inverse of $g$, and $g^{-n} = (g^{-1})^n$.
In mathematics, a common theme is to require that objects satisfy certain minimalistic properties, with certain examples in mind, but then ignore the examples on paper and try to deduce as much as you can just from the properties alone. (Math olympiad veterans are likely familiar with "functional equations" in which knowing a single property about a function is enough to determine the entire function.) Let's try to do this here, and see what we can conclude just from knowing .
It is a law in Guam and 37 other states that I now state the following proposition.
Let $G$ be a group.
(a) The identity of a group is unique.
(b) The inverse of any element is unique.
(c) For any $g \in G$, ${(g^{-1})}^{-1} = g$.
*Proof.* This is mostly just some formal manipulations, and you needn't feel bad skipping it on a first read.
(a) If $1$ and $1'$ are identities, then $1 = 1 \star 1' = 1'$.
(b) If $h$ and $h'$ are inverses to $g$, then $1_G = g \star h
\implies h' = (h' \star g) \star h = 1_G \star h = h$.
(c) Trivial; omitted.
◻
Now we state a slightly more useful proposition.
Let $G$ be a group, and $a,b \in G$. Then $(ab)^{-1} = b^{-1} a^{-1}$.
*Proof.* Direct computation. We have $$(ab)(b^{-1} a^{-1})
= a (bb^{-1}) a^{-1} = aa^{-1} = 1_G.$$ Similarly, $(b^{-1} a^{-1})(ab) = 1_G$ as well. Hence $(ab)^{-1} = b^{-1} a^{-1}$. ◻
Finally, we state a very important lemma about groups, which highlights why having an inverse is so valuable.
Let $G$ be a group, and pick a $g \in G$. Then the map $G \to G$ given by $x \mapsto gx$ is a bijection.
Check this by showing injectivity and surjectivity directly. (If you don't know what these words mean, consult .)
Let $G = (\mathbb{Z}/7\mathbb{Z})^\times$ (as in ) and pick $g=3$. The above lemma states that the map $x \mapsto 3 \cdot x$ is a bijection, and we can see this explicitly: $$\begin{align*}
1 &\overset{\times 3}{\longmapsto} 3 \pmod 7 \\
2 &\overset{\times 3}{\longmapsto} 6 \pmod 7 \\
3 &\overset{\times 3}{\longmapsto} 2 \pmod 7 \\
4 &\overset{\times 3}{\longmapsto} 5 \pmod 7 \\
5 &\overset{\times 3}{\longmapsto} 1 \pmod 7 \\
6 &\overset{\times 3}{\longmapsto} 4 \pmod 7.
\end{align*}$$
The fact that the map is injective is often called the **cancellation law**. (Why do you think so?)
You don't need to worry about this for a few chapters, but I'll bring it up now anyways. In most of our examples up until now the operation $\star$ was thought of like multiplication of some sort, which is why $1 = 1_G$ was a natural notation for the identity element.
But there are groups like $\mathbb{Z} = (\mathbb{Z},+)$ where the operation $\star$ is thought of as addition, in which case the notation $0 = 0_G$ might make more sense instead. (In general, whenever an operation is denoted $+$, the operation is almost certainly commutative.) We will eventually start doing so too when we discuss rings and linear algebra. | 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 only differ in the names of the elements. Think about what this might mean formally for a moment.
Specifically the map $$\phi \colon \mathbb{Z} \to 10 \mathbb{Z} \text{ by } x \mapsto 10 x$$ is a bijection of the underlying sets which respects the group operation. In symbols, $$\phi(x + y) = \phi(x) + \phi(y).$$ In other words, $\phi$ is a way of re-assigning names of the elements without changing the structure of the group. That's all just formalism for capturing the obvious fact that $(\mathbb{Z},+)$ and $(10 \mathbb{Z}, +)$ are the same thing.
Now, let's do the general definition.
Let $G = (G, \star)$ and $H = (H, \ast)$ be groups. A bijection $\phi \colon G \to H$ is called an **isomorphism** if $$\phi(g_1 \star g_2) = \phi(g_1) \ast \phi(g_2) \quad
\text{for all $g_1, g_2 \in G$}.$$ If there exists an isomorphism from $G$ to $H$, then we say $G$ and $H$ are **isomorphic** and write $G \cong H$.
Note that in this definition, the left-hand side $\phi(g_1 \star g_2)$ uses the operation of $G$ while the right-hand side $\phi(g_1) \ast \phi(g_2)$ uses the operation of $H$.
Let $G$ and $H$ be groups. We have the following isomorphisms.
(a) $\mathbb{Z} \cong 10 \mathbb{Z}$, as above.
(b) There is an isomorphism $$G \times H \cong H \times G$$ by the map $(g,h) \mapsto (h,g)$.
(c) The identity map $\mathrm{id} \colon G \to G$ is an isomorphism, hence $G \cong G$.
(d) There is another isomorphism of $\mathbb{Z}$ to itself: send every $x$ to $-x$.
As a nontrivial example, we claim that $\mathbb{Z}/6\mathbb{Z} \cong (\mathbb{Z}/7\mathbb{Z})^\times$. The bijection is $$\phi(\text{$a$ mod $6$}) = \text{$3^a$ mod $7$}.$$
- This map is a bijection by explicit calculation: $$(3^0, 3^1, 3^2, 3^3, 3^4, 3^5)
\equiv (1,3,2,6,4,5) \pmod 7.$$ (Technically, I should more properly write $3^{0 \bmod 6} = 1$ and so on to be pedantic.)
- Finally, we need to verify that this map respects the group operation. In other words, we want to see that $\phi(a+b) = \phi(a) \phi(b)$ since the operation of $\mathbb{Z}/6\mathbb{Z}$ is addition while the operation of $(\mathbb{Z}/7\mathbb{Z})^\times$ is multiplication. That's just saying that $3^{a+b \bmod 6} \equiv 3^{a \bmod 6} 3^{b \bmod 6} \pmod 7$, which is true.
More generally, for any prime $p$, there exists an element $g \in (\mathbb{Z}/p\mathbb{Z})^\times$ called a **primitive root** modulo $p$ such that $1, g, g^2, \dots, g^{p-2}$ are all different modulo $p$. One can show by copying the above proof that $$\mathbb{Z}/c\mathbb{Z}{p-1} \cong (\mathbb{Z}/p\mathbb{Z})^\times \text{ for all primes $p$}.$$ The example above was the special case $p=7$ and $g=3$.
Assuming the existence of primitive roots, establish the isomorphism $\mathbb{Z}/c\mathbb{Z}{p-1} \cong (\mathbb{Z}/p\mathbb{Z})^\times$ as above.
It's not hard to see that $\cong$ is an equivalence relation (why?). Moreover, because we really only care about the structure of groups, we'll usually consider two groups to be the same when they are isomorphic. So phrases such as "find all groups" really mean "find all groups up to isomorphism". | 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 only differ in the names of the elements. Think about what this might mean formally for a moment.
Specifically the map $$\phi \colon \mathbb{Z} \to 10 \mathbb{Z} \text{ by } x \mapsto 10 x$$ is a bijection of the underlying sets which respects the group operation. In symbols, $$\phi(x + y) = \phi(x) + \phi(y).$$ In other words, $\phi$ is a way of re-assigning names of the elements without changing the structure of the group. That's all just formalism for capturing the obvious fact that $(\mathbb{Z},+)$ and $(10 \mathbb{Z}, +)$ are the same thing.
Now, let's do the general definition.
Let $G = (G, \star)$ and $H = (H, \ast)$ be groups. A bijection $\phi \colon G \to H$ is called an **isomorphism** if $$\phi(g_1 \star g_2) = \phi(g_1) \ast \phi(g_2) \quad
\text{for all $g_1, g_2 \in G$}.$$ If there exists an isomorphism from $G$ to $H$, then we say $G$ and $H$ are **isomorphic** and write $G \cong H$.
Note that in this definition, the left-hand side $\phi(g_1 \star g_2)$ uses the operation of $G$ while the right-hand side $\phi(g_1) \ast \phi(g_2)$ uses the operation of $H$.
Let $G$ and $H$ be groups. We have the following isomorphisms.
(a) $\mathbb{Z} \cong 10 \mathbb{Z}$, as above.
(b) There is an isomorphism $$G \times H \cong H \times G$$ by the map $(g,h) \mapsto (h,g)$.
(c) The identity map $\mathrm{id} \colon G \to G$ is an isomorphism, hence $G \cong G$.
(d) There is another isomorphism of $\mathbb{Z}$ to itself: send every $x$ to $-x$.
As a nontrivial example, we claim that $\mathbb{Z}/6\mathbb{Z} \cong (\mathbb{Z}/7\mathbb{Z})^\times$. The bijection is $$\phi(\text{$a$ mod $6$}) = \text{$3^a$ mod $7$}.$$
- This map is a bijection by explicit calculation: $$(3^0, 3^1, 3^2, 3^3, 3^4, 3^5)
\equiv (1,3,2,6,4,5) \pmod 7.$$ (Technically, I should more properly write $3^{0 \bmod 6} = 1$ and so on to be pedantic.)
- Finally, we need to verify that this map respects the group operation. In other words, we want to see that $\phi(a+b) = \phi(a) \phi(b)$ since the operation of $\mathbb{Z}/6\mathbb{Z}$ is addition while the operation of $(\mathbb{Z}/7\mathbb{Z})^\times$ is multiplication. That's just saying that $3^{a+b \bmod 6} \equiv 3^{a \bmod 6} 3^{b \bmod 6} \pmod 7$, which is true.
More generally, for any prime $p$, there exists an element $g \in (\mathbb{Z}/p\mathbb{Z})^\times$ called a **primitive root** modulo $p$ such that $1, g, g^2, \dots, g^{p-2}$ are all different modulo $p$. One can show by copying the above proof that $$\mathbb{Z}/c\mathbb{Z}{p-1} \cong (\mathbb{Z}/p\mathbb{Z})^\times \text{ for all primes $p$}.$$ The example above was the special case $p=7$ and $g=3$.
Assuming the existence of primitive roots, establish the isomorphism $\mathbb{Z}/c\mathbb{Z}{p-1} \cong (\mathbb{Z}/p\mathbb{Z})^\times$ as above.
It's not hard to see that $\cong$ is an equivalence relation (why?). Moreover, because we really only care about the structure of groups, we'll usually consider two groups to be the same when they are isomorphic. So phrases such as "find all groups" really mean "find all groups up to isomorphism". | 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** just to mean that $\left\lvert G \right\rvert$ is finite.
For a prime $p$, $\left\lvert (\mathbb{Z}/p\mathbb{Z})^\times \right\rvert = p-1$. In other words, the order of $(\mathbb{Z}/p\mathbb{Z})^\times$ is $p-1$. As another example, the order of the symmetric group $S_n$ is $n!$ and the order of the dihedral group $D_{2n}$ is $2n$.
The **order of an element** $g \in G$ is the smallest positive integer $n$ such that $g^n = 1_G$, or $\infty$ if no such $n$ exists. We denote this by $\ord g$.
The order of $-1$ in $\mathbb{Q}^\times$ is $2$, while the order of $1$ in $\mathbb{Z}$ is infinite.
Find the order of each of the six elements of $\mathbb{Z}/6\mathbb{Z}$, the cyclic group on six elements. (See if you've forgotten what $\mathbb{Z}/6\mathbb{Z}$ means.)
If you know olympiad number theory, this coincides with the definition of an order of a residue mod $p$. That's why we use the term "order" there as well. In particular, a primitive root is precisely an element $g \in (\mathbb{Z}/p\mathbb{Z})^\times$ such that $\ord g = p-1$.
You might also know that if $x^n \equiv 1 \pmod p$, then the order of $x \pmod p$ must divide $n$. The same is true in a general group for exactly the same reason.
If $g^n = 1_G$ then $\ord g$ divides $n$.
Also, you can show that any element of a finite group has a finite order. The proof is just an olympiad-style pigeonhole argument. Consider the infinite sequence $1_G, g, g^2, \dots$, and find two elements that are the same.
Let $G$ be a finite group. For any $g \in G$, $\ord g$ is finite.
What's the last property of $(\mathbb{Z}/p\mathbb{Z})^\times$ that you know from olympiad math? We have Fermat's little theorem: for any $a \in (\mathbb{Z}/p\mathbb{Z})^\times$, we have $a^{p-1} \equiv 1 \pmod p$. This is no coincidence: exactly the same thing is true in a more general setting.
Let $G$ be any finite group. Then $x^{\left\lvert G \right\rvert} = 1_G$ for any $x \in G$.
Keep this result in mind! We'll prove it later in the generality of .
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** just to mean that $\left\lvert G \right\rvert$ is finite.
For a prime $p$, $\left\lvert (\mathbb{Z}/p\mathbb{Z})^\times \right\rvert = p-1$. In other words, the order of $(\mathbb{Z}/p\mathbb{Z})^\times$ is $p-1$. As another example, the order of the symmetric group $S_n$ is $n!$ and the order of the dihedral group $D_{2n}$ is $2n$.
The **order of an element** $g \in G$ is the smallest positive integer $n$ such that $g^n = 1_G$, or $\infty$ if no such $n$ exists. We denote this by $\ord g$.
The order of $-1$ in $\mathbb{Q}^\times$ is $2$, while the order of $1$ in $\mathbb{Z}$ is infinite.
Find the order of each of the six elements of $\mathbb{Z}/6\mathbb{Z}$, the cyclic group on six elements. (See if you've forgotten what $\mathbb{Z}/6\mathbb{Z}$ means.)
If you know olympiad number theory, this coincides with the definition of an order of a residue mod $p$. That's why we use the term "order" there as well. In particular, a primitive root is precisely an element $g \in (\mathbb{Z}/p\mathbb{Z})^\times$ such that $\ord g = p-1$.
You might also know that if $x^n \equiv 1 \pmod p$, then the order of $x \pmod p$ must divide $n$. The same is true in a general group for exactly the same reason.
If $g^n = 1_G$ then $\ord g$ divides $n$.
Also, you can show that any element of a finite group has a finite order. The proof is just an olympiad-style pigeonhole argument. Consider the infinite sequence $1_G, g, g^2, \dots$, and find two elements that are the same.
Let $G$ be a finite group. For any $g \in G$, $\ord g$ is finite.
What's the last property of $(\mathbb{Z}/p\mathbb{Z})^\times$ that you know from olympiad math? We have Fermat's little theorem: for any $a \in (\mathbb{Z}/p\mathbb{Z})^\times$, we have $a^{p-1} \equiv 1 \pmod p$. This is no coincidence: exactly the same thing is true in a more general setting.
Let $G$ be any finite group. Then $x^{\left\lvert G \right\rvert} = 1_G$ for any $x \in G$.
Keep this result in mind! We'll prove it later in the generality of . | 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 subgroup of $\GL_n(\mathbb{R})$. And this definition generalizes in exactly the way you expect.
Let $G = (G, \star)$ be a group. A **subgroup** of $G$ is exactly what you would expect it to be: a group $H = (H, \star)$ where $H$ is a subset of $G$. It's a **proper subgroup** if $H \neq G$.
To specify a group $G$, I needed to tell you both what the set $G$ was and the operation $\star$ was. But to specify a subgroup $H$ of a given group $G$, I only need to tell you who its elements are: the operation of $H$ is just inherited from the operation of $G$.
(a) $2\mathbb{Z}$ is a subgroup of $\mathbb{Z}$, which is isomorphic to $\mathbb{Z}$ itself!
(b) Consider again $S_n$, the symmetric group on $n$ elements. Let $T$ be the set of permutations $\tau \colon \{1, \dots, n\} \to \{1, \dots, n\}$ for which $\tau(n) = n$. Then $T$ is a subgroup of $S_n$; in fact, it is isomorphic to $S_{n-1}$.
(c) Consider the group $G \times H$ () and the elements $\left\{ (g, 1_H) \mid g \in G \right\}$. This is a subgroup of $G \times H$ (why?). In fact, it is isomorphic to $G$ by the isomorphism $(g,1_H) \mapsto g$.
For any group $G$, the trivial group $\{1_G\}$ and the entire group $G$ are subgroups of $G$.
Next is an especially important example that we'll talk about more in later chapters.
Let $x$ be an element of a group $G$. Consider the set $$\left<x\right> = \left\{ \dots, x^{-2}, x^{-1}, 1, x, x^2, \dots \right\}.$$ This is also a subgroup of $G$, called the subgroup generated by $x$.
If $\ord x = 2015$, what is the above subgroup equal to? What if $\ord x = \infty$?
Finally, we present some non-examples of subgroups.
Consider the group $\mathbb{Z} = (\mathbb{Z}, +)$.
(a) The set $\left\{ 0,1,2,\dots \right\}$ is not a subgroup of $\mathbb{Z}$ because it does not contain inverses.
(b) The set $\{ n^3 \mid n \in \mathbb{Z} \}
= \{ \dots, -8, -1, 0, 1, 8, \dots \}$ is not a subgroup because it is not closed under addition; the sum of two cubes is not in general a cube.
(c) The empty set $\varnothing$ is not a subgroup of $\mathbb{Z}$ because it lacks an identity element.
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 subgroup of $\GL_n(\mathbb{R})$. And this definition generalizes in exactly the way you expect.
Let $G = (G, \star)$ be a group. A **subgroup** of $G$ is exactly what you would expect it to be: a group $H = (H, \star)$ where $H$ is a subset of $G$. It's a **proper subgroup** if $H \neq G$.
To specify a group $G$, I needed to tell you both what the set $G$ was and the operation $\star$ was. But to specify a subgroup $H$ of a given group $G$, I only need to tell you who its elements are: the operation of $H$ is just inherited from the operation of $G$.
(a) $2\mathbb{Z}$ is a subgroup of $\mathbb{Z}$, which is isomorphic to $\mathbb{Z}$ itself!
(b) Consider again $S_n$, the symmetric group on $n$ elements. Let $T$ be the set of permutations $\tau \colon \{1, \dots, n\} \to \{1, \dots, n\}$ for which $\tau(n) = n$. Then $T$ is a subgroup of $S_n$; in fact, it is isomorphic to $S_{n-1}$.
(c) Consider the group $G \times H$ () and the elements $\left\{ (g, 1_H) \mid g \in G \right\}$. This is a subgroup of $G \times H$ (why?). In fact, it is isomorphic to $G$ by the isomorphism $(g,1_H) \mapsto g$.
For any group $G$, the trivial group $\{1_G\}$ and the entire group $G$ are subgroups of $G$.
Next is an especially important example that we'll talk about more in later chapters.
Let $x$ be an element of a group $G$. Consider the set $$\left<x\right> = \left\{ \dots, x^{-2}, x^{-1}, 1, x, x^2, \dots \right\}.$$ This is also a subgroup of $G$, called the subgroup generated by $x$.
If $\ord x = 2015$, what is the above subgroup equal to? What if $\ord x = \infty$?
Finally, we present some non-examples of subgroups.
Consider the group $\mathbb{Z} = (\mathbb{Z}, +)$.
(a) The set $\left\{ 0,1,2,\dots \right\}$ is not a subgroup of $\mathbb{Z}$ because it does not contain inverses.
(b) The set $\{ n^3 \mid n \in \mathbb{Z} \}
= \{ \dots, -8, -1, 0, 1, 8, \dots \}$ is not a subgroup because it is not closed under addition; the sum of two cubes is not in general a cube.
(c) The empty set $\varnothing$ is not a subgroup of $\mathbb{Z}$ because it lacks an identity element. | 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 $4$ are
- $\mathbb{Z}/4\mathbb{Z}$, the cyclic group on four elements,
- $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, called the Klein Four Group.
5. The only group of order $5$ is $\mathbb{Z}/5\mathbb{Z}$.
6. The groups of order six are
- $\mathbb{Z}/6\mathbb{Z}$, the cyclic group on six elements.
- $S_3$, the permutation group of three elements. This is the first non-abelian group.
Some of you might wonder where $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is. All I have to say is: Chinese remainder theorem!
You might wonder where $D_6$ is in this list. It's actually isomorphic to $S_3$.
7. The only group of order $7$ is $\mathbb{Z}/7\mathbb{Z}$.
8. The groups of order eight are more numerous.
- $\mathbb{Z}/8\mathbb{Z}$, the cyclic group on eight elements.
- $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
- $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
- $D_8$, the dihedral group with eight elements, which is not abelian.
- A non-abelian group $Q_8$, called the *quaternion group*. It consists of eight elements $\pm 1$, $\pm i$, $\pm j$, $\pm k$ with $i^2=j^2=k^2=ijk=-1$.
9. The groups of order nine are
- $\mathbb{Z}/9\mathbb{Z}$, the cyclic group on nine elements.
- $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.
10. The groups of order $10$ are
- $\mathbb{Z}/10\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ (again Chinese remainder theorem).
- $D_{10}$, the dihedral group with $10$ elements. This group is non-abelian.
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 $4$ are
- $\mathbb{Z}/4\mathbb{Z}$, the cyclic group on four elements,
- $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, called the Klein Four Group.
5. The only group of order $5$ is $\mathbb{Z}/5\mathbb{Z}$.
6. The groups of order six are
- $\mathbb{Z}/6\mathbb{Z}$, the cyclic group on six elements.
- $S_3$, the permutation group of three elements. This is the first non-abelian group.
Some of you might wonder where $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is. All I have to say is: Chinese remainder theorem!
You might wonder where $D_6$ is in this list. It's actually isomorphic to $S_3$.
7. The only group of order $7$ is $\mathbb{Z}/7\mathbb{Z}$.
8. The groups of order eight are more numerous.
- $\mathbb{Z}/8\mathbb{Z}$, the cyclic group on eight elements.
- $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
- $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
- $D_8$, the dihedral group with eight elements, which is not abelian.
- A non-abelian group $Q_8$, called the *quaternion group*. It consists of eight elements $\pm 1$, $\pm i$, $\pm j$, $\pm k$ with $i^2=j^2=k^2=ijk=-1$.
9. The groups of order nine are
- $\mathbb{Z}/9\mathbb{Z}$, the cyclic group on nine elements.
- $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.
10. The groups of order $10$ are
- $\mathbb{Z}/10\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ (again Chinese remainder theorem).
- $D_{10}$, the dihedral group with $10$ elements. This group is non-abelian. | 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, you want to include enough constraints that your objects are "nice". On the other hand, if you include too many constraints, then your definition applies to too few objects.
So, for example, we include "associative" because that makes our lives easier and most operations we run into are associative. In particular, associativity is required for the inverse of an element to necessarily be unique. However we don't include "commutative", because examples below show that there are lots of non-abelian groups we care about. (But we introduce another name "abelian" because we still want to keep track of it.)
Another comment: a good motivation for the inverse axioms is that you get a large amount of *symmetry*. The set of positive integers with addition is not a group, for example, because you can't subtract $6$ from $3$: some elements are "larger" than others. By requiring an inverse element to exist, you get rid of this issue. (You also need identity for this; it's hard to define inverses without it.)
Even more abstruse comment: shows that groups are actually shadows of symmetric groups. This makes rigorous the notion that "groups are very symmetric".
What is the joke in the following figure? (Source: [@img:snsd].)
{height="8cm"}
Orders.
The point is that $\heartsuit$ is a group, $G \subsetneq \heartsuit$ a subgroup and $G \cong \heartsuit$. This can only occur if $\left\lvert \heartsuit \right\rvert = \infty$; otherwise, a proper subgroup would have strictly smaller size than the original.
Prove Lagrange's theorem for orders in the special case that $G$ is a finite abelian group.
Copy the proof of Fermat's little theorem, using .
Let $\{g_1, g_2, \dots, g_n\}$ denote the elements of $G$. For any $g \in G$, this is the same as the set $\{gg_1, \dots, gg_n\}$. Taking the entire product and exploiting commutativity gives $g^n \cdot g_1g_2 \dots g_n = g_1g_2 \dots g_n$, hence $g^n=1$.
Show that $D_6 \cong S_3$ but $D_{24} \not\cong S_4$.
For the former, decide where the isomorphism should send $r$ and $s$, and the rest will follow through. For the latter, look at orders.
One can check manually that $D_6 \cong S_3$, using the map $r \mapsto (1 \; 2 \; 3)$ and $s \mapsto (1 \; 2)$. (The right-hand sides are in "cycle notation", as mentioned in .) On the other hand $D_{24}$ contains an element of order $12$ while $S_4$ does not.
Let $p$ be a prime. Show that if $G$ is a group of order $p$ then $G \cong \mathbb{Z}/p\mathbb{Z}$.
Generated groups.
Let $G$ be a group of order $p$, and $1 \neq g \in G$. Look at the group $H$ generated by $g$ and use Lagrange's theorem.
Find a subgroup $H$ of $S_8$ which is isomorphic to $D_8$, and write the isomorphism explicitly.
Let $G$ be a finite group.[^1] Show that there exists a positive integer $n$ such that
(a) (Cayley's theorem) $G$ is isomorphic to some subgroup of the symmetric group $S_n$.
(b) (Representation Theory) $G$ is isomorphic to some subgroup of the general linear group $\GL_n(\mathbb{R})$. (This is the group of invertible $n \times n$ matrices.)
Use $n = \left\lvert G \right\rvert$.
The idea is that each element $g \in G$ can be thought of as a permutation $G \to G$ by $x \mapsto gx$.
Find the smallest integer $n$ such that the symmetric group $S_n$ has a subgroup isomorphic to the dihedral group $D_{2018}$ of order $2018$.
For the lower bound, consider orders (note that $1009$ is prime). For the upper bound, consider a $1009$-gon.
The answer is $n = 1009$. This solution uses the fact that $1009$ is prime.
To show that no smaller $m$ is possible, note that $D_{2018}$ has elements of order $1009$, a prime. Since $S_n$ has no elements of this order for $n < 1009$, we need $n \ge 1009$.
To give a construction from $n = 1009$, note that $D_{2018}$ can be thought of the symmetries of a $1009$-gon. If one labels the vertices of the $1009$-gon by $S \coloneqq \{1,2,\dots,1009\}$, then elements of $D_{2018}$ induces permutations on $S$, and the set of permutations achieved is the desired subgroup.
There are $n$ markers, each with one side white and the other side black. In the beginning, these $n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it.
Prove that if $n \equiv 1 \pmod 3$ it's impossible to reach a state with only two markers remaining. (In fact the converse is true as well.)
Draw inspiration from $D_6$.
We have $www = bb$, $bww = wb$, $wwb = bw$, $bwb = ww$. Interpret these as elements of $D_6$.
Let $p$ be a prime and $F_1 = F_2 = 1$, $F_{n+2} = F_{n+1} + F_n$ be the Fibonacci sequence. Show that $F_{2p(p^2-1)}$ is divisible by $p$.
Look at the group of $2 \times 2$ matrices mod $p$ with determinant $\pm 1$. | 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 highly unsettled by this theorem when I first heard of it, but in hindsight it is not so surprising. | 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, you want to include enough constraints that your objects are "nice". On the other hand, if you include too many constraints, then your definition applies to too few objects.
So, for example, we include "associative" because that makes our lives easier and most operations we run into are associative. In particular, associativity is required for the inverse of an element to necessarily be unique. However we don't include "commutative", because examples below show that there are lots of non-abelian groups we care about. (But we introduce another name "abelian" because we still want to keep track of it.)
Another comment: a good motivation for the inverse axioms is that you get a large amount of *symmetry*. The set of positive integers with addition is not a group, for example, because you can't subtract $6$ from $3$: some elements are "larger" than others. By requiring an inverse element to exist, you get rid of this issue. (You also need identity for this; it's hard to define inverses without it.)
Even more abstruse comment: shows that groups are actually shadows of symmetric groups. This makes rigorous the notion that "groups are very symmetric".
What is the joke in the following figure? (Source: [@img:snsd].)
{height="8cm"}
Orders.
The point is that $\heartsuit$ is a group, $G \subsetneq \heartsuit$ a subgroup and $G \cong \heartsuit$. This can only occur if $\left\lvert \heartsuit \right\rvert = \infty$; otherwise, a proper subgroup would have strictly smaller size than the original.
Prove Lagrange's theorem for orders in the special case that $G$ is a finite abelian group.
Copy the proof of Fermat's little theorem, using .
Let $\{g_1, g_2, \dots, g_n\}$ denote the elements of $G$. For any $g \in G$, this is the same as the set $\{gg_1, \dots, gg_n\}$. Taking the entire product and exploiting commutativity gives $g^n \cdot g_1g_2 \dots g_n = g_1g_2 \dots g_n$, hence $g^n=1$.
Show that $D_6 \cong S_3$ but $D_{24} \not\cong S_4$.
For the former, decide where the isomorphism should send $r$ and $s$, and the rest will follow through. For the latter, look at orders.
One can check manually that $D_6 \cong S_3$, using the map $r \mapsto (1 \; 2 \; 3)$ and $s \mapsto (1 \; 2)$. (The right-hand sides are in "cycle notation", as mentioned in .) On the other hand $D_{24}$ contains an element of order $12$ while $S_4$ does not.
Let $p$ be a prime. Show that if $G$ is a group of order $p$ then $G \cong \mathbb{Z}/p\mathbb{Z}$.
Generated groups.
Let $G$ be a group of order $p$, and $1 \neq g \in G$. Look at the group $H$ generated by $g$ and use Lagrange's theorem.
Find a subgroup $H$ of $S_8$ which is isomorphic to $D_8$, and write the isomorphism explicitly.
Let $G$ be a finite group.[^1] Show that there exists a positive integer $n$ such that
(a) (Cayley's theorem) $G$ is isomorphic to some subgroup of the symmetric group $S_n$.
(b) (Representation Theory) $G$ is isomorphic to some subgroup of the general linear group $\GL_n(\mathbb{R})$. (This is the group of invertible $n \times n$ matrices.)
Use $n = \left\lvert G \right\rvert$.
The idea is that each element $g \in G$ can be thought of as a permutation $G \to G$ by $x \mapsto gx$.
Find the smallest integer $n$ such that the symmetric group $S_n$ has a subgroup isomorphic to the dihedral group $D_{2018}$ of order $2018$.
For the lower bound, consider orders (note that $1009$ is prime). For the upper bound, consider a $1009$-gon.
The answer is $n = 1009$. This solution uses the fact that $1009$ is prime.
To show that no smaller $m$ is possible, note that $D_{2018}$ has elements of order $1009$, a prime. Since $S_n$ has no elements of this order for $n < 1009$, we need $n \ge 1009$.
To give a construction from $n = 1009$, note that $D_{2018}$ can be thought of the symmetries of a $1009$-gon. If one labels the vertices of the $1009$-gon by $S \coloneqq \{1,2,\dots,1009\}$, then elements of $D_{2018}$ induces permutations on $S$, and the set of permutations achieved is the desired subgroup.
There are $n$ markers, each with one side white and the other side black. In the beginning, these $n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it.
Prove that if $n \equiv 1 \pmod 3$ it's impossible to reach a state with only two markers remaining. (In fact the converse is true as well.)
Draw inspiration from $D_6$.
We have $www = bb$, $bww = wb$, $wwb = bw$, $bwb = ww$. Interpret these as elements of $D_6$.
Let $p$ be a prime and $F_1 = F_2 = 1$, $F_{n+2} = F_{n+1} + F_n$ be the Fibonacci sequence. Show that $F_{2p(p^2-1)}$ is divisible by $p$.
Look at the group of $2 \times 2$ matrices mod $p$ with determinant $\pm 1$. | 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 highly unsettled by this theorem when I first heard of it, but in hindsight it is not so surprising. | 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 should draw lots of pictures as you learn about new objects, like convergent sequences, open sets, closed sets, and so on.
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 should draw lots of pictures as you learn about new objects, like convergent sequences, open sets, closed sets, and so on.
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 $d(x,y) \ge 0$ with equality if and only if $x=y$.
- The function $d$ should satisfy the **triangle inequality**: for all $x,y,z \in M$, $$d(x,z) + d(z,y) \ge d(x,y).$$
Just like with groups, we will abbreviate $(M,d)$ as just $M$.
(a) The real line $\mathbb{R}$ is a metric space under the metric $d(x,y) = \left\lvert x-y \right\rvert$.
(b) The interval $[0,1]$ is also a metric space with the same distance function.
(c) In fact, any subset $S$ of $\mathbb{R}$ can be made into a metric space in this way.
(a) We can make $\mathbb{R}^2$ into a metric space by imposing the Euclidean distance function $$d\left( (x_1, y_1), (x_2, y_2) \right) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$$
(b) Just like with the first example, any subset of $\mathbb{R}^2$ also becomes a metric space after we inherit it. The unit disk, unit circle, and the unit square $[0,1]^2$ are special cases.
It is also possible to place the **taxicab distance** on $\mathbb{R}^2$: $$d\left( (x_1, y_1), (x_2, y_2) \right) =
\left\lvert x_1-x_2 \right\rvert + \left\lvert y_1-y_2 \right\rvert.$$ For now, we will use the more natural Euclidean metric.
We can generalize the above examples easily. Let $n$ be a positive integer.
(a) We let $\mathbb{R}^n$ be the metric space whose points are points in $n$-dimensional Euclidean space, and whose metric is the Euclidean metric $$d\left(
\left( a_1, \dots, a_n \right), \left( b_1, \dots, b_n \right)
\right)
= \sqrt{(a_1-b_1)^2 + \dots + (a_n-b_n)^2}.$$ This is the $n$-dimensional **Euclidean space**.
(b) The open **unit ball** $B^{n}$ is the subset of $\mathbb{R}^n$ consisting of those points $\left( x_1, \dots, x_n \right)$ such that $x_1^2 + \dots + x_n^2 < 1$.
(c) The **unit sphere** $S^{n-1}$ is the subset of $\mathbb{R}^n$ consisting of those points $\left( x_1, \dots, x_n \right)$ such that $x_1^2 + \dots + x_n^2 = 1$, with the inherited metric. (The superscript $n-1$ indicates that $S^{n-1}$ is an $n-1$ dimensional space, even though it lives in $n$-dimensional space.) For example, $S^1 \subseteq \mathbb{R}^2$ is the unit circle, whose distance between two points is the length of the chord joining them. You can also think of it as the "boundary" of the unit ball $B^n$.
We can let $M$ be the space of continuous functions $f \colon [0,1] \to \mathbb{R}$ and define the metric by $d(f,g) = \int_0^1 \left\lvert f-g \right\rvert \; dx$. (It admittedly takes some work to check $d(f,g) = 0$ implies $f=g$, but we won't worry about that yet.)
Here is a slightly more pathological example.
Let $S$ be any set of points (either finite or infinite). We can make $S$ into a **discrete space** by declaring $$d(x,y)
=
\begin{cases}
1 & \text{if $x \neq y$} \\
0 & \text{if $x = y$}.
\end{cases}$$ If $\left\lvert S \right\rvert = 4$ you might think of this space as the vertices of a regular tetrahedron, living in $\mathbb{R}^3$. But for larger $S$ it's not so easy to visualize...
Any connected simple graph $G$ can be made into a metric space by defining the distance between two vertices to be the graph-theoretic distance between them. (The discrete metric is the special case when $G$ is the complete graph on $S$.)
Check the conditions of a metric space for the metrics on the discrete space and for the connected graph.
From now on, we will refer to $\mathbb{R}^n$ with the Euclidean metric by just $\mathbb{R}^n$. Moreover, if we wish to take the metric space for a subset $S \subseteq \mathbb{R}^n$ with the inherited metric, we will just write $S$. | 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 $d(x,y) \ge 0$ with equality if and only if $x=y$.
- The function $d$ should satisfy the **triangle inequality**: for all $x,y,z \in M$, $$d(x,z) + d(z,y) \ge d(x,y).$$
Just like with groups, we will abbreviate $(M,d)$ as just $M$.
(a) The real line $\mathbb{R}$ is a metric space under the metric $d(x,y) = \left\lvert x-y \right\rvert$.
(b) The interval $[0,1]$ is also a metric space with the same distance function.
(c) In fact, any subset $S$ of $\mathbb{R}$ can be made into a metric space in this way.
(a) We can make $\mathbb{R}^2$ into a metric space by imposing the Euclidean distance function $$d\left( (x_1, y_1), (x_2, y_2) \right) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$$
(b) Just like with the first example, any subset of $\mathbb{R}^2$ also becomes a metric space after we inherit it. The unit disk, unit circle, and the unit square $[0,1]^2$ are special cases.
It is also possible to place the **taxicab distance** on $\mathbb{R}^2$: $$d\left( (x_1, y_1), (x_2, y_2) \right) =
\left\lvert x_1-x_2 \right\rvert + \left\lvert y_1-y_2 \right\rvert.$$ For now, we will use the more natural Euclidean metric.
We can generalize the above examples easily. Let $n$ be a positive integer.
(a) We let $\mathbb{R}^n$ be the metric space whose points are points in $n$-dimensional Euclidean space, and whose metric is the Euclidean metric $$d\left(
\left( a_1, \dots, a_n \right), \left( b_1, \dots, b_n \right)
\right)
= \sqrt{(a_1-b_1)^2 + \dots + (a_n-b_n)^2}.$$ This is the $n$-dimensional **Euclidean space**.
(b) The open **unit ball** $B^{n}$ is the subset of $\mathbb{R}^n$ consisting of those points $\left( x_1, \dots, x_n \right)$ such that $x_1^2 + \dots + x_n^2 < 1$.
(c) The **unit sphere** $S^{n-1}$ is the subset of $\mathbb{R}^n$ consisting of those points $\left( x_1, \dots, x_n \right)$ such that $x_1^2 + \dots + x_n^2 = 1$, with the inherited metric. (The superscript $n-1$ indicates that $S^{n-1}$ is an $n-1$ dimensional space, even though it lives in $n$-dimensional space.) For example, $S^1 \subseteq \mathbb{R}^2$ is the unit circle, whose distance between two points is the length of the chord joining them. You can also think of it as the "boundary" of the unit ball $B^n$.
We can let $M$ be the space of continuous functions $f \colon [0,1] \to \mathbb{R}$ and define the metric by $d(f,g) = \int_0^1 \left\lvert f-g \right\rvert \; dx$. (It admittedly takes some work to check $d(f,g) = 0$ implies $f=g$, but we won't worry about that yet.)
Here is a slightly more pathological example.
Let $S$ be any set of points (either finite or infinite). We can make $S$ into a **discrete space** by declaring $$d(x,y)
=
\begin{cases}
1 & \text{if $x \neq y$} \\
0 & \text{if $x = y$}.
\end{cases}$$ If $\left\lvert S \right\rvert = 4$ you might think of this space as the vertices of a regular tetrahedron, living in $\mathbb{R}^3$. But for larger $S$ it's not so easy to visualize...
Any connected simple graph $G$ can be made into a metric space by defining the distance between two vertices to be the graph-theoretic distance between them. (The discrete metric is the special case when $G$ is the complete graph on $S$.)
Check the conditions of a metric space for the metrics on the discrete space and for the connected graph.
From now on, we will refer to $\mathbb{R}^n$ with the Euclidean metric by just $\mathbb{R}^n$. Moreover, if we wish to take the metric space for a subset $S \subseteq \mathbb{R}^n$ with the inherited metric, we will just write $S$. | 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 that $x_n$ **converges** to $x$ if the following condition holds: for all $\varepsilon > 0$, there is an integer $N$ (depending on $\varepsilon$) such that $d(x_n, x) < \varepsilon$ for each $n \ge N$. This is written $$x_n \to x$$ or more verbosely as $$\lim_{n \to \infty} x_n = x.$$ We say that a sequence converges in $M$ if it converges to a point in $M$.
You should check that this definition coincides with your intuitive notion of "converges".
If the parent space $M$ is understood, we will allow ourselves to abbreviate "converges in $M$" to just "converges". However, keep in mind that convergence is defined relative to the parent space; the "limit" of the space must actually be a point in $M$ for a sequence to converge.
size(9cm); Drawing(\"x_1\", (-9,0.1), dir(90)); Drawing(\"x_2\", (-6,0.8), dir(90)); Drawing(\"x_3\", (-5,-0.3), dir(90)); Drawing(\"x_4\", (-2, 0.8), dir(90)); Drawing(\"x_5\", (-1.7, -0.7), dir(-90)); Drawing(\"x_6\", (-0.6, -0.3), dir(225)); Drawing(\"x_7\", (-0.4, 0.3), dir(90)); Drawing(\"x_8\", (-0.25, -0.24), dir(-90)); Drawing(\"x_9\", (-0.12, 0.1), dir(45)); dot(\"$x$\", (0,0), dir(-45), blue); draw(CR(origin, 1.5), blue+dashed);
Consider the sequence $x_1 = 1$, $x_2 = 1.4$, $x_3 = 1.41$, $x_4 = 1.414$, ....
(a) If we view this as a sequence in $\mathbb{R}$, it converges to $\sqrt 2$.
(b) However, even though each $x_i$ is in $\mathbb{Q}$, this sequence does NOT converge when we view it as a sequence in $\mathbb{Q}$!
What are the convergent sequences in a discrete metric space?
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 that $x_n$ **converges** to $x$ if the following condition holds: for all $\varepsilon > 0$, there is an integer $N$ (depending on $\varepsilon$) such that $d(x_n, x) < \varepsilon$ for each $n \ge N$. This is written $$x_n \to x$$ or more verbosely as $$\lim_{n \to \infty} x_n = x.$$ We say that a sequence converges in $M$ if it converges to a point in $M$.
You should check that this definition coincides with your intuitive notion of "converges".
If the parent space $M$ is understood, we will allow ourselves to abbreviate "converges in $M$" to just "converges". However, keep in mind that convergence is defined relative to the parent space; the "limit" of the space must actually be a point in $M$ for a sequence to converge.
size(9cm); Drawing(\"x_1\", (-9,0.1), dir(90)); Drawing(\"x_2\", (-6,0.8), dir(90)); Drawing(\"x_3\", (-5,-0.3), dir(90)); Drawing(\"x_4\", (-2, 0.8), dir(90)); Drawing(\"x_5\", (-1.7, -0.7), dir(-90)); Drawing(\"x_6\", (-0.6, -0.3), dir(225)); Drawing(\"x_7\", (-0.4, 0.3), dir(90)); Drawing(\"x_8\", (-0.25, -0.24), dir(-90)); Drawing(\"x_9\", (-0.12, 0.1), dir(45)); dot(\"$x$\", (0,0), dir(-45), blue); draw(CR(origin, 1.5), blue+dashed);
Consider the sequence $x_1 = 1$, $x_2 = 1.4$, $x_3 = 1.41$, $x_4 = 1.414$, ....
(a) If we view this as a sequence in $\mathbb{R}$, it converges to $\sqrt 2$.
(b) However, even though each $x_i$ is in $\mathbb{Q}$, this sequence does NOT converge when we view it as a sequence in $\mathbb{Q}$!
What are the convergent sequences in a discrete metric space? | 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 $\left\lvert x-p \right\rvert < \delta
> \implies
> \left\lvert f(x) - f(p) \right\rvert < \varepsilon$.
Can you guess what the corresponding definition for metric spaces is?
All we have to do is replace the absolute values with the more general distance functions: this gives us a definition of continuity for any function $M \to N$.
Let $M = (M, d_M)$ and $N = (N, d_N)$ be metric spaces. A function $f \colon M \to N$ is **continuous** at a point $p \in M$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $$d_M(x,p) < \delta \implies d_N(f(x), f(p)) < \varepsilon.$$ Moreover, the entire function $f$ is continuous if it is continuous at every point $p \in M$.
Notice that, just like in our definition of an isomorphism of a group, we use the metric of $M$ for one condition and the metric of $N$ for the other condition.
This generalization is nice because it tells us immediately how we could carry over continuity arguments in $\mathbb{R}$ to more general spaces like $\mathbb{C}$. Nonetheless, this definition is kind of cumbersome to work with, because it makes extensive use of the real numbers (epsilons and deltas). Here is an equivalent condition.
A function $f \colon M \to N$ of metric spaces is continuous at a point $p \in M$ if and only if the following property holds: if $x_1$, $x_2$, ... is a sequence in $M$ converging to $p$, then the sequence $f(x_1)$, $f(x_2)$, ... in $N$ converges to $f(p)$.
*Proof.* One direction is not too hard:
Show that $\varepsilon$-$\delta$ continuity implies sequential continuity at each point.
Conversely, we will prove if $f$ is not $\varepsilon$-$\delta$ continuous at $p$ then it does not preserve convergence.
If $f$ is not continuous at $p$, then there is a "bad" $\varepsilon > 0$, which we now consider fixed. So for each choice of $\delta = 1/n$, there should be some point $x_n$ which is within $\delta$ of $p$, but which is mapped more than $\varepsilon$ away from $f(p)$. But then the sequence $x_n$ converges to $p$, and $f(x_n)$ is always at least $\varepsilon$ away from $f(p)$, contradiction. ◻
Example application showcasing the niceness of sequential continuity:
Let $f \colon M \to N$ and $g \colon N \to L$ be continuous maps of metric spaces. Then their composition $g \circ f$ is continuous.
*Proof.* Dead simple with sequences: Let $p \in M$ be arbitrary and let $x_n \to p$ in $M$. Then $f(x_n) \to f(p)$ in $N$ and $g(f(x_n)) \to g(f(p))$ in $L$, QED. ◻
Let $M$ be any metric space and $D$ a discrete space. When is a map $f \colon D \to M$ continuous? | 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 **homeomorphism** if it is a bijection, and both $f \colon M \to N$ and its inverse $f^{-1} \colon N \to M$ are continuous. We say $M$ and $N$ are **homeomorphic**.
Needless to say, homeomorphism is an equivalence relation.
You might be surprised that we require $f^{-1}$ to also be continuous. Here's the reason: you can show that if $\phi$ is an isomorphism of groups, then $\phi^{-1}$ also preserves the group operation, hence $\phi^{-1}$ is itself an isomorphism. The same is not true for continuous bijections, which is why we need the new condition.
(a) There is a continuous bijection from $[0,1)$ to the circle, but it has no continuous inverse.
(b) Let $M$ be a discrete space with size $|\mathbb{R}|$. Then there is a continuous function $M \to \mathbb{R}$ which certainly has no continuous inverse.
Note that this is the topologist's definition of "same" -- homeomorphisms are "continuous deformations". Here are some examples.
(a) Any space $M$ is homeomorphic to itself through the identity map.
(b) The old saying: a doughnut (torus) is homeomorphic to a coffee cup. (Look this up if you haven't heard of it.)
(c) The unit circle $S^1$ is homeomorphic to the boundary of the unit square. Here's one bijection between them, after an appropriate scaling:
:::: center
::: asy
size(1.5cm); draw(unitcircle); pair A = (1.4, 1.4); pair B = rotate(90)\*A; pair C = rotate(90)\*B; pair D = rotate(90)\*C; draw(A--B--C--D--cycle); dot(origin); pair P = Drawing(dir(70)); pair Q = Drawing(extension(origin, P, A, B)); draw(origin--Q, dashed);
:::
::::
It may have seemed strange that our metric function on $S^1$ was the one inherited from $\mathbb{R}^2$, meaning the distance between two points on the circle was defined to be the length of the chord. Wouldn't it have made more sense to use the circumference of the smaller arc joining the two points?
In fact, it doesn't matter: if we consider $S^1$ with the "chord" metric and the "arc" metric, we get two homeomorphic spaces, as the map between them is continuous.
The same goes for $S^{n-1}$ for general $n$.
Surprisingly, the open interval $(-1,1)$ is homeomorphic to the real line $\mathbb{R}$! One bijection is given by $$x \mapsto \tan(x \pi/2)$$ with the inverse being given by $t \mapsto \frac2\pi \arctan(t)$.
This might come as a surprise, since $(-1,1)$ doesn't look that much like $\mathbb{R}$; the former is "bounded" while the latter is "unbounded". | 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 **homeomorphism** if it is a bijection, and both $f \colon M \to N$ and its inverse $f^{-1} \colon N \to M$ are continuous. We say $M$ and $N$ are **homeomorphic**.
Needless to say, homeomorphism is an equivalence relation.
You might be surprised that we require $f^{-1}$ to also be continuous. Here's the reason: you can show that if $\phi$ is an isomorphism of groups, then $\phi^{-1}$ also preserves the group operation, hence $\phi^{-1}$ is itself an isomorphism. The same is not true for continuous bijections, which is why we need the new condition.
(a) There is a continuous bijection from $[0,1)$ to the circle, but it has no continuous inverse.
(b) Let $M$ be a discrete space with size $|\mathbb{R}|$. Then there is a continuous function $M \to \mathbb{R}$ which certainly has no continuous inverse.
Note that this is the topologist's definition of "same" -- homeomorphisms are "continuous deformations". Here are some examples.
(a) Any space $M$ is homeomorphic to itself through the identity map.
(b) The old saying: a doughnut (torus) is homeomorphic to a coffee cup. (Look this up if you haven't heard of it.)
(c) The unit circle $S^1$ is homeomorphic to the boundary of the unit square. Here's one bijection between them, after an appropriate scaling:
:::: center
::: asy
size(1.5cm); draw(unitcircle); pair A = (1.4, 1.4); pair B = rotate(90)\*A; pair C = rotate(90)\*B; pair D = rotate(90)\*C; draw(A--B--C--D--cycle); dot(origin); pair P = Drawing(dir(70)); pair Q = Drawing(extension(origin, P, A, B)); draw(origin--Q, dashed);
:::
::::
It may have seemed strange that our metric function on $S^1$ was the one inherited from $\mathbb{R}^2$, meaning the distance between two points on the circle was defined to be the length of the chord. Wouldn't it have made more sense to use the circumference of the smaller arc joining the two points?
In fact, it doesn't matter: if we consider $S^1$ with the "chord" metric and the "arc" metric, we get two homeomorphic spaces, as the map between them is continuous.
The same goes for $S^{n-1}$ for general $n$.
Surprisingly, the open interval $(-1,1)$ is homeomorphic to the real line $\mathbb{R}$! One bijection is given by $$x \mapsto \tan(x \pi/2)$$ with the inverse being given by $t \mapsto \frac2\pi \arctan(t)$.
This might come as a surprise, since $(-1,1)$ doesn't look that much like $\mathbb{R}$; the former is "bounded" while the latter is "unbounded". | 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 $M \times N$: $$\begin{align*}
d_{\text{max}} ( p_1, p_2 )
&:= \max \left\{ d_M(x_1, x_2), d_N(y_1, y_2) \right\} \\
d_{\text{Euclid}} ( p_1, p_2 )
&:= \sqrt{d_M(x_1,x_2)^2 + d_N(y_1, y_2)^2} \\
d_{\text{taxicab}} \left( p_1, p_2 \right)
&:= d_M(x_1, x_2) + d_N(y_1, y_2).
\end{align*}$$ All of these are good candidates. We are about to see it doesn't matter which one we use:
Verify that $$d_{\text{max}}(p_1,p_2)
\le d_{\text{Euclid}}(p_1, p_2)
\le d_{\text{taxicab}}(p_1, p_2)
\le 2d_{\text{max}}(p_1, p_2).$$ Use this to show that the metric spaces we obtain by imposing any of the three metrics are homeomorphic, with the homeomorphism being just the identity map.
Hence we will usually simply refer to *the* metric on $M \times N$, called the **product metric**. It will not be important which of the three metrics we select.
If $M = N = \mathbb{R}$, we get $\mathbb{R}^2$, the Euclidean plane. The metric $d_{\text{Euclid}}$ is the one we started with, but using either of the other two metric works fine as well.
The product metric plays well with convergence of sequences.
We have $(x_n, y_n) \to (x,y)$ if and only if $x_n \to x$ and $y_n \to y$.
*Proof.* We have $d_{\text{max}} \left( (x,y), (x_n, y_n) \right)
= \max\left\{ d_M(x, x_n), d_N(y, y_n) \right\}$ and the latter approaches zero as $n \to \infty$ if and only if $d_M(x,x_n) \to 0$ and $d_N(y, y_n) \to 0$. ◻
Let's see an application of this:
The addition and multiplication maps are continuous maps $\mathbb{R} \times \mathbb{R} \to \mathbb{R}$.
*Proof.* For multiplication: for any $n$ we have $$\begin{align*}
x_n y_n &= \left( x + (x_n-x) \right)
\left( y + (y_n-y) \right) \\
&= xy + y(x_n-x) + x(y_n-y) + (x_n-x)(y_n-y) \\
\implies \left\lvert x_n y_n - xy \right\rvert
&\le \left\lvert y \right\rvert
\left\lvert x_n - x \right\rvert
+ \left\lvert x \right\rvert
\left\lvert y_n - y \right\rvert
+ \left\lvert x_n - x \right\rvert
\left\lvert y_n - y \right\rvert.
\end{align*}$$ As $n \to \infty$, all three terms on the right-hand side tend to zero. The proof that $+ \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is continuous is similar (and easier): one notes for any $n$ that $$|(x_n + y_n) - (x+y)| \le |x_n-x| + |y_n-y|$$ and both terms on the right-hand side tend to zero as $n \to \infty$. ◻
covers the other two operations, subtraction and division. The upshot of this is that, since compositions are also continuous, most of your naturally arising real-valued functions will automatically be continuous as well. For example, the function $\frac{3x}{x^2+1}$ will be a continuous function from $\mathbb{R} \to \mathbb{R}$, since it can be obtained by composing $+$, $\times$, $\div$. | 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 $p \in M$, we define $$M_r(p) := \left\{ x \in M: d(x,p) < r \right\}.$$ The set $M_r(p)$ is called an **$r$-neighborhood** of $p$.
size(4cm); bigblob(\"$M$\"); pair p = Drawing(\"p\", (0.3,0.1), dir(-90)); real r = 1.8; draw(CR(p,r), dashed); label(\"$M_r(p)$\", p+r\*dir(-65), dir(-65)); draw(p--(p+r\*dir(130))); label(\"$r$\", midpoint(p--(p+r\*dir(130))), dir(40));
We can rephrase convergence more succinctly in terms of $r$-neighborhoods. Specifically, a sequence $(x_n)$ converges to $x$ if for every $r$-neighborhood of $x$, all terms of $x_n$ eventually stay within that $r$-neighborhood.
Let's try to do the same with functions.
In terms of $r$-neighborhoods, what does it mean for a function $f \colon M \to N$ to be continuous at a point $p \in M$?
Essentially, we require that the pre-image of every $\varepsilon$-neighborhood has the property that some $\delta$-neighborhood exists inside it. This motivates:
A set $U \subseteq M$ is **open** in $M$ if for each $p \in U$, some $r$-neighborhood of $p$ is contained inside $U$. In other words, there exists $r>0$ such that $M_r(p) \subseteq U$.
Note that a set being open is defined *relative to* the parent space $M$. However, if $M$ is understood we can abbreviate "open in $M$" to just "open".
<figure id="fig:example_open" data-latex-placement="ht">
<div class="asy">
<p>size(5cm); draw(unitcircle, dashed); pair P = Drawing("p", (0.6,0.2), dir(-90)); draw(CR(P, 0.3), dotted); MP("x^2+y^2<1", dir(45), dir(45));</p>
</div>
<figcaption>The set of points <span class="math inline"><em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> < 1</span> in <span class="math inline">$\mathbb{R}^2$</span> is open in <span class="math inline">$\mathbb{R}^2$</span>.</figcaption>
</figure>
(a) Any $r$-neighborhood of a point is open.
(b) Open intervals of $\mathbb{R}$ are open in $\mathbb{R}$, hence the name! This is the prototypical example to keep in mind.
(c) The open unit ball $B^n$ is open in $\mathbb{R}^n$ for the same reason.
(d) In particular, the open interval $(0,1)$ is open in $\mathbb{R}$. However, if we embed it in $\mathbb{R}^2$, it is no longer open!
(e) The empty set $\varnothing$ and the whole set of points $M$ are open in $M$.
(a) The closed interval $[0,1]$ is not open in $\mathbb{R}$. There is no $\varepsilon$-neighborhood of the point $0$ which is contained in $[0,1]$.
(b) The unit circle $S^1$ is not open in $\mathbb{R}^2$.
What are the open sets of the discrete space?
Here are two quite important properties of open sets.
(a) The intersection of finitely many open sets is open.
(b) The union of open sets is open, even if there are infinitely many.
Convince yourself this is true.
Exhibit an infinite collection of open sets in $\mathbb{R}$ whose intersection is the set $\{0\}$. This implies that infinite intersections of open sets are not necessarily open.
The whole upshot of this is:
A function $f \colon M \to N$ of metric spaces is continuous if and only if the pre-image of every open set in $N$ is open in $M$.
*Proof.* I'll just do one direction...
Show that $\delta$-$\varepsilon$ continuity follows from the open set condition.
Now assume $f$ is continuous. First, suppose $V$ is an open subset of the metric space $N$; let $U = f\pre(V)$. Pick $x \in U$, so $y = f(x) \in V$; we want an open neighborhood of $x$ inside $U$.
size(12cm); bigblob(\"$N$\"); pair Y = Drawing(\"y\", origin, dir(75)); real eps = 1.5; draw(CR(Y, eps), dotted); label(\"$\varepsilon$\", Drawing(Y--(Y+eps\*dir(255)))); label(\"$V$\", Drawing(shift(-0.5,0)\*rotate(190)\*scale(3.2,2.8)\*unitcircle, dashed)); add(shift( (13,0) ) \* CC()); label(\"$f$\", Drawing( (4.5,0)--(8,0), EndArrow));
bigblob(\"$M$\"); real delta = 1.1; pair X = Drawing(\"x\", (-1.5,-0.5), dir(-45)); label(\"$\delta$\", Drawing(X--(X+delta\*dir(155)))); draw(CR(X, delta), dotted); label(\"$U = f^{\text{pre}}(V)$\", Drawing(shift(-1.5,-0.3)\*rotate(235)\*scale(2.4,1.8)\*unitcircle, dashed));
As $V$ is open, there is some small $\varepsilon$-neighborhood around $y$ which is contained inside $V$. By continuity of $f$, we can find a $\delta$ such that the $\delta$-neighborhood of $x$ gets mapped by $f$ into the $\varepsilon$-neighborhood in $N$, which in particular lives inside $V$. Thus the $\delta$-neighborhood lives in $U$, as desired. ◻ | 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 suppose that $x_n$ converges to $x$ in $M$. Then $x \in S$ as well.
Same caveat: we abbreviate "closed in $M$" to just "closed" if the parent space $M$ is understood.
Here's another way to phrase it. The **limit points** of a subset $S \subseteq M$ are defined by $$\lim S := \left\{ p \in M :
\exists (x_n) \in S \text{ such that } x_n \to p \right\}.$$ Thus $S$ is closed if and only if $S = \lim S$.
Prove that $\lim S$ is closed even if $S$ isn't closed. (Draw a picture.)
For this reason, $\lim S$ is also called the **closure** of $S$ in $M$, and denoted $\overline S$. It is simply the smallest closed set which contains $S$.
(a) The empty set $\varnothing$ is closed in $M$ for vacuous reasons: there are no sequences of points with elements in $\varnothing$.
(b) The entire space $M$ is closed in $M$ for tautological reasons. (Verify this!)
(c) The closed interval $[0,1]$ in $\mathbb{R}$ is closed in $\mathbb{R}$, hence the name. Like with open sets, this is the prototypical example of a closed set to keep in mind!
(d) In fact, the closed interval $[0,1]$ is even closed in $\mathbb{R}^2$.
Let $S=(0,1)$ denote the open interval. Then $S$ is not closed in $\mathbb{R}$ because the sequence of points $$\frac12, \;
\frac14, \;
\frac18, \;
\dots$$ converges to $0 \in \mathbb{R}$, but $0 \notin (0,1)$.
I should now warn you about a confusing part of this terminology. Firstly, **"most" sets are neither open nor closed**.
The half-open interval $[0,1)$ is neither open nor closed in $\mathbb{R}$.
Secondly, it's **also possible for a set to be both open and closed**; this will be discussed in .
The reason for the opposing terms is the following theorem:
Let $M$ be a metric space, and $S \subseteq M$ any subset. Then the following are equivalent:
- The set $S$ is closed in $M$.
- The complement $M \setminus S$ is open in $M$.
Prove this theorem! You'll want to draw a picture to make it clear what's happening: for example, you might take $M = \mathbb{R}^2$ and $S$ to be the closed unit disk.
Let $M = (M,d)$ be a metric space. Show that $$d \colon M \times M \to \mathbb{R}$$ is itself a continuous function (where $M \times M$ is equipped with the product metric).
Consider $\mathbb{Q}$ and $\mathbb{N}$ as metric spaces (each with the obvious metric $d(x,y) = |x-y|$). Are these spaces homeomorphic?
No. There is not even a continuous injective map from $\mathbb{Q}$ to $\mathbb{N}$.
Two possible approaches, one using metric definition and one using open sets.
Metric approach: I claim there is no injective map from $\mathbb{Q}$ to $\mathbb{N}$ that is continuous. Indeed, suppose $f$ was such a map and $f(x) = n$. Then, choose $\varepsilon = 1/2$. There should be a $\delta > 0$ such that everything with $\delta$ of $x$ in $\mathbb{Q}$ should land within $\varepsilon$ of $n \in \mathbb{N}$ --- i.e., is equal to $n$. This is a blatant contradiction of injectivity.
Open set approach: In $\mathbb{Q}$, no singleton set is open, whereas in $\mathbb{N}$, they all are (in fact $\mathbb{N}$ is discrete). As you'll see at the start of , with the new and improved definition of "homeomorphism", we found out that the structure of open sets on $\mathbb{Q}$ and $\mathbb{N}$ are different, so they are not homeomorphic.
Show that subtraction is a continuous map $- \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, and division is a continuous map $\div \colon \mathbb{R} \times \mathbb{R}_{>0} \to \mathbb{R}$.
You can do this with bare hands. You can also use composition.
For subtraction, the map $x \mapsto -x$ is continuous so you can view it as a composed map
Similarly, if you are willing to believe $x \mapsto 1/x$ is a continuous function, then division is composition
If for some reason you are suspicious that $x \mapsto 1/x$ is continuous, then here is a proof using sequential continuity. Suppose $x_n \to x$ with $x_n > 0$ and $x > 0$ (since $x$ needs to be in $\mathbb{R}_{>0}$ too). Then $$\left\lvert \frac{1}{x} - \frac 1{x_n} \right\rvert
= \frac{\left\lvert x_n-x \right\rvert}{\left\lvert x x_n \right\rvert}.$$ If $n$ is large enough, then $\left\lvert x_n \right\rvert > x/2$; so the denominator is at least $x^2/2$, and hence the whole fraction is at most $\frac{2}{x^2} \left\lvert x_n-x \right\rvert$, which tends to zero as $n \to \infty$.
Exhibit a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f$ is continuous at $x \in \mathbb{R}$ if and only if $x=0$.
$\pm x$ for good choices of $\pm$.
Let $f(x) = x$ for $x \in \mathbb{Q}$ and $f(x) = -x$ for irrational $x$.
Prove that a function $f \colon \mathbb{R} \to \mathbb{R}$ which is strictly increasing must be continuous at some point.
Project gaps onto the $y$-axis. Use the fact that uncountably many positive reals cannot have finite sum. | 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, that can only happen if the function misses all $y$-values in the interval $(f(x)-\varepsilon_x, f(x))$ or $(f(x), f(x)+\varepsilon_x)$ (or both).
Projecting these missing intervals to the $y$-axis you find uncountably many intervals (one for each $x \in [0,1]$) all of which are disjoint. In particular, summing the $\varepsilon_x$ you get that a sum of uncountably many positive reals is $1$.
But in general it isn't possible for an uncountable family $\mathcal F$ of positive reals to have finite sum. Indeed, just classify the reals into buckets $\frac1k \le x < \frac1{k-1}$. If the sum is actually finite then each bucket is finite, so the collection $\mathcal F$ must be countable, contradiction.
Someone on the Internet posted the question "is $1/x$ a continuous function?", leading to great controversy on Twitter. How should you respond?
First answer the following question: "is $1/x$ a function?".
Like most Internet "debates" about math, the question revolves around sloppy definitions. The original posed question (which is ill-formed) is
> \(1\) Is $1/x$ a continuous function?
To make it well-formed, I want to *first* bring up the question:
> \(2\) Is $1/x$ a function?
Technically, this question is *also* ill-formed because it never specifies the domain of the function, which is part of the data needed to specify a function. One reasonable guess what the asker meant would be $\mathbb{R} \setminus \{0\}$, i.e. the set of nonzero real numbers, in which case we get the question
> (2') Does $1/x$ define a function from $\mathbb{R} \setminus \{0\}$ to $\mathbb{R}$?
which has the firm answer YES.
On the other hand, it does *not* make sense to try to define $1/x$ as a function on $\mathbb{R}$. The definition a function requires you to specify an output value for every input, so at least if you want a real-valued function[^2], there isn't any way to construe $1/x$ as a function on all of $\mathbb{R}$.
Now, returning to (1), we can now ask a well-formed question
> (1') Does $1/x$ describe a continuous function from $\mathbb{R} \setminus \{0\} \to \mathbb{R}$?
which again has the firm answer YES.
Of course, you could also consider a question like "does $1/x$ describe a continuous function $\mathbb{R} \to \mathbb{R}$?". However, this feels misleading: it would be like asking "is $\sqrt{2}$ an even integer?". The question doesn't make sense to begin with because $\sqrt2$ isn't an integer, and "even" is an adjective used for integers, so trying to ask whether it applies to $\sqrt2$ is a [type-error](https://qchu.wordpress.com/2013/05/28/the-type-system-of-mathematics/). Similarly, "continuous" is an adjective used for functions; it doesn't make sense to ask whether it applies to something that isn't a function.
See <https://twitter.com/davidcpvm/status/1481024944830046209> for the Twitter post (in Spanish) and the accompanying Reddit post (one of several) at <https://www.reddit.com/r/math/comments/s82vf8>.
[^1]: Also, "metric" is a fun word to say.
[^2]: Those of you that know what $\mathbb{RP}^1$ is could consider it as a function $\mathbb{RP}^1 \to \mathbb{RP}^1$ if you insisted; but it's continuous in that case too. | 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 suppose that $x_n$ converges to $x$ in $M$. Then $x \in S$ as well.
Same caveat: we abbreviate "closed in $M$" to just "closed" if the parent space $M$ is understood.
Here's another way to phrase it. The **limit points** of a subset $S \subseteq M$ are defined by $$\lim S := \left\{ p \in M :
\exists (x_n) \in S \text{ such that } x_n \to p \right\}.$$ Thus $S$ is closed if and only if $S = \lim S$.
Prove that $\lim S$ is closed even if $S$ isn't closed. (Draw a picture.)
For this reason, $\lim S$ is also called the **closure** of $S$ in $M$, and denoted $\overline S$. It is simply the smallest closed set which contains $S$.
(a) The empty set $\varnothing$ is closed in $M$ for vacuous reasons: there are no sequences of points with elements in $\varnothing$.
(b) The entire space $M$ is closed in $M$ for tautological reasons. (Verify this!)
(c) The closed interval $[0,1]$ in $\mathbb{R}$ is closed in $\mathbb{R}$, hence the name. Like with open sets, this is the prototypical example of a closed set to keep in mind!
(d) In fact, the closed interval $[0,1]$ is even closed in $\mathbb{R}^2$.
Let $S=(0,1)$ denote the open interval. Then $S$ is not closed in $\mathbb{R}$ because the sequence of points $$\frac12, \;
\frac14, \;
\frac18, \;
\dots$$ converges to $0 \in \mathbb{R}$, but $0 \notin (0,1)$.
I should now warn you about a confusing part of this terminology. Firstly, **"most" sets are neither open nor closed**.
The half-open interval $[0,1)$ is neither open nor closed in $\mathbb{R}$.
Secondly, it's **also possible for a set to be both open and closed**; this will be discussed in .
The reason for the opposing terms is the following theorem:
Let $M$ be a metric space, and $S \subseteq M$ any subset. Then the following are equivalent:
- The set $S$ is closed in $M$.
- The complement $M \setminus S$ is open in $M$.
Prove this theorem! You'll want to draw a picture to make it clear what's happening: for example, you might take $M = \mathbb{R}^2$ and $S$ to be the closed unit disk.
Let $M = (M,d)$ be a metric space. Show that $$d \colon M \times M \to \mathbb{R}$$ is itself a continuous function (where $M \times M$ is equipped with the product metric).
Consider $\mathbb{Q}$ and $\mathbb{N}$ as metric spaces (each with the obvious metric $d(x,y) = |x-y|$). Are these spaces homeomorphic?
No. There is not even a continuous injective map from $\mathbb{Q}$ to $\mathbb{N}$.
Two possible approaches, one using metric definition and one using open sets.
Metric approach: I claim there is no injective map from $\mathbb{Q}$ to $\mathbb{N}$ that is continuous. Indeed, suppose $f$ was such a map and $f(x) = n$. Then, choose $\varepsilon = 1/2$. There should be a $\delta > 0$ such that everything with $\delta$ of $x$ in $\mathbb{Q}$ should land within $\varepsilon$ of $n \in \mathbb{N}$ --- i.e., is equal to $n$. This is a blatant contradiction of injectivity.
Open set approach: In $\mathbb{Q}$, no singleton set is open, whereas in $\mathbb{N}$, they all are (in fact $\mathbb{N}$ is discrete). As you'll see at the start of , with the new and improved definition of "homeomorphism", we found out that the structure of open sets on $\mathbb{Q}$ and $\mathbb{N}$ are different, so they are not homeomorphic.
Show that subtraction is a continuous map $- \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, and division is a continuous map $\div \colon \mathbb{R} \times \mathbb{R}_{>0} \to \mathbb{R}$.
You can do this with bare hands. You can also use composition.
For subtraction, the map $x \mapsto -x$ is continuous so you can view it as a composed map
Similarly, if you are willing to believe $x \mapsto 1/x$ is a continuous function, then division is composition
If for some reason you are suspicious that $x \mapsto 1/x$ is continuous, then here is a proof using sequential continuity. Suppose $x_n \to x$ with $x_n > 0$ and $x > 0$ (since $x$ needs to be in $\mathbb{R}_{>0}$ too). Then $$\left\lvert \frac{1}{x} - \frac 1{x_n} \right\rvert
= \frac{\left\lvert x_n-x \right\rvert}{\left\lvert x x_n \right\rvert}.$$ If $n$ is large enough, then $\left\lvert x_n \right\rvert > x/2$; so the denominator is at least $x^2/2$, and hence the whole fraction is at most $\frac{2}{x^2} \left\lvert x_n-x \right\rvert$, which tends to zero as $n \to \infty$.
Exhibit a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f$ is continuous at $x \in \mathbb{R}$ if and only if $x=0$.
$\pm x$ for good choices of $\pm$.
Let $f(x) = x$ for $x \in \mathbb{Q}$ and $f(x) = -x$ for irrational $x$.
Prove that a function $f \colon \mathbb{R} \to \mathbb{R}$ which is strictly increasing must be continuous at some point.
Project gaps onto the $y$-axis. Use the fact that uncountably many positive reals cannot have finite sum. | 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, that can only happen if the function misses all $y$-values in the interval $(f(x)-\varepsilon_x, f(x))$ or $(f(x), f(x)+\varepsilon_x)$ (or both).
Projecting these missing intervals to the $y$-axis you find uncountably many intervals (one for each $x \in [0,1]$) all of which are disjoint. In particular, summing the $\varepsilon_x$ you get that a sum of uncountably many positive reals is $1$.
But in general it isn't possible for an uncountable family $\mathcal F$ of positive reals to have finite sum. Indeed, just classify the reals into buckets $\frac1k \le x < \frac1{k-1}$. If the sum is actually finite then each bucket is finite, so the collection $\mathcal F$ must be countable, contradiction.
Someone on the Internet posted the question "is $1/x$ a continuous function?", leading to great controversy on Twitter. How should you respond?
First answer the following question: "is $1/x$ a function?".
Like most Internet "debates" about math, the question revolves around sloppy definitions. The original posed question (which is ill-formed) is
> \(1\) Is $1/x$ a continuous function?
To make it well-formed, I want to *first* bring up the question:
> \(2\) Is $1/x$ a function?
Technically, this question is *also* ill-formed because it never specifies the domain of the function, which is part of the data needed to specify a function. One reasonable guess what the asker meant would be $\mathbb{R} \setminus \{0\}$, i.e. the set of nonzero real numbers, in which case we get the question
> (2') Does $1/x$ define a function from $\mathbb{R} \setminus \{0\}$ to $\mathbb{R}$?
which has the firm answer YES.
On the other hand, it does *not* make sense to try to define $1/x$ as a function on $\mathbb{R}$. The definition a function requires you to specify an output value for every input, so at least if you want a real-valued function[^2], there isn't any way to construe $1/x$ as a function on all of $\mathbb{R}$.
Now, returning to (1), we can now ask a well-formed question
> (1') Does $1/x$ describe a continuous function from $\mathbb{R} \setminus \{0\} \to \mathbb{R}$?
which again has the firm answer YES.
Of course, you could also consider a question like "does $1/x$ describe a continuous function $\mathbb{R} \to \mathbb{R}$?". However, this feels misleading: it would be like asking "is $\sqrt{2}$ an even integer?". The question doesn't make sense to begin with because $\sqrt2$ isn't an integer, and "even" is an adjective used for integers, so trying to ask whether it applies to $\sqrt2$ is a [type-error](https://qchu.wordpress.com/2013/05/28/the-type-system-of-mathematics/). Similarly, "continuous" is an adjective used for functions; it doesn't make sense to ask whether it applies to something that isn't a function.
See <https://twitter.com/davidcpvm/status/1481024944830046209> for the Twitter post (in Spanish) and the accompanying Reddit post (one of several) at <https://www.reddit.com/r/math/comments/s82vf8>.
[^1]: Also, "metric" is a fun word to say.
[^2]: Those of you that know what $\mathbb{RP}^1$ is could consider it as a function $\mathbb{RP}^1 \to \mathbb{RP}^1$ if you insisted; but it's continuous in that case too. | 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 tightly, and shake vigorously.
Using just the element $x$,
we get a pretty explosion that produces the subgroup above.
What happens if we put two elements $x$, $y$ in the box?
Among the elements that get produced are things like
\[ xyxyx, x^2y^9x^-5y^3, y^-2015, \]
Essentially, I can create any finite product of $x$, $y$, $x$, $y$.
This leads us to define:
Let $S$ be a subset of $G$.
The subgroup generated by $S$,
denoted $<S>$,
is the set of elements which can be written as a
finite product of elements in $S$ (and their inverses).
If $<S> = G$ then we say $S$ is a set of
generators for $G$,
as the elements of $S$ together create all of $G$.
Why is the condition ``and their inverses''
not necessary if $G$ is a finite group?
(As usual, assume Lagrange's theorem.)
Consider $1$ as an element of $ = (, +)$.
We see $<1> = $, meaning $\1\$ generates $$.
It's important that $-1$, the inverse of $1$ is also allowed:
we need it to write all integers as the sum of $1$ and $-1$.
This gives us an idea for a way to try and express groups compactly.
Why not just write down a list of generators for the groups?
For example, we could write
\[ < a > \]
meaning that $$ is just the group generated by one element.
There's one issue: the generators usually satisfy certain properties.
For example, consider $100$.
It's also generated by a single element $x$,
but this $x$ has the additional property that $x^100 = 1$.
This motivates us to write
\[ 100 = < x x^100 = 1 >. \]
I'm sure you can see where this is going.
All we have to do is specify a set of generators and
relations between the generators,
and say that two elements are equal if and only if
you can get from one to the other using relations.
Such an expression is appropriately called a group presentation.
The dihedral group of order $2n$ has a presentation
\[ D_2n = < r, s
r^n = s^2 = 1, rs = sr >. \]
Thus each element of $D_2n$ can be written uniquely in the form $r^$
or $sr^$, where $ = 0, 1, , n-1$.
The Klein four group,
isomorphic to $2 2$, is given by the presentation
\[ < a,b a^2=b^2=1, ab=ba >. \]
[Free group]
The free group on $n$ elements is the group
whose presentation has $n$ generators and no relations at all.
It is denoted $F_n$, so
\[
F_n = < x_1, x_2, , x_n >.
\]
In other words, $F_2 = <a,b>$ is the set of strings
formed by appending finitely many copies of $a$, $b$, $a$, $b$ together.
Notice that $F_1 $.
One might unfortunately notice that ``subgroup generated by $a$ and $b$''
has exactly the same notation as the free group $<a,b>$.
We'll try to be clear based on context which one we mean.
Presentations are nice because they provide a compact way to write down groups.
They do have some shortcomings, though.%
Actually, determining whether two elements of a presentation are equal is undecidable.
In fact, it is undecidable to even determine if a group is finite from its presentation.
[Presentations can look very different]
The same group can have very different presentations.
For instance consider
\[ D_2n = < x,y x^2=y^2=1, (xy)^n=1 >. \]
(To see why this is equivalent, set $x=s$, $y=rs$.)
Homomorphisms
$.
How can groups talk to each other?
Two groups are ``the same'' if we can write an isomorphism between them.
And as we saw, two metric spaces are ``the same''
if we can write a homeomorphism between them.
But what's the group analogy of a continuous map?
We simply drop the ``bijection'' condition.
Let $G = (G, )$ and $H = (H, )$ be groups.
A group homomorphism is a map $ G H$
such that for any $g_1, g_2 G$ we have
\[ (g_1 g_2) = (g_1) (g_2). \]
(Not to be confused with ``homeomorphism'' from
last chapter: note the spelling.)
[Examples of homomorphisms]
Let $G$ and $H$ be groups.
Any isomorphism $G H$ is a homomorphism.
In particular, the identity map $G G$ is a homomorphism.
The trivial homomorphism $G H$ sends
everything to $1_H$.
There is a homomorphism from $$ to $100$ by
sending each integer to its residue modulo $100$.
There is a homomorphism from $$ to itself by $x 10x$
which is injective but not surjective.
There is a homomorphism from $S_n$ to $S_n+1$ by ``embedding'':
every permutation on $\1,,n\$ can be thought of as a permutation
on $\1,,n+1\$ if we simply let $n+1$ be a fixed point.
A homomorphism $: D_12 D_6$
is given by $s_12 s_6$
and $r_12 r_6$.
Specifying a homomorphism $ G$ is the same as
specifying just the image of the element $1 $. Why?
The last two examples illustrate something: suppose we have a presentation of $G$.
To specify a homomorphism $G H$, we only have to specify where each generator of $G$ goes, in such a way that the relations are all satisfied.
Important remark:
the right way to think about an isomorphism is as a ``bijective homomorphism''.
To be explicit,
Show that $G H$ if and only if there exist
homomorphisms $ G H$ and $ H G$
such that $ = _H$ and $ = _G$.
So the definitions of homeomorphism of metric spaces
and isomorphism of groups are not too different.
Some obvious properties of homomorphisms follow.
Let $ G H$ be a homomorphism.
Then $(1_G) = 1_H$ and $(g) = (g)$.
Boring, and I'm sure you could do it yourself if you wanted to. | 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 $$ is in fact a subgroup of $G$.
We also have the following important fact, which we also encourage the reader to verify.
[Kernel determines injectivity]
The map $$ is injective if and only if $ = \1_G\$.
To make this concrete, let's compute the kernel of each of our examples.
[Examples of kernels]
The kernel of any isomorphism $G H$ is trivial,
since an isomorphism is injective.
In particular, the kernel of the identity map $G G$ is $\1_G\$.
The kernel of the trivial homomorphism $G H$
(by $g 1_H$) is all of $G$.
The kernel of the homomorphism $ 100$ by $n n$
is precisely \[ 100 = \, -200, -100, 0, 100, 200, \. \]
The kernel of the map $ $ by $x 10x$ is trivial: $\0\$.
There is a homomorphism from $S_n$ to $S_n+1$ by ``embedding'',
but it also has trivial kernel because it is injective.
A homomorphism $ D_12 D_6$
is given by $s_12 s_6$ and $r_12 r_6$.
You can check that
\[ = \ 1, r_12^3 \ 2. \]
Exercise below.
Fix any $g G$.
Suppose we have a homomorphism $ G$ by $n g^n$.
What is the kernel?
Show that for any homomorphism $: G H$,
the image $(G)$ is a subgroup of $H$.
Hence, we'll be especially interested in the case where $$ is surjective.
Cosets and modding out
The next few sections are a bit dense.
If this exposition doesn't work for you, try ref:gowers.
Let $G$ and $Q$ be groups, and suppose there exists
a surjective homomorphism \[ G Q. \]
In other words, if $$ is injective then $ G Q$ is a bijection,
and hence an isomorphism.
But suppose we're not so lucky and $$ is bigger than just $\1_G\$.
What is the correct interpretation of a more general homomorphism?
Let's look at the special case where $ 100$ is ``modding out by $100$''.
We already saw that the kernel of this map is
\[
= 100 = \ , -200, -100, 0, 100, 200, \.
\]
Recall now that $ $ is a subgroup of $G$.
What this means is that $$ is indifferent to the subgroup $100$ of $$:
\[ (15) = (2000 + 15) = (-300 + 15) = (700 + 15) = . \]
So $100$ is what we get when we ``mod out by $100$''. Cool.
In other words, let $G$ be a group and $ G Q$
be a surjective homomorphism with kernel $N G$.
We claim that $Q$ should be thought of as the quotient of $G$ by $N$.
To formalize this, we will define a so-called
quotient group $G/N$
in terms of $G$ and $N$ only (without referencing $Q$)
which will be naturally isomorphic to $Q$.
For motivation, let's give a concrete description of $Q$ using just $$ and $G$.
Continuing our previous example, let $N = 100$ be our subgroup of $G$.
Consider the sets
N &= \ , -200, -100, 0, 100, 200, \ \\
1+N &= \ , -199, -99, 1, 101, 201, \ \\
2+N &= \ , -198, -98, 2, 102, 202, \ \\
& \\
99+N &= \ , -101, -1, 99, 199, 299, \.
The elements of each set all have the same image when we apply $$,
and moreover any two elements in different sets have different images.
Then the main idea is to notice that
We can think of $Q$ as the group
whose elements are the sets above.
Thus, given $$ we define an equivalence relation $_N$
on $G$ by saying $x _N y$ for $(x) = (y)$.
This $_N$ divides $G$ into several equivalence classes in $G$
which are in obvious bijection with $Q$, as above.
Now we claim that we can write these equivalence classes very explicitly.
% Also, for each $g G$ define $ g$ to be the equivalence class of $g$ under $_$.
Show that $x _N y$ if and only if $x = yn$ for some $n N$
(in the mod $100$ example, this means they ``differ by some multiple of $100$'').
Thus for any $g G$, the equivalence class of $_N$ which contains $g$
is given explicitly by \[ gN \ gn n N \. \]
%Note that
%\[ (x) = (y)
% 1_Q = (x)(y) = (xy)
% xy . \]
%Hence another way to describe $ x$ is
%\[ g = gN = \ gn n N \. \]
Here's the word that describes the types of sets we're running into now.
Let $H$ be any subgroup of $G$ (not necessarily the kernel of some homomorphism).
A set of the form $gH$ is called a left coset of $H$.
Although the notation might not suggest it,
keep in mind that $g_1N$ is often equal to $g_2N$ even if $g_1 g_2$.
In the ``mod $100$'' example, $3+N = 103+N$.
In other words, these cosets are sets.
This means that if I write ``let $gH$ be a coset'' without telling you what $g$ is,
you can't figure out which $g$ I chose from just the coset itself.
If you don't believe me, here's an example of what I mean:
\[ x+100 = \ , -97, 3, 103, 203, \ x = ?. \]
There's no reason to think I picked $x=3$. (I actually picked $x=-13597$.)
remark:coset_warning
Given cosets $g_1H$ and $g_2H$,
you can check that the map $x g_2g_1 x$ is a bijection between them.
So actually, all cosets have the same cardinality.
So, long story short,
Elements of the group $Q$ are naturally identified with left cosets of $N$.
In practice, people often still prefer to picture elements of $Q$ as single points
(for example it's easier to think of $2$ as $\0,1\$
rather than $\ \,-2,0,2,\, \,-1,1,3,\ \$).
If you like this picture,
then you might then draw $G$ as a bunch of equally tall fibers (the cosets),
which are then ``collapsed'' onto $Q$.
size(9cm);
for (int i=1; i<=6; ++i)
draw( (i,0)--(i,3.4) );
dot( (i,0.6) );
dot( (i,1) );
dot( (i,1.7) );
dot( (i,2.4) );
dot( (i,2.8) );
dot( (i,3) );
label(rotate(90)*scale(1.4)*"$$", (i+0.02,-0.9), dir(90));
dot( (i,-1) );
label("$G$", (0.5, 3));
label("$Q$", (0.5,-1)); | An Infinitely Large Napkin | napkin | general | advanced | Basic Abstract Algebra | Homomorphisms and quotient groups | 07_quotient.md | 1 | 1,568 |
End of preview. Expand
in Data Studio
olympiad-books-open-source
Chunked content from 12 open-source mathematics textbooks, suitable for retrieval (RAG), embedding, and math reasoning research.
Source code: github.com/yoonholee/olympiad-books-open-source-pipeline
Books
| Book | Author(s) | License | Source |
|---|---|---|---|
| An Infinitely Large Napkin | Evan Chen | CC BY-SA 4.0 / GPL v3 | GitHub |
| Mathematical Reasoning: Writing and Proof | Ted Sundstrom | CC BY-NC-SA 3.0 | GitHub |
| Exploring Combinatorial Mathematics | Richard Grassl, Oscar Levin | GFDL | GitHub |
| Discrete Mathematics: An Open Introduction | Oscar Levin | CC BY-SA 4.0 | GitHub |
| Abstract Algebra: Theory and Applications | Tom Judson | GFDL | GitHub |
| Applied Combinatorics | Mitchel Keller, William Trotter | CC BY-SA 4.0 | GitHub |
| Combinatorics Through Guided Discovery | Kenneth Bogart | GFDL | GitHub |
| A First Course in Linear Algebra | Robert Beezer | GFDL | GitHub |
| Elementary Number Theory | William Stein | Free (Springer agreement) | GitHub |
| Basic Analysis | Jiri Lebl | CC BY-NC-SA 4.0 + CC BY-SA 4.0 | GitHub |
| An Introduction to Proof via Inquiry-Based Learning | Dana Ernst | CC BY-SA 4.0 | GitHub |
| Open Logic | Open Logic Project | CC BY 4.0 | GitHub |
Schema
| Field | Type | Description |
|---|---|---|
text |
string | Chunk content (markdown with LaTeX math) |
book |
string | Full book title |
book_key |
string | Short identifier (e.g. napkin, aata, openlogic) |
subject |
string | Math subject area: general, abstract-algebra, combinatorics, discrete-math, linear-algebra, logic, number-theory, proofs, real-analysis |
level |
string | intro (undergrad) or advanced (upper-division / graduate) |
part |
string | Part/unit grouping within a book (e.g. "Group Theory", "Single Variable") |
chapter |
string | Chapter title (e.g. "Groups", "Real Numbers") |
section |
string | Section title (empty if chunk is in chapter intro) |
source_file |
string | Original markdown filename |
chunk_id |
int | Sequential chunk index within chapter |
tokens_est |
int | Estimated token count (~3.5 chars/token) |
Stats
| Book | Chunks | Median tokens | Total tokens |
|---|---|---|---|
| An Infinitely Large Napkin | 794 | 1446 | 945K |
| Open Logic | 480 | 1464 | 645K |
| Basic Analysis | 408 | 1056 | 418K |
| A First Course in Linear Algebra | 370 | 1184 | 389K |
| Abstract Algebra: Theory and Applications | 330 | 1080 | 335K |
| Mathematical Reasoning | 274 | 1014 | 270K |
| Applied Combinatorics | 229 | 1001 | 226K |
| Discrete Mathematics | 205 | 1087 | 214K |
| Combinatorics Through Guided Discovery | 124 | 930 | 112K |
| Elementary Number Theory | 116 | 1056 | 118K |
| Exploring Combinatorial Mathematics | 90 | 1156 | 94K |
| An Introduction to Proof via IBL | 86 | 1491 | 112K |
| Total | 3506 | 3.88M |
Chunking method
Content-aware chunking from source files (LaTeX/PreTeXt), not PDF extraction:
- Section boundaries (
##/###headers) always start a new chunk - Math environments (Theorem, Definition, Example, Proof, etc.) are kept intact
- Paragraph splits used only when a block exceeds the max token limit
- Math blocks (
$$...$$) are never split - Small adjacent chunks merged up to the max token limit (default 512-1536 tokens)
Topics covered
- Napkin: Abstract algebra, topology, linear algebra, representation theory, complex analysis, measure theory, differential geometry, algebraic number theory, algebraic topology, category theory, algebraic geometry, set theory
- Open Logic: Propositional logic, first-order logic, modal logic, intuitionistic logic, computability, incompleteness, set theory, model theory
- Basic Analysis: Real numbers, sequences, series, continuous functions, derivatives, integrals, metric spaces, multivariable calculus
- First Course in Linear Algebra: Systems of equations, vectors, matrices, determinants, eigenvalues, linear transformations, vector spaces
- Abstract Algebra: Groups, cyclic groups, permutation groups, cosets, isomorphisms, normal subgroups, homomorphisms, rings, integral domains, fields, vector spaces, Galois theory
- Mathematical Reasoning: Logic, proof techniques, set theory, functions, equivalence relations, number theory
- Applied Combinatorics: Strings/sequences, induction, generating functions, graph theory, inclusion-exclusion, network flows, partially ordered sets
- Discrete Mathematics: Logic, graph theory, counting, sequences, mathematical structures
- Combinatorics Through Guided Discovery: Counting, generating functions, distribution problems, graph theory, Polya enumeration
- Elementary Number Theory: Prime numbers, congruences, quadratic reciprocity, continued fractions, elliptic curves
- Exploring Combinatorial Mathematics: Pascal's triangle, binomial coefficients, generating functions, Stirling numbers, partitions, inclusion-exclusion
- Introduction to Proof via IBL: Logic, set theory, induction, real number axioms, topology of reals, sequences, continuity
Usage
from datasets import load_dataset
ds = load_dataset("yoonholee/olympiad-books-open-source", split="train")
# Filter by book
napkin = ds.filter(lambda x: x["book_key"] == "napkin")
# Filter by subject
algebra = ds.filter(lambda x: x["subject"] == "abstract-algebra")
# Filter by level
intro = ds.filter(lambda x: x["level"] == "intro")
# Filter by part within a book
groups = ds.filter(lambda x: x["book_key"] == "aata" and x["part"] == "Group Theory")
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