What's a Quotient Group, Really? Part 1

I realize that most of my posts for the past, er, few months have been about some pretty hefty duty topics. Today, I'd like to dial it back a bit and chat about some basic group theory! So let me ask you a question: When you hear the words "quotient group," what do you think of? In case you'd like a little refresher, here's the definition: 

Definition: Let $G$ be a group and let $N$ be a normal subgroup of $G$. Then $G/N=\{gN:g\in G\}$ is the set of all cosets of $N$ in $G$ and is called the quotient group of $N$ in $G$.
Personally, I think answering the question "What is a quotient group?" with the words "the set of all cosets" isn't very enlightening or satisfying. Here's what I think is a more intuitive answer:
But let me explain what I mean by "sorta." Recall that belonging to a subgroup $N$ simply means you satisfy a special property:

  • $n\mathbb{Z}\subset\mathbb{Z}$ means: "You live in $n\mathbb{Z}$ iff you're an integer and a multiple of $n$."
  • $SL_n(F)\subset GL_n(F)$ means: "You live in $SL_n(F)$ iff you're an invertible $n\times n$ matrix with entries in a field $F$ and your determinant is 1."
  • $\ker\phi\subset G$ means: "You live in $\ker\phi$ iff you get sent to the identity $e\in H$ under a homomorphism $\phi:G\to H$."
  • $Z(G)\subset G$ means: "You live in $Z(G)$ (the center of $G$) iff you commute with every element of $G$."
  • $[G,G]\subset G$ means: "You live in $[G,G]$ (the commutator subgroup) iff you look like a finite product of things of the form $ghg^{-1}h^{-1}$ where $g,h\in G$."
  • and so on...

So when you hear something like, "Form the quotient $G/N$..." or "Mod out $G$ by the subgroup $N$..." what the speaker really means is, "Consider all the elements of $G$ that don't satisfy the property of belonging to $N$." But in general, there are many ways to fail to satisfy the property to be in $N$. So there's a little more to say here. To get an idea for this, we can imagine all the elements of $G$ taking an online survey:

Now suppose we were to collect the survey results and sort the elements of $G$ according to their answers. The story might go something like this:

[Mathematician enters room full of elements of $G$ chatting quietly amongst themselves]

Hi folks. How are we today? Doin' well? Great. Listen, would those of you who answered "yes" to question #1 please raise your hand? Fantastic, hi there. Thank you. Now, if you would, please huddle together in a single pile. Yes, just like that. You're doin' fine, folks, just fine. Alright, from now on we will refer to you collectively as "$N$" or - on a good day - we might also call you "the trivial coset." But we no longer care about ya'll as individuals. Sorry. You'll get used to it.

[Mathematician turns her attention to the folks not in $N$]

Hey there, everyone. Would you please raise your hand if you selected "not too badly" for question #2? Great, how you folks doin'? Good. Look, although none of you satisfy the property to belong to $N$, you do satisfy a different property: You all fail not too badly (ntb). Congrats! Now please form your own huddle over in that corner. Quickly now, folks. Okay perfect. Listen, we no longer care about you individually - ya'll are all indistinuishable to us. For this reason, we'll refer to you as "(ntb)N" or sometimes "the coset ntb."

[Mathematician addresses remaining elements in the room]

Hi there, ya'll, thanks for waiting. Would those of you who fail to belong to $N$ "pretty badly" (pb) please form your own pile? Sure, you can stand in that corner. That's right, go ahead. Now because you all possess the special property of 'failing pretty badly,' you're all the same to us, and so we'll just call all of you "(pb)N" or "the coset pb."

Alright now, I see ya'll who are "not even close" (nec) to meeting the requirements of belonging to $N$ have already huddled together. Thanks so much, folks. Now now, stop all that crying! It's not such a bad thing. You, too, satisfy a very special property: you all fail really badly. Isn't that great? It sure is. So we'll collectively refer to you all as "(nec)N" or "the coset nec."

[Mathematcian happily exists the room]

[Group elements resume quiet chatter]

So you see? We can organize the entire group $G$ based on the how the elements relate to the subgroup $N$. Those who belong to $N$, well, belong to $N$. And those who don't can be sorted together according to how badly they miss the mark. Of course, labels like "not too badly" and "pretty badly" and "not even close" are rather fabricated, and there can certainly be more than three options. In fact, it's better to replace "how badly they fail" with just "how they fail." But in any case, this is the bird's-eye-view. Those little organization piles are precisely the cosets of $G/N$.

And taking our analogy one step further, this action of 'administering the survey,' i.e. of organizing the members of $G$ according to their relationship to $N$, is precisely what the so-called natural projection homomorphism $\varphi:G\to G/N$ is doing! (Here, $\varphi$ sends an element $g$ to the coset $gN$.) As I like to tell my college algebra students, functions are like verbs! They do things according to some rule. And the same thing is true of group homomorphisms such as $\varphi$. It tells the elements of $G$ to get organized - that's the verb - according to the rule: "If you fail *this* badly, then go stand in the appropriate coset."

Well, I hope this was a little helpful! We'll continue this discussion next time by looking at the quotient group $\mathbb{Z}/n\mathbb{Z}$. I'll also say a word or two about the other examples listed at the beginning of this post.

Until then!

The Sierpinski Space and Its Special Property


Last time we chatted about a pervasive theme in mathematics, namely that objects are determined by their relationships with other objects, or more informally, you can learn a lot about an object by studying its interactions with other things. Today I'd to give an explicit illustration of this theme in the case when


"objects" = topological spaces


"relationships with other objects" = continuous functions


The goal of this post, then, is to convince you that

The topology on a space X is completely determined by the set* of all continuous functions to X.

But what do I mean by "is completely determined by"? Well, suppose $Z$ is any topological space, and let hom$(Z,X)$ denote the set of all continuous functions from $Z$ to $X$. Then the above means the following:

#1: the topology on X dictates what hom(Z,X) must be. 


#2: hom(Z,X) dictates what the topology on X must be.

Now what exactly do these two notions mean? To answer this, I think it will help to look at a concrete example. In fact, let's consider the case when $X$ is one of the simplest topological spaces out there---the Sierpinski space.
Start with a set $S$ with two elements, say $\{0,1\}$. We can turn this set into a topological space---called the Sierpinski space---by declaring the open sets to be $\emptyset$, $1$ and $S$. We'll call this the Sierpinski topology. (Incidentally, there are only three possible topologies on $\{0,1\}$--the discrete one, the indiscrete one, and this one.) Notice that what we call the elements isn't so important. That is, you can replace "0" and "1" by "red" and "blue" or "dog" and "cat," if you like. Either way, we can illustrate the Sierpinski space as shown on the right.
lookin' a little bit like the Target dog....

lookin' a little bit like the Target dog....

Now suppose $Z$ is any topological space and observe that for any open set $U$ in $Z$, we can construct a function $f_U$ from $Z$ to the Sierpinski space $S$ which sends every point in $U$ to 1 and all other points to 0: $$f_U(z)= \begin{cases} 1, &\text{if $z\in U$,}\\ 0, &\text{if $z\not\in U$}. \end{cases}$$ Even better, such a map $f_U$ is continuous! The preimage of each open set in $S$ is open in Z: $f_U^{-1}(\emptyset)=\emptyset$, $f_U^{-1}(\{1\})=U,$ and $f_U^{-1}(S)=Z$. On the flip side, if $f:Z\to S$ is any continuous function then we can consider the subset of $Z$ $$U_f=f^{-1}(\{1\})=\{z\in Z:f(z)=1\}.$$ Since $f$ is continuous and $\{1\}$ is open in $S$, the set $U_f$ is necessarily open! These two constructions reveal that there is a bijection of sets:**
And in fact, the Sierpinski topology on $S$ is completely characterized by this property! That is, the Sierpinski topology on a two-point set $S$ is the ONLY topology (up to homeomorphism) on $S$ that satisfies the property: "continuous functions from any space $Z$ to $S$ are in one-to-one correspondence with the open sets of $Z$."*** This follows from the following two observations that we alluded to earlier:

#1: The topology on S dictates what hom(Z,S) must be.

That is, if we endow $S$ with the Sierpinski topology then the maps of hom$(Z,S)$ must be in one-to-one correspondence with the open sets of $Z$. We saw this above. But what if we were to give $S$ the indiscrete topology? Or the discrete topology? The the set hom$(Z,S)$ will change accordingly. Indeed, suppose $S$ has the indiscrete topology. Then every function $Z\to S$ is continuous! In other words hom$(Z,S)$ is the set of all functions $f:Z\to S$, and the number of such $f$ may exceed the number of open subsets of $Z$.

Suppose now that $S$ has the discrete topology. Then number of continuous functions $f:Z\to S$ may be smaller than the number of open subsets of $Z$. To see this, suppose $U\subset Z$ is open and consider the function $f_U:Z\to S$ that we defined earlier. This map is continuous if and only if $f_U^{-1}(\{1\})=U$ is open---which it is---AND if $f_U^{-1}(\{0\})=Z\smallsetminus U$ is open---which it may not be.

This is a helpful observation: whenever a space $X$ (any $X$, not just $S$) has the indiscrete topology, it's easy to be continuous! In fact, every function to $X$ will be continuous. But if $X$ has the discrete topology, it's much harder for functions to $X$ to be continuous. Fewer open sets in $X$ = more continuous functions to $X$. More open sets in $X$ = fewer continuous functions to $X$.

#2: hom(Z,S) dictates what the topology on S must be.

To see this, suppose $\tau$ is any topology on the set $S$. If for any space $Z$ the maps of hom$(Z,S)$ are in one-to-one correspondence with the open subsets of $Z$, I claim the topology $\tau$ MUST be the Sierpinski topology. But this follows from our conversation above! If $\tau$ is the Sierpinski topology, then the claim holds. But if $\tau$ is either the indiscrete or the discrete topology, we've just seen that the continuous maps may not be in bijection with the open subsets of $Z$.
Pretty cool, huh? The Sierpinski topology is just the right topology to put on a two-point space so that continuous maps from any space $Z$ correspond exactly with the open sets of $Z$. And the punchline is that we can play a similar game on any topological space $X$ to discover that

the data of the topology on a space X is "encoded" in hom(Z,X)

(or hom$(X,Z)$)! In other words, a topological space is completely determined by the continuous functions to it.
Those with a little knowledge of category theory may like to know that today's theme---and the theme of our last post---is a consequence of the following proposition (which is not terribly hard to prove and is actually fun to try!):

Proposition: Let $\mathsf{C}$ be a locally small category. The following are equivalent:

  1. $f:X\to Y$ is an isomorphism.
  2. For all objects $Z$ in $\mathsf{C}$, $f^*:\text{hom}(Y,Z)\to\text{hom}(X,Z)$ is an isomorphism.
  3. For all objects $Z$ in $\mathsf{C},$ $f_*:\text{hom}(Z,X)\to\text{hom}(Z,Y)$ is an isomorphism.
Here $f^*$ is called the pullback of $f$ and it sends a morphism $g\in\text{hom}(Y,Z)$ to the morphism $g\circ f\in\text{hom}(X,Z)$. Likewise, $f_*$ is the pushforward of $f$ and it sends a morphism $h\in\text{hom}(Z,X)$ to the morphism $f\circ h\in\text{hom}(Z,Y)$.
For concreteness, it might help to think of $X$ and $Y$ as topological spaces and hom$(Z,X)$ as the set of continuous functions to $X$ from $Z$. The proposition then tells us that two spaces $X$ and $Y$ have the same topology (i.e. are homeomorphic) if and only if the set of continuous functions to (or from) $X$ is the same as the set of continuous functions to (or from) $Y$. That is to say, a topological space $X$ is completely determined by the set of all continuous maps to it!

But notice the proposition holds for any (locally small) category! Thus we recover the statement: an object is completely determined by the set of morphisms to (or from) it. In short,

objects are completely determined by their relationships to other objects!



*If you're concerned with the word "set" here, note that the category of all topological spaces and continuous functions is a locally small category, that is for any two spaces X and Y, the collection of continuous functions between them is a bona fide set.

**In fact, if we view Z and S as plain sets (and not topological spaces), notice all functions from Z to S look like indicator functions on the subsets of Z. So the set of all functions from Z to S is in bijection with the set of all subsets of Z. This is a helpful thing to keep in mind.

***This is actually a key fact. What's important about the Sierpinski topology---or any topology, for that matter---is not so much its definition but rather the property that the space possesses once it's endowed with the topology.

The Most Obvious Secret in Mathematics

obvious secret.jpg

Yes, I agree. The title for this post is a little pretentious. It's certainly possible that there are other mathematical secrets that are more obvious than this one, but hey, I got your attention, right? Good. Because I'd like to tell you about an overarching theme in mathematics - a mathematical mantra, if you will. A technique that mathematicians use all the time to, well, do math. 

I'm calling it a 'secret' because until recently, I've rarely (if ever?) heard it stated explicitlyThis suggests to me that it's one of those things that folks assume you'll just eventually pick up. Hopefully. Like some sort of unspoken rule of mathematics. But a few weeks ago while chatting with my advisor*, I finally heard this unspoken rule uttered! Explicitly. Repeatedly, in fact. And at that time I realized it needs to be ushered further into the spotlight. Today's post, then, is my invitation to you to come listen in on that conversation.

So enough with the chit-chat! What's the secret? Here 'tis:


A mathematical object is determined by its relationships to other objects. 


Practically speaking, this suggests that


an often fruitful way to discover properties of an object is NOT to investigate the object itself, but rather to study the collection of maps to or from the object.


  Or to be a little less formal,


you can learn a lot about an object by studying its interactions with other things.


By "object" I mean things like sets or groups or measurable spaces or vector spaces or topological spaces or.... And by "maps" I mean the appropriate version of 'function':  functions, group homomorphisms, measurable functions, linear transformations, continuous functions, and so on.

So now do you see why I'm calling this an obvious secret? We students have been using this technique - though perhaps unknowingly - since we were mathematical infants! We've known about functions since grade school. We've labored over properties of real-valued functions and their (anti)derivatives throughout Calculus. We became well-acquainted with linear transformations and their corresponding matrices in linear algebra. We battled with homomorphisms during the first week of undergrad abstract algebra. We finally learned the real definition of a continuous map in point-set topology. The list goes on and on. 

See how pervasive this idea is? It's obvious!

And that's my point.

Because have you ever stopped to really think about it?

At first glance, perhaps it seems a little odd that "best" way to study an object is to divert your attention away from the object and focus on something else. But we do this all the time. Take people-watching, for instance. You can learn a lot about a person simply by looking at how they relate to the folks around them. And the same is true in mathematics.


I've hinted at this theme briefly in a previous post, but I'd like to list a few examples to further convince you. Keep in mind, though, that this is a philosophy that permeates throughout all of mathematics. So what I'm sharing below is peanuts compared to what's out there. But I hope it's enough to illustrate the idea.

In analysis...

One word: sequences! Recall (or observe) that a sequence $\{x_n\}=\{x_1,x_2,\ldots\}$ is - yes, a long list of numbers but ultimately - a function $\phi:\mathbb{N}\to\mathbb{R}$, where $x_n=\phi(n)$. By using sequences to 'probe' the real line $\mathbb{R}$, we learn that $\mathbb{R}$ has no "holes" - if you point your infinitesimally small finger anywhere on the real line, you'll always land on a real number. This property of $\mathbb{R}$ is called completeness, and it is investigated by special types of sequences called Cauchy sequences. Another good example is curvature. Need to measure how much a curve or surface is bending in space? Then you'll want to think about second derivatives which, assuming the curve/surface is "nice enough," are themselves continuous functions to $\mathbb{R}$!**

In group theory...

By looking at homomorphisms from arbitrary groups to special types of groups called symmetric groups, we discover that the raison d'être of a group is to shuffle things around! This is captured in Cayley's Theorem, a major result in group theory, which says that every group is isomorphic to a group of permutations or, less formally, a group is to math what a verb is to language. In fact, this (not the theorem, per se, but the idea) is historically how groups were first understood and is precisely what motivated Galois to lay down the foundations of the discipline of mathematics that bears his name. You might recall that we've chatted previously about the verb-like behavior of groups in this non-technical introduction to Galois theory.

In topology...

Want to know if your topolgical space $X$ is connected? Just check that any continuous map from it to $\{0,1\}$ is constant! Want to determine how many 'holes' $X$ has? Study continuous functions from the circle, $S^1$, into it! This leads to the fundamental group, $\pi_1$. Want to know many higher dimensional 'holes' there are? Look at continuous functions from the $n$-sphere into it! This leads to the higher homotopy groups, $\pi_n$. Want to know what the topology on any given space is? Simply look at the collection of continuous functions from it to a little two-point space! In fact, this last example is really quite paradigmatic, and I'd like to elaborate a bit more. So stay tuned for next time!

In the mean time, how many examples of today's not-so-secret secret can you think of? I'd love to hear 'em. Let me know in the comments below!



*Yes! I have an advisor now! And since my written qualifying exams are out of the way, the next thing on my to-do list is passing the oral qual. I've also picked up a teaching assignment this year. For both of these reasons, blogging has been - and may continue to be - a little bit slow. But although my posts may become less frequent, I'm hoping the content will be richer. I'm almost positive they'll be more topology/category-flavored, too.

** This is really a statement about differential geometry rather than analysis, for it generalizes nicely for things called manifolds. In fact, the whole premise behind differential geometry is a great example of today's theme. The idea is that globally, a manifold $M$ may be so complicated and wonky that we don't have many tools to probe it with. But - following the old adage, How do you eat an elephant? One bite at a time. - the impossible becomes possible if we just consider $M$ little patches at a time. Why? Because locally manifolds look exactly like Euclidean space, $\mathbb{R}^n$. (Take the earth, for example. Even though it's round, it looks flat locally.) And since we have tons of tools at our disposal in $\mathbb{R}^n$ (like calculus!), we can apply them to the little patches of our manifold too.

Resources for Intro-Level Graduate Courses

Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.

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A Ramble About Qualifying Exams

A Ramble About Qualifying Exams

Today I'm talking about about qualifying exams! But no, I won't be dishing out advice on preparing for these exams. Tons of excellent advice is readily available online, so I'm not sure I can contribute much that isn't already out there. However, it's that very web-search that has prompted me to write this post. You see, before I started graduate school I had heard of these rites-of-passage called the qualifying exams. And to be frank, I thought they sounded terrifying....

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Good Reads: The Princeton Companion to Mathematics

Good Reads: The Princeton Companion to Mathematics

Next up on Good Reads: The Princeton Companion to Mathematics, edited by Fields medalist Timothy GowersThis book is an exceptional resource! With over 1,000 pages of mathematics explained by the experts for the layperson, it's like an encyclopedia for math, but so much more. Have you heard about category theory but aren't sure what it is? There's a chapter for that! Seen the recent headlines about the abc conjecture but don't know what it's about? There's a chapter for that! Need a crash course in general relativity and Einstein's equations, or the P vs. NP conjecture, or C*-algebras, or the Riemann zeta function, or Calabi-Yau manifolds? There are chapters for all of those and more

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Clever Homotopy Equivalences

Clever Homotopy Equivalences

You know the routine. You come across a topological space X and you need to find its fundamental group. Unfortunately, X is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space Y that is homotopy equivalent to X and whose fundamental group is much easier to compute. And voila! Since X and Y are homotopy equivalent, you know that the fundamental group of X is isomorphic to the fundamental group of Y. Mission accomplished.

Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.

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Snippets of Mathematical Candor

Snippets of Mathematical Candor

A while ago I wrote a post in response to a great Slate article reminding us that math - like writing - isn't something that anyone is good at without (at least a little!) effort. As the article's author put it, "no one is born knowing the axiom of completeness." Since then, I've come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I've decided to share them here on the blog for a little morale-boosting.

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