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!