What's a Quotient Group, Really? Part 2

Today we're resuming our informal chat on quotient groups. Previously we said that belonging to a (normal, say) subgroup $N$ of a group $G$ just means you satisfy some property. For example, $5\mathbb{Z}\subset\mathbb{Z}$ means "You belong to $5\mathbb{Z}$ if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in $G$ either

#1) belongs to N            or            #2) doesn't belong to N

and noting that the elements of $G$ can be grouped together according to HOW they satisfy either #1 or #2. The resulting 'piles' are precisely the cosets $gN$ of $G/N$. And the actual process of creating the piles is what the so-called "natural projection homomorphism" $\phi:G\to G/N$ is doing when it sends an element $g$ to the coset $gN$. The representative $g$ simply indicates/describes how #1 or #2 is satisfied.

Notice there is only one way to satisfy #1---you simply belong to $N$---but in general there can be many ways to satisfy #2, i.e. there can be many ways to fail to be in $N$. That's why $G/N$ has exactly one "trivial" coset but often more than one "nontrivial" coset. This reminds me of the saying,

There are many ways to be wrong, but only one way to be right!

and is why I like to think of $G/N$ as roughly all the things in $G$ that don't belong to $N$. Sure, it's true that $G/N$ also contains all the folks that do belong to $N$ (after all, $N\in G/N$!), but there's only one way to satisfy this property, and therefore there's nothing interesting to talk about. It's trivial.

As an example, let's look more closely at the case when $G=\mathbb{Z}$ and $N=5\mathbb{Z}$.

A Simple Example

We might think of $\mathbb{Z}/5\mathbb{Z}$ as all the integers that aren't multiples of 5, i.e. of those who fail to belong to the subgroup $5\mathbb{Z}$. Though to be efficient, we'll consider some integers as being the 'same' if they fail to belong to $5\mathbb{Z}$ in the same way:
  • All the folks in $\{\ldots,-9,-4,1,6,7,\ldots\}$ aren't in $5\mathbb{Z}$ precisely because they're off by 1. So we call them the coset $[1]=1+5\mathbb{Z}$.
  • All the folks in $\{\ldots,-8,-3,2,7,12,\ldots\}$ aren't in $5\mathbb{Z}$ precisely because they're off by 2. So we call them the coset $[2]=2+5\mathbb{Z}[2]$.
  • All the folks in $\{\ldots,-7,-2,3,8,13,\ldots\}$ aren't in $5\mathbb{Z}$ precisely because they're off by 3. So we call them the coset $[3]=3+5\mathbb{Z}$.
  • All the folks in $\{\ldots,-6,-1,4,9,14,\ldots\}$
  • aren't $5\mathbb{Z}$ precisely because they're off by 4. So we call them the coset $[4]=4+5\mathbb{Z}$.
Of course everyone in $\{\ldots,-10,-5,0,5,10,\ldots\}$ is a multiple of 5. They're all off by 0! So we call them $0+5\mathbb{Z}$ or simply $[0]$. So you see? Every integer is either divisible by 5 or it's not. If it's not, then we can ask the additional question, "Why?" And since there are four possible answers---either it's off by 1 or 2 or 3 or 4---we get four non-trivial cosets. This is why $\mathbb{Z}/5\mathbb{Z}$ is a group of order five: there's exactly one way to be a member of $5\mathbb{Z}$ but four ways to not be.
You probably noticed that the difference between any two integers in $[1]=\{\ldots,-9,-4,1,6,7,\ldots\}$ is 5. And the same observation holds for any two integers in $[2]$, and any two integers in $[3]$, and so on. This notion of "taking the difference" is precisely how we determine which group elements belong in which coset. Two elements are thought of as "the same" whenever their difference lies in the normal subgroup that we're modding out by.

This is exactly what's going on when your textbook says something like, "Suppose $N$ is normal in $G$ and let $a,b\in G$. Then two cosets $aN$ and $bN$ are equal if and only if $b^{-1}a\in N$." Notice that if $G$ is abelian, then we write $b^{-1}a$ as $-b+a$, i.e. "$b$ inverse plus $a$." But when $G$ is non-abelian, we replace "plus" by "times" to obtain $b^{-1}a$. This is the multiplicative version of a difference.

Next week we'll close out this mini-series by taking an intuitive look at a few more quotient groups:
  • $G/\ker\phi$ where $\phi:G\to H$ is any group homomorphism
  • $GL_n(F)/SL_n(F)$ where $F$ is a field, $GL_n(F)$ is the general linear group, and $SL_n(F)$ is the special linear group
  • $G/Z(G)$ where $G$ is any group and $Z(G)$ is its center