# What's a Quotient Group, Really? Part 2

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

*how*#1 or #2 is satisfied.

*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,

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

*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

*in the same way*:

- All the folks in $\{\ldots,-9,-4,1,6,11,\ldots\}$
*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 $[2]=2+5\mathbb{Z}$. - 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 $[4]=4+5\mathbb{Z}$.

*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.

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