# The ba­sic idea

Let’s say we have a group. Maybe the group is kinda large and un­wieldy, and we want to find an eas­ier way to think about it. Or maybe we just want to fo­cus on a cer­tain as­pect of the group. Some of the ac­tions will change things in ways we just don’t re­ally care about, or don’t mind ig­nor­ing for now. So let’s cre­ate a group ho­mo­mor­phism that will map all these ac­tions to the iden­tity ac­tion in a new group. The image of this ho­mo­mor­phism will be a group much like the first, ex­cept that it will ig­nore all the effects that come from those ac­tions that we’re ig­nor­ing—just what we wanted! This new group is called the quo­tient group.

# Definition

We start with our group $$G$$. The ac­tions we want to ig­nore form a group $$N$$, which must be a nor­mal sub­group of $$G$$. The quo­tient group is then called $$G/N$$, and has a canon­i­cal ho­mo­mor­phism $$\phi: G \rightarrow G/N$$ which maps $$g \in G$$ to the coset $$gN$$.

## The di­vi­sor group

In the defi­ni­tion, we re­quire the di­vi­sor $$N$$, to be a nor­mal sub­group of $$G$$. Why? Well first, let’s see why re­quiring $$N$$ to be a group makes sense. Re­mem­ber that $$N$$ has the ac­tions whose effects we want to ig­nore. So it makes sense that it should con­tain the iden­tity ac­tion, which has no effect. It also is rea­son­able that it would be closed un­der the group op­er­a­tion—do­ing two things we don’t care about shouldn’t change any­thing we care about. To­gether, these two prop­er­ties im­ply it is a sub­group: $$N \le G$$.

A sub­group is great, but it isn’t quite good enough by it­self to work here. That’s be­cause we want the quo­tient group to pre­serve the over­all struc­ture of the group, i.e. it should pre­serve the group mul­ti­pli­ca­tion. In other words, there needs to be a group ho­mo­mor­phism $$\phi$$ from $$G$$ to $$G/N$$. Since $$N$$ is the sub­group of things we want to ig­nore, all its ac­tions should get mapped to the iden­tity ac­tion un­der this ho­mo­mor­phism. That means it’s the ker­nel of the ho­mo­mor­phism $$\phi$$, which means it’s a nor­mal sub­group: $$N \trianglelefteq G$$.

## Cosets

What ex­actly are the el­e­ments of the new group? They are equiv­alence classes of ac­tions, the sets $$gN = \{gn : n \in N\}$$ where $$g \in G$$, also known as a coset. The iden­tity el­e­ment is the set $$N$$ it­self. Mul­ti­pli­ca­tion is defined by $$g_1N \cdot g_2N = (g_1g_2)N$$.

# Gen­er­al­izes the idea of a quotient

What gives a quo­tient group the right to call it­self a quo­tient? If $$G$$ and $$N$$ both have finite or­der, then $$|G/N| = |G|/|N|$$, which can be proved by the fact that $$G/N$$ con­sists of the cosets of $$N$$ in $$G$$, and that these cosets are the same size, and par­ti­tion $$G$$.

# Example

Sup­pose you have a col­lec­tion of ob­jects, and you need to split them into two equal groups. So you are try­ing to de­ter­mine un­der what cir­cum­stances chang­ing the num­ber of ob­jects will af­fect this prop­erty. You no­tice that chang­ing the size of the col­lec­tion by cer­tain num­bers such as 0, 2, 4, 24, and −6 doesn’t af­fect this prop­erty.

The set of differ­ent size changes can be mod­eled as the ad­di­tive group of in­te­gers $$\mathbb Z$$. The changes that don’t af­fect this prop­erty also form a group: $$2\mathbb Z = \{2n : n\in \mathbb Z\}$$. Ex­er­cise: ver­ify that this is a nor­mal sub­group of $$\mathbb Z$$.

This sub­group gives us two cosets: $$0 + 2\mathbb Z$$ and $$1 + 2\mathbb Z$$ (re­mem­ber that $$+$$ is the group op­er­a­tion in this ex­am­ple), which are the el­e­ments of our quo­tient group. We will give them their con­ven­tional names: $$\text{even}$$ and $$\text{odd}$$, and we can ap­ply the coset mul­ti­pli­ca­tion rule to see that $$\text{even}+ \text{even} = \text{even}$$, $$\text{even} + \text{odd} = \text{odd}$$, and $$\text{odd} + \text{odd} = \text{even}$$.

In­stead of think­ing about spe­cific num­bers, and how they will change our abil­ity to split our col­lec­tion of ob­jects into two equal groups, we now have re­duced the prob­lem to its essence. Only the par­ity mat­ters, and it fol­lows the sim­ple rules of the quo­tient group we dis­cov­ered.