Quotient group

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.

See also

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.