What's the motivation for coset and quotient structure?

[kof9595995]

I'm curious why people develop these objects. Although I've seen some proofs of theorems using coset(or quotient space somtimes), it remains mysterious to me how people come up with these in the first palce. So what's the motivation for inventing coset or quotient space, logical and historical?

 

[micromass]

Hi kof9595995!

I suppose the prime reason to come up with quotient structures is to lose information about the group to make problems easier. The idea with quotient groups is to identify information.

Let me start with an example. One of the oldest problems in mathematics is solving Diophantine equations. One such a problem gives us numbers m and n and asks us to find integers x and y such that xm+yn=1. Now, why should such integers x and y even exist? Well, they don't, at least, they don't always exist.

Something we can do is to observe that yn is divisible by n. Thus if a solution should exist, then xm-1 is divisible by n, thus if we divide xm by n, we are left with remainder -1. This is becoming fairly complicated, so the trick to do is to identify all multiples of n with each other. So we get

$$...=-2n=-n=0=n=2n=3n=4n=...$$

saying that xm-1 is divisible by n is then just saying that xm-1=0 (mod n) (=saying mod n indicated that we did this identification). Thus xm=1 (mod n) and we see now that we need to find an inverse of m (mod n). It appears that such an inverse only exists if m and n are coprime, but the point is that we only knew this by making the identification n=0.

In general, when given a group G, it might contain too much information. So we just identify information to make working with the things easier.

Another example are complex numbers. When forming complex numbers, we "adjoin an element i with i²=1". What we actually do is form the polynomial ring $\mathbb{R}[X]$ and identifying X² with 1. So we form the quotient $\mathbb{R}[X]/(X^2+1)$.

 

[Deveno]

the same thing happens in two other seemingly different settings.

imagine you have a line in the plane, and you want to describe some other line parallel to it. well, all you need to do is say "how far you shift it over". one way of specifying this, is to pick some other line NOT parallel to the first line, and just say how far along the 2nd line you go.

in other words, instead of thinking of the plane as "two dimensions of space", we can pick a one-dimensional subset, and divide the plane into "cosets" of this line (parallel lines to our original line). this set of parallel lines "acts" just like a line itself.

in technical terms, if V is a vector space, with a subspace W, we can decompose V into a direct sum: V is "isomorphic" to W⊕(V/W). we can pick a point in V, by first specifying a point in W, and then specifying "which copy" of W in V it lies in. we can do this in any dimension: in 3 dimensions, think of a plane, and then the cosets are parallel planes stacked on the original plane like a deck of cards. or if we have a line, the cosets are parellel lines (like a stack of (really, really thin) straws), if we pick a point on an intersecting plane, it is like saying "where" on "which" straw we're at.

in topology, you have a similar phenomenon. up close, a square and a cylinder seem the same (if you were living on a large cylinder, you might have no idea you were). however, the square has 4 edges, and the cylinder only has 2 (which never meet). we can model the cylinder by "modding out" the extra edges of a square (of great practical value when it comes to making maps). in fact, methods very similar to this are actually used by people who DO make maps. paper is flat, Earth is not. so you can imagine a globe as a quotient of a flat map of the world, we "mod out the edges".

even more basic, we can talk about doing the same thing just with sets. we can define an equivalence relation on a set, and call everything in an equivalence class "the same" (equivalent). we often do this intuitively when it comes to financial transactions. your five dollars aren't exactly the same dollars as my five dollars, but they are "equivalent".

in all of this, the basic principle is the same: "forget" some of the information, so that we can focus on what remains.

 

[kof9595995]

the same thing happens in two other seemingly different settings.

imagine you have a line in the plane, and you want to describe some other line parallel to it. well, all you need to do is say "how far you shift it over". one way of specifying this, is to pick some other line NOT parallel to the first line, and just say how far along the 2nd line you go.

in other words, instead of thinking of the plane as "two dimensions of space", we can pick a one-dimensional subset, and divide the plane into "cosets" of this line (parallel lines to our original line). this set of parallel lines "acts" just like a line itself.

in technical terms, if V is a vector space, with a subspace W, we can decompose V into a direct sum: V is "isomorphic" to W⊕(V/W). we can pick a point in V, by first specifying a point in W, and then specifying "which copy" of W in V it lies in. we can do this in any dimension: in 3 dimensions, think of a plane, and then the cosets are parallel planes stacked on the original plane like a deck of cards. or if we have a line, the cosets are parellel lines (like a stack of (really, really thin) straws), if we pick a point on an intersecting plane, it is like saying "where" on "which" straw we're at.

Then in this example, what are we trying to "forget"?

 

[Kreizhn]

These guys have all obviously done a great job of explaining it to you, but it may be worth taking a look at the universal property of quotients. ]

Essentially, let's look at A/~. What we've done here is define an equivalence relation on A via ~. Then the quotient A/~ is the set of equivalence classes of A under ~. In the case of groups for example, cosets define equivalence classes. If G is a group, there is a bijection between the set of equivalence relations satisfying $a \sim b \Leftrightarrow a^{-1}b \in H$, H a subgroup of G, and the set of subgroups of G. In such a case, the equivalence classes are given by the (say left) cosets of H.

Now of course in groups, we require that left- and right cosets agree resulting in requiring normal groups for quotients. If you would like a really good example of how cosets and quotient groups can be used, take a look at this http://link.aps.org/doi/10.1103/PhysRevA.63.032308"

Intuitively, the author wants to show that the time-optimal trajectory in a system that drifts in one direction but can travel infinitely fast in other directions is parameterized in terms of the drift alone. This is done by creating cosets of the spaces on which we can travel infinitely fast (that is, we define an equivalence relation in that a ~ b if the infimum of the time to get from a to b is zero). The author then quotients this space, and shows that the time-optimal trajectory depends only on the drift.

Be careful reading this though. This works with Lie groups, so that while we can take cosets, the quotient by subgroups is only a differentiable manifold, not a Lie group (since the subgroups are not normal).

 

[Deveno]

Then in this example, what are we trying to "forget"?

when we consider the plane made up of "cosets of lines", we are forgetting an entire dimension. in other words, we're not concern about points on a given parallel line, they are all considered "equivalent". we just want to know "which" line we're on. this is like "removing a variable" from a (linear) system, by considering it constant.

 

[kof9595995]

I see, I think I understand it better now, thank you all.