Looking for Explanation of a Quotient Group

So I have an elementary understanding of group theory and the goals of studying sets and operations on them but I noticed that along the way, my understanding of a quotient group is severely flawed.

I understand what a so called normal subgroup is, but could someone please give an in depth explanation of what a quotient group is and the corresponding homomorphism properties?

Thanks so much!!

also an explanation of a coset would be much appreciated! I think that is where my understanding breaks down.

HallsofIvy
Homework Helper
How can you "understand what a normal subgroup is" if you do not understand what cosets are?

Let G be a group with subgroup H. If a is a member of the group, then, since H contains the identity, it is contained in the set of "a times all members of H". That is a "left coset" if a is multiplied on the left, a "right coset" if a is multiplied on the right.

For example, suppose G is the rotation group on 4 elements, (e, a, b, c) with e the identity. Let H be the subgroup (e, b). The coset aH (a left coset since a is on the left) is {ae= a, ab= c}. bH= {be= b, bb= e}, cH= {ce= c, cb= a} and, of course, eH= {ee= e, eb= b}. That is, the left cosets are just {e, b}= H itself, and {a, c}. There are two cosets because G has |G|= 4 elements and H has |H|= 2. The number of cosets is |G|/|H|= 4/2= 2. Since G is commutative, of course the right cosets are identical.

If the group is NOT commutative, then the left and right cosets are not necessarily the same. If they are then we say that H is a "normal" subgroup of G. Obviously, every subgroup of a commutative group is normal.

The left (or right) cosets "partition" a group- every element is in one and only one coset. Now, we define an operation on those cosets: to "multiply" two cosets, choose a "representative" of each coset- that is, an arbitrary member of and multiply them. Their product lies in the group and so in one coset. We identify the product with that coset. For example, the cosets above were E= {e, b} and A= {a, c}. To multiply E with itself, we choose either of the elements of E. Here, since E contains only two members, it is easy to show all possibilities: We might choose e both times: ee= e which is in E. We might choose e the first time and b the second: eb= b which is in E. We might choose b the first time and e the second: be= b which is in E. Finally, we might choose b both times: bb= e which is in E. That is EE= E.
For EA, we might, say, choose e and a: ea= a which is in A. Or we might choose b and c: bc= a which is in A. In fact no matter what "representatives" we choose from E and A, the product is in A: EA= A.

For AA we might choose aa= b, ac= e, ca= e, or cc=- b. In any case, we got a member of E: AA= E. That is, those two subsets, with this operation, form a group of order 2.

For an arbitrary group and subgroup, it is NOT necessarily true that using different representatives of the same coset will always result in a product in the same coset. One can prove that will happen if an only the right and left cosets are the same- that the subgroup is a normal subgroup. In that case we can alway produce a new group in which the member are the cosets and the operation is this "representative" multiplication. That new group is the "quotient" group, typically denoted G/H. The name and representation as a fraction is just because its order is |G|/|H|.

Deveno
if you think of a group as being like a big "blob", made out of its elements (particles), then a subgroup is like a "chunk" containing containing the identity element (which lies in the very middle of the blob). the idea is, can we make a smaller group out of bigger chunks than the particles?

and those chunks are what cosets are. we name the chunks by picking out a certain element, and calling it a representative of that coset. but this is somewhat arbitrary, a coset is a chunk, and might have several equivalent names.

now our blob isn't amorphous, it has an internal structure (like crystals do), a certain regularity imposed on it by the fact that we have a multiplication. we can only make chunks that are the same size as a subgroup.

abelian groups are the nicest, because we can go back and forth without paying attention to which "direction" came first (multiplication is like saying: start at point a, and then use point b as an instruction to travel to point ab). in fact, abelian groups are "almost" like vector spaces, they're just not "big enough" usually to have the full force of geometry.

for example, the integers, can be thought of as lying on a line. what happens when we "mod out nZ (the multiples of n)"? well, we make all the multiples of n indistinguishable. so all these "dots" (spaced every n dots apart), all contract to a single point (the "origin", or "0").

and instead of a line, now we have dots arranged in a circle, with n stacked on top of 0, n+1 stacked on top of 1, n+2 stacked on top of 2, and so forth (the line "wraps around the circle" infinitlely many times, so each "dot" is now a coset composed of an infinite number of dots). so it's no longer important what each number sitting on a dot used to be, the only thing that matters now is "how far from 0 it is".

we actually have something that acts like this: a clock. for this reason, integers mod n, are sometimes called "clock numbers". the circular nature of this group gives rise to the name cyclic.

now, with a general group, G, and a subgroup H, we try to mimic this behavior of the integers mod n, as much as we can. we want the cosets, or translates of H by g, gH, to act like k + nZ (move the 0-dot (clump) by k). moreover, it would be nice if the "clumps" had a group structure (thus replacing a complicated group structure with a simpler one, which might be easier to deal with).

so we would like (xH)(yH) = xyH. that would make everything nice and simple for us.

well, let's see what goes wrong:

we pick an element xh in xH, and an element yh' in yH, and multiply them:

xhyh' = xy....? hmm, the trouble is, that hy might not be equal to anything like yh". the left coset of H that contains y might not be the same as the right coset that contains y. this is a problem, it stops us dead in our tracks. and since there are LOTS of non-abelian groups, this is bound to come up over and over again.

what to do? well, if H is a subgroup that "kinda commutes" with everything, in the sense that yH = Hy for any y in G, that will fix everything. then hy = yh", and we can continue:

xhyh' = x(hy)h' = x(yh")h' = xy(h"h')...success! in coset form:

xHyH = x(Hy)H = x(yH)H = xy(HH) = xyH (since for any subgroup, HH = H).

personally, i think "quotient group" is a bad name, because nothing like "division" is going on. so what G/H really means, is G, in H-sized bites. and (mostly because we have non-abelian groups, which are the troublemakers of group-land), H has to be a special kind of group, called normal, or else the "clumps" don't behave well.

mathwonk