 So in order to turn our set of cosets into a real group, we have to introduce one more important concept which is that of a normal subgroup. And we can view this as the following. Our goal is to fail to produce a group and see what we have to do in order to succeed. So to recap our cosets, 0, 1 and so on and so forth up to n minus 1 of nc, they do actually form a group when we define the sum of the cosets by the sum of the coset representatives. And the question at hand is can we do this for other groups? And so if we look at our process, what we did is we started out by finding a subgroup of h. We found the cosets, we defined a binary operation on the cosets, we proved that this binary operation was well defined, and then we verified that our group properties actually held. So we often learn more when we fail to do something than when we succeed, so let's see if we can make this process of producing a group crash. So let's go step by step. Form a subgroup, well, that seems to be pretty error-proof. It seems like we can always find subgroups. So we're probably not going to crash at that point. We can always find cosets. Once you have a subgroup, you can always find the cosets. So again, we're not likely to cause that process to crash at this step. How about that coset multiplication? So given two cosets where I'm going to actually start writing them in the form that we use for cosets a, h, b, h. Well, if I want to multiply these two cosets together, I can always define this in terms of what I do to the elements of the group that produce the coset. And again here we're distinguishing between this symbol which indicates the operation on the cosets and this symbol which indicates the operation in the group. And I can always define coset multiplication that way. So that doesn't seem like that's going to cause us to crash. And after that we proved the operation was well-defined and this is where we might run into a point of failure. No problem, no problem, no problem. This is the first place we may actually run into problems. So the first point of failure might be that our coset multiplication is not well-defined. So let's see if we can investigate and see when this might happen. So let g be a group with binary operation plus and let h be a subgroup. We'll define our coset multiplication. If I take two cosets and try to multiply them, what I'm going to do is I'm going to take the group elements that produce the cosets and apply the group operation to them. And in order for coset multiplication to be well-defined it's necessary that whenever I have two cosets equal that the results of the coset multiplication are going to be equal. Now as long as we remember that coset operation is different from the group operation we actually don't need these operator symbols. We can just drop them out so instead of a h coset times b h equals a group plus b h. I can just write a h b h equals a b h and a prime b h and so on. But it's important to remember that this is a group operation, this is the coset operation and they are different. And again our goal is to break this so that we see what we actually need to ensure. And so that means we want to make sure that coset multiplication is not well-defined, that I should be able to take two identical cosets but get a different product when I use the representatives that are different. So now that we know how we can fail to produce a group let's see how we can avoid it. So I have some subgroup h and I have cosets a h b h and so on. And I'm going to let a prime h and a h b be the same coset. And I know that a prime has to be a h k so that's one of our coset lemmas. And if you don't remember that you should prove it. And I want to make sure that a prime b h is the same as a b h because these two cosets are the same. I want to make sure the product of these two cosets is the same as the product of these two cosets. So I'll start with what I want, product equals product and we'll work our way back. So again if two cosets are equal I know the thing that generates the coset has to be the element of the other coset. Which means that this has to be a b times something. Now I know that a prime equals a h k so I can drop that in h k b equals a b h j. And so now I have this and because this is a group I can multiply by the inverse of a that drops out h k b equals b h j. And this seems to be a good place to start. If I have this then I get this. So it seems that coset multiplication will be well defined provided that for any b I can satisfy this relation h k b equals b h j. Well that's a little complicated because I don't know what h k and h j are. On the other hand I'm not really concerned with what they are I just need to know that they exist. So we could simply require that such values must exist and here's the catch for any value of b in the groups. These things have to exist regardless of what b actually is. And the easiest way we can do that is to just require that b h and h b, this left coset and the corresponding right coset, we can require that those have to be the same thing. And that seems to be the requirement so we now have the following definition. Let h be any subgroup of g. We say that h is a normal subgroup if for any element of g the left and right cosets are the same. Now it should be easy to prove the following which is to say you should actually try and prove them yourself. All subgroups of an abelian group are normal and likewise if h is a normal subgroup then h and its cosets form a group under coset multiplication. Now this particular group because again this coset formation is a lot like finding the quotient g divided by h, we call that the factor group or the quotient group. Well we're in an interesting position now. If a subgroup h of g is normal we can form the factor group g over h. We say that normality is a sufficient condition. If we have it we can guarantee the existence of the factor group. But a mathematician might ask can we make do with less? I know we can get this if we have a normal subgroup. Can we do anything like this if we don't have a normal subgroup? And so we're looking for a weaker condition. Well let's see if we actually need normality. Since every subgroup of an abelian group is normal we have to consider the non-abelian groups and so the simplest one we know of is p3 and so is coset multiplication well defined? Well let's find a non-normal subgroup and after some effort we find this one consisting of e and the transposition ab. Turns out that's not normal. And it has cosets abc times s and acv times s. So here's the cosets and since s itself is eab our group p3 has been partitioned into one, two, three cosets. So let's see what happens. We'll try coset multiplication and let's take the product of these two, this with itself. And so our definition of coset multiplication will multiply the representatives abc times acb and that gives us the set s once again. On the other hand in this set abcs is this element ac and I can use any element of a coset as a representative. So abcs is the same as acs and likewise in acbs is this element bc and I can use that as my coset representative. And so this product and this product these are the same cosets but when I do the product I get ac times vs and so what that's going to give me is acbs and these are different and so that says that my coset multiplication is not well-defined if my subgroup is not normal. And if I put that together what that tells us is that normality is enough for the formation of factor groups it's also necessary and we say that normality is a necessary and sufficient condition it is the weakest possible condition that will allow us to form a factor group.