 So, now that we have the cosets forming a set of elements that we can compare, we can try and define a binary operation on these cosets and see if we can form a group. So, let's start off again with our subgroup nz, and again, this subgroup nz and its cosets form a quotient thing, with that we describe as z over divide by nz, and we can turn this into a group if we can define a binary operation, where I can take two of the cosets and form it into a third coset, and again the other thing we have to do is just defining the binary operation is not enough, we do have to make sure that we have the other group requirements. So, let's think about that. These cosets are represented in terms of p, q, and r. All of these things are elements of the set of integers. So, I might try to define the coset addition in terms of the addition of p and q. And one caution here, one thing we should be careful with here, is that when we write plus here, it actually means the group operation on the set of cosets, that is to say in z divide by nz. On the other hand, when I write the plus here, p and q are both integers, so this is ordinary integer addition. Now, one way we do have to distinguish between the two is we can introduce our special symbol circle plus to represent the binary operation in our set of cosets, so I'll replace those, and we can leave plus to indicate the addition of integers. Now, so let's define our coset addition as follows. Take a coset, take a coset, circle plus them together. What you get is the coset that is represented by the sum of the representatives. And so, we've defined our binary operation, well, not quite. So remember, anytime we try to define a binary operation on equivalence classes, we do have to worry that we have something that is not well defined. So, I've defined coset plus coset to equal this, but what happens if I have a different representative for the same coset? Well, then first of all, again, when I do that coset sum, I'm going to be adding the coset representatives, and the thing I have to worry about is because P and P prime are different than P prime plus Q is different from P plus Q. So, conceivably, this coset sum might be different from this coset sum, and so we want to make sure they're the same because I want to make sure that the sum does not depend on the individual representative. So, we have to check to see if our coset sum is well defined, and we'll go ahead and do that. So, let's see. We'll try and show that this is going to be true as long as P and P prime represent the same coset. So, suppose P and P prime 2 represent the same coset. As we've done before, we're going to start with what we want and see if we can get there. We want these two to be the same things. Let's go ahead and write that down as our goal, what we have. Let's see. So, there's a definition of coset sum. Now, since these two cosets are the same thing, then the difference is a multiple of n. Remember that these cosets are drawn from the subgroup nz, where z is the set of integers and is some defined number. So, two cosets are equivalent if their difference is a multiple of n. So, that gives me some information. I know that the difference is n times something. And let's see. Well, also down here, I know that if these two cosets are the same, then their difference also has to be a multiple of n. And I do a little bit of algebra. I do a little bit more algebra. And yeah, it seems like I can get from here to here without any difficulty. So, I can start with the two cosets are the same. The difference is a multiple of n. And that gets me here. And it seems like I can go all the way to the end. And we may want to add a couple of extra notes just to make sure that we can actually take the steps that we've undertaken. So, I know the difference is a multiple of n. That's from our lemma. That's just algebra in our set of integers. All I've done is I've replaced the a with the b, and I'm allowed to do that. A little bit more algebra. Again, everything inside is an integer. So, I'm allowed to do algebra, more algebra. Again, if two things differ by a multiple of n, then they are going to represent the same coset. That's again from our lemma. And our definition of circle plus is straight out of here. And so, here's our proof that we have a operation that is well defined, except the mathematician inside your head should be asking, do we need to prove that this is true when I change the second term? The other mathematician in your head should answer, yes, absolutely we need to prove it. Now, you might get the impression that mathematicians, their heads are a little bit crowded with other mathematicians talking to them. And, well, that's true. Well, for me anyway. I don't always listen to the voices in my head. But if it's a mathematician asking, I will try to answer it. Well, so now I want to prove that this and this are equal. And so I could follow the pattern of the previous proof. I could just go through all of these steps once again, and then prove that coset addition is, in fact, well defined. But it's often useful to find a different way of proving the same thing. So again, I want to prove that P plus Q prime is the same as P plus Q. And again, I have to take this step backwards because I can't really do anything with it. And, well, here's something interesting. P and Q are and Q prime. These are all integers. And one of the things I know is that the addition, which is what this symbol is, commutes. I don't know whether this is a commutative operation because this is coset addition and I have no guarantee that it works. But this is just the addition of ordinary integers. And I know I have commutativity. So I can reverse these two. I can reverse these two and I can disassemble. I can deconstruct my coset addition definition. This is Q prime plus P equals Q plus P. And that's exactly what I proved in the previous theorem, previous result. So this is by the previous result. If I change the first term to something that's the same, if I change the first representative, it doesn't change anything. I still get the same sum and then everything else follows through by definition or algebra or by the definition of coset addition. Now, why did we go through all this trouble? Well, because plus and Z is commutative, the key step was interchanging the sums here. But because plus and Z is commutative and because circle plus is the sum in Z by definition, I know that I can reverse the sum and I can get the reversal of the coset addition. This little proof here suggested we could actually do this because we made use of that commutativity in Z. And so by looking for a different proof of a result, we found something new, which is that circle plus is also a commutative operation. And this leads us to the following result. When we put everything together, the cosets of Z divided by NZ, where we define coset addition, this way, we get an abelian group out of it. Now, you should prove that all of the group properties are in fact present, that we have the identity inverse and so on and so forth. We'll leave that as an exercise, but here's the bigger question. Can we generalize this process to groups other than the group of integers? Can we go further and make similar quotient groups from subgroups? And we'll take a look at that next.