 Well, what else can we do when we find the multiplication table for a group? Well, let's consider a different case. It's helpful to have the following uniqueness lemma. Let G be a group and A, B, C, B elements in G. And first of all, remember we already proved the identity element E is unique. The inverse is also unique. If A, B equals A, C, if I have two products that are equal, then B is equal to C. That's sometimes referred to as left cancellation because we're dropping this term on the left. Because groups aren't necessarily commutative, we have to consider the product in the other order. If B, A equals C, A, then again B equals C. And again, because I'm dropping the factor on the right, sometimes this is called right cancellation. And because this is a lemma, you should be able to prove all four of these things. They're very important group properties. It's also helpful to have another idea. And here's how this one comes about. If it swims like a duck, wax like a duck, and flies like a duck, it's probably a duck. And what that means is that we can go on to what we might call the duck lemma. I suppose I have a group with identity E and I have two elements in that group. Then if A, B equals A, if A times B is equal to A, well, A, sorry, B, looks a lot like the identity element. Well, it turns out it is. And again, since we're not guaranteed commutativity, we have to consider the product in the reverse order. If B, A is E, then B, again, looks like the identity element. And one more, if A times B is E, then B has to be A inverse. If B looks like the inverse, it actually is the inverse. And again, because commutativity isn't guaranteed, if I left multiply by B and get the identity, then B has to be the inverse again. And again, these are things you should be able to prove relatively easily. So let's book a table for four. So we want to find the multiplication table for a group with four elements, and we'll call the identity E and we'll name the elements A, B, and C. So first of all, because E is the identity, we can start by filling in the column and row that corresponds to multiplication by the identity and multiplication of the identity with something else. So we can fill in that first row and column. So the next thing we want to do is find what A times A is. And let's go through our possibilities. There are four of them because we have four elements of the group. A times A could be the identity element. A times A could be A, well actually we know that's not true. Make sure you understand why that can't be possible. Make sure you're able to prove that that can't be possible. A times A might be B and A times A might be C. And what that means is there's three buses we can board. We can board the A times A equals E bus. We can board the A times A equals B bus. Or we could board the A times A equals C bus. Now it's possible one or more of these buses may have a flat tire, but we won't know until we actually get on the bus. So let's pick one and see where it takes us. So suppose A times A equals E. So we'll get on the bus labeled A times A equals E. And let's see if we can fill out the rest of the table. So let's go ahead and list those possibilities for A times B. So A times B could be E, A, B, or C. Well, A times B equals E, well that means B is A inverse, but I already know what the inverse is. The inverse is unique, so this first product is not a possibility. A times B equals A, well then A, then B looks a lot like the identity element because here's A times something giving you A again. So the identity is unique, so we can't have that. And for the same reason A times B equals B is not a possibility. Because that would make A the identity element. And that means that A times B is the only possibility here. A times B has to be equal to C. So A times B has to be C, we'll fill that in. And now let's consider A times C. Well again, we'll go through our possibilities. A times C could be E, well it can't be because I've already have A inverse. A times C could be A, well that makes C the identity and we can't have that. A times C equal to B, well I see no obvious reason why that can't be, so let's hold on to that. A times C equal to C, again that makes A look an awful lot like the identity element and we can't have that. So that says A times C equals B is our only possibility for that product, so I'll fill that in. Now we'll move on to the next row. Well this is B times A. Well I know B times A equals A times B because, well no, we don't actually know that we cannot assume commutativity. B times A equals A times B, we cannot assume that these two things are equal because we cannot assume commutativity. We have to find out what it is through some other means. So let's check the bus schedule. B times A equals E, well that makes B and A inverses of each other, but I already know what A inverse is. A times A gives you E, so A is A inverse, it can't be B, so that doesn't work. B times A equals A, well that makes B look like the identity element, can't have that. B times A equals B, that makes A look like the identity element, can't have that. B times A equals C, there's no obvious reason why that's wrong, so we can take that as our only allowable choice. B times A has to be equal to C because all the other possibilities are prohibited. For B times B, well let's see. Well again we'll go through our possibilities. B times B could be E, no reason why that can't be. B times B could be A, yeah that's a possibility as well. B times B equals B, well we know that can't be the case because B is not the identity. B times B equals C, yeah maybe that works too. Now one thing to realize here is we've just reached the end of the A times A equal E line, so we assumed A times A equals E and that got us all of these entries, but now we don't know what this is, there's three possibilities for it, and so the idea is that this bus has taken us to another station and now there's one, two, three possible buses and we can get on any one of those buses and see where it takes us. Well let's pick one. Let's pick the B times B equals E bus, so we get on this bus and again we need to find B times C and again we'll go through our possibilities. B times C could be E, well again that makes B and C inverses, but I already know what the inverse of B is, so that won't work. B times C equals A, no obvious reason why we can't have that, so that's a possibility. B times C equals B, well that makes C the identity, nope. B times C equals C, well that makes B the identity, nope. So that means B times C has to be A. How about C times A, well again C times A is the identity, but I already know what the inverse of A is. A times A gives you E, C times A can't also give you the identity. C times A equals A, nope. C times A equals B, maybe. C times A equals C, nope. And so C times A equals B is again our only possibility. C times B, well again C times B equals the identity, well that makes C equal to B inverse, but we already know what B inverses. B times B gives us the identity, so C can't be the inverse, so that doesn't work. C times B equals A, maybe. C times B equals B, nope. C times B equals C, nope. And so C times B equal to A is again our only possibility. And finally, because G is supposed to be a group, C times something has to give you the identity, which means that this product here has to give us E. And this completes our multiplication table where we boarded the A times A equals E bus and got to here. We then boarded the B times B equal to E bus and got to here. So there's our multiplication table where we made these two assumptions. If you want to think like a mathematician, the question to ask is, well, what if I made different choices? What if I got on different buses? Would I get a different multiplication table? And we'll look at that.