 One of the more challenging aspects of mathematics is trying to figure out a way of representing something. And so with groups, one of the ways we do that is we represent groups using a multiplication table. So this is a fairly common way of representing small groups. It's not quite as useful once our groups have like thousands of elements, but we can construct these multiplication tables. So for example, let's say I have a group with three elements and it has to have an identity, I'll call that E, and I have two other elements, A and B, and because it has three elements, A and B have to actually be different elements. That kind of sounds obvious, but as we'll see, that's a little bit tricky. So we can represent the group by giving the table of all possible products of the elements. So what I have is I'll set up my multiplication table, I have my three elements, E, A and B, E, A and B, and I'm going to set up this table for this product, P times Q. Now remember, it's important to keep in mind that groups are not in general commutative. The group operation is not going to be commutative. So we have to read our table carefully. This is for P times Q. So our first factor is specified by the row. Our second factor is specified by the column. So one important idea here is that because E, A and B are supposed to form a group with three elements with identity E, the entries in this table can't be completely arbitrary. They are constrained by the requirement that we are dealing with a group. And what this does is this means that our multiplication table in many ways is going to be set by the fact that we have a group. So let's tackle this as a problem. Let G be a group with three elements, E, our identity, A and B, and let's produce a multiplication table for the group. So our multiplication table looks like that. And remember, our assumption is that E is the identity. So E times E has to be E. We also know that, again, because E is the identity, A times E has to be A, and B times E has to be B. So we can fill in those entries as well. And, again, E is the identity. So we know that E times A has to be A, E times B has to be B. So we can fill in those entries as well. And so note that the column with the multiplication by E and the row, which is E times any element of the group, those values are predetermined by the fact that E is the identity of the group. So we have quite a bit of the table filled in just from the fact that we have a group. Now, what about this entry here? This is A times A. Well, let's think about that. There are three possibilities for what A times A is. A times A might be A, it might be B, or it might be E. And these are the three possibilities because these are the three elements of the group. So how do we decide which one it is? And, well, here's a way of thinking about it. We have three buses in front of us. And to decide which one we want to use, well, let's get on a bus and see where it takes us. And if we don't like where the bus takes us, we'll go back and not get on that bus. So let's take a look at the bus labeled A times A equals A. Well, if we get on that bus, then A to the third also has to equal A. You might want to think about why that's true. And in general, A to the power N has to equal A. But we know there is a value K where A to the power K has to be the identity element. And so that says, well, A to the N is A. Well, that says that A and E have to be the same thing. But the identity of a group is unique. And so A can't be equal to E, which means that A has to be E, but A can't be equal to E. This bus has a flat tire and is going to get us nowhere. So we don't want to get on this bus. We don't want to get on the bus that says A times A equals A. Well, what about the A times A equals B bus? Well, let's get on that bus and see where it takes us. Now, since G is a group, A needs to have an inverse, which means that A times something has to give you the identity. Well, A times E isn't it. A times A isn't it. That means A times B has to be the identity. So we can fill in that entry. And because the inverse commutes with the entry, I know that A times B has to give you E. That means that B times A also has to give you E, so I can fill that in. How about B times B? Well, let's think about that again. B times B can't be B for the same reason that A times A can't be A. B times B can't be E because, well, B times A was the identity, which means that A is the inverse of B. And so B can't be the inverse. So B times B has to be A. That's our only choice. And so if we get on the A times A equals B bus, we can complete our multiplication table for the group. Now, to think like a mathematician, don't stop with just one solution. In this particular case, we found this multiplication table by boarding the A times A equals B bus, and we're able to fill out the multiplication table from that point. But there was another possibility. There was a third bus waiting at the station for us, which was the A times A equals E bus. Now, what happens if we get on that bus? Now, the first thing to notice here is that because A times A gives us the identity, then A is its own inverse. A inverse is the same as A. So we need to find what A times B is. And so, again, let's go through our possibilities. A times B, well, it can't be the identity because then B would be A inverse. You should be able to explain why that's the case. And the inverse has to be unique. A times B can't be B either because this would make A the identity. Again, you should be able to prove that. And the identity is unique. And finally, A times B can't be A because then that would make B the identity. And again, the identity is unique. And so, A times B can't be E, can't be B, can't be A. Well, there's nothing else it could be. There's only three elements in the group. And what that means is there's no possible value for the product A times B. If we get on board this bus, it almost immediately has a flat tire. And we want to stay off this bus because we'll be able to get nowhere. What that means is the only possible bus that works, the only possible product is A times A equals B. And this is the only possible multiplication table for a group with three elements.