 So these product groups are really, it's one of the easier concepts I think that we've dealt with. So just to cement, just our easier ideas, but also to cement understanding, we've got to understand these things. Let's just look at an example and what I'm going to do is the product group with two cyclic groups. And the cyclic groups are both going to be the same and they're going to be the cyclic groups in two elements. So let's just look at C2 and that would be this set of only two elements and let's use the symbols one and two. Remember, we're not talking numbers here, these are just symbols and they have this group operation which is this permutation. What are the two permutations? Well, if we look at the two permutations, it's just one goes to one and two goes to two. That's the identity element and then the second one is one goes to two and two goes to one and that would be the second one. Let's give these two permutations just so there's no confusion. Let's make it one, two and one, two. So one goes to one and two goes to two and the other one is one, two, one, two, one goes to two and two goes to one. So let's just give them these are just symbols and let's give them names. Obviously this one we could name E because that is going to be the identity permutation and in most textbooks this will be called the tau, this by convention. So I have these two elements that I have and let's look at this now as not this. Let's make our group then contain these two elements and it is the composition of those two. So if I were to do a Cayley's table on that, let's make it a bit bigger. So you can see there's E, there's tau and there's E, there's tau. What happens if I do the identity element? Well, that's just going to stay the same tau. And what happens if I compose tau with tau? So that means one goes to two and two goes to one. So that means one stays with one and two goes to one and one goes to two. So two stays with two so I just have the identity element there. Now I'm interested in this, I'm interested in the product of these two groups. So these two cyclic groups, so my cyclic group remembers this. This was actually just to mention that this is the set and the set is actually these two. If we just look at the two elements, it's these two elements that we permute and it's these that make up this group. Let's not confuse those two. So what's happening here? I want C2 and C2. Let's look at the Cartesian product of the two sets. The Cartesian product of the two sets. So the Cartesian product of these two sets, that is just going to equal. So I'm going to have this C and say C star such that C is an element of C2 and C star is the element of the other C2. And I call it the other C2 and I'll show you why in a moment, but just leave it there. It's an element of C2. So what do we have? So it's going to be the identity element, identity element, identity element and tau. Tau and identity element and tau and tau. As simple as that. So that would be the four that we do have. So let's have a look at that. Let's compose all of these together. What will happen if this is now what I'm dealing with? E and tau, tau and identity element and tau and tau. And the same I've got to do on this side. Remember? And we have tau and E and we have tau and tau. Now this is our identity element. Our identity element. So the binary operation of those two, this composition is going to be very easy. So we can just do this. It takes a lot of time. You can play this and fast forward. And we have E and tau and tau and E and tau and tau. Now for the rest of the 16. So we've got nine left to go. What happens if I do these two? So remember this is E tau composed with E tau. So this binary operation. And what happens there? It's E composed with E, the identity element. Composed with the identity element, tau composed with tau. So that's what we're looking at. That is how we defined the product of two groups. So what do I have? Well, if we remember from that E, that remains E. The identity element and tau composed with tau. That's also the identity element. So that's just how you do that. Now you can really fast forward. And let's do this. So tau, tau composed with E is tau. E composed with tau is tau. tau with E is tau. tau with tau is E. And see that we have, remember our Cayley's theorem. So we have all of them there. Let's do these two. So E, I'm going to say E is identity element and tau. That's tau. tau and E, that's tau. And we have tau and tau, which is identity element. Identity element, identity element. tau and tau is identity element. tau and this is and tau is that. And now we have identity element and tau, which is tau. tau and tau, which is the identity element. We have tau and tau, which is the identity element. E, identity element and tau is tau and we have tau and tau which is identity element, identity element. I hope you can see there in the corner. So that's the great composition and the composition of all of these. So you can see very simple from the definition that we had how this happens and where it flows from here. Now I've got to clean the board because I've got to show you one more thing. So there we have a clean board. These were just symbols and that's why I mentioned before that's another C2. So let's make C2 and we just have the elements, the set of A and B and the other C2 we make D and E. So if we get the union of these two, we're going to have A, B, C and D. Now we have these four elements. Why keep doing this? There are these four elements now. And let's just consider what happens with these four group elements. So what we're saying here is if we get the union of these two and just for the sake of clarity it's not one, two and one, two so that we can see there are four different things. What happens when we do this, remember this is an element in a group. What happens to that? What we're saying that is a permutation on this union of these two sets. So if I have this and this, what am I going to get? It's A, B, C, D so we're going to have another, you know, it's another, it's a permutation and just means this E refers to these first two and those two refers to those two. If I look at E tau, the permutation that I'm looking at, so my permutation here is just, if I write it in this format it's just A because that means A goes to A, B goes to B, C goes to C, D goes to D and that is just the identity permutation there. So this would be A goes to A, B goes to B, but C goes to D and D goes to C. C goes to D and D goes to C. So this one is CD. That is the group permutation there. If I look at tau E, identity element, then that is just going to be A, B because C goes to C and D goes to D and if I look at tau, tau, what that would be is A, B because A goes to B, B goes to A and C goes to D and D goes to C. So you see it's these four permutations, the four permutations on this set and that is isomorphic to another set which we haven't discussed before but just be aware of it and that's the Klein 4, the Klein 4 group. So this is isomorphic, this set of permutations is isomorphic to this group, the Klein 4 group. But it's clear to see that if we just combine the C2 and C2, these two sets of these two cyclic groups and then we do those four, this is an element of a group, element of a group, element of a group, element of a group, referring to set permutations and you can clearly see what the set permutations mean when we write it like this and just as a bonus remember that's isomorphic to the Klein 4, to the Klein 4 group. So really I think I've shown you that there's no difficulty here with the product groups, they're actually quite, quite simple.