 So before we carry on, I want us to have a closer look at conjugation. We've seen conjugation before. And remember that is where conjugation of an element, say A, is this if we take G composed with A composed with the inverse of the first term conjugation. But can we see conjugation of the set that makes up a group or group on a set where that set is the set that makes up the group? Let me show you. Let's consider G, the group. So I have a set that makes up my group and a binary operation. And I'm going to have a set A. But this time I'm going to let the set A just be exactly the set that makes up the group. Now what happens if we do conjugation here? So I'm going to keep on calling all the elements G, elements of G. By that I mean the set in the group or the group itself. And I'm going to keep on using elements A, elements of the set A. But in your mind just remember that this set is exactly the same as that set. And I want to show that I can actually conjugation. I can set up conjugation as a group action. And remember we would write a group action as G dot A. And I want to define this group action as being this conjugation. And if I can do that, that'd be a very nice thing to do. Because that means I can see every element in the group as a permutation of the elements in that set. So listen very carefully, this is a very important thing. So I can see every element in this group. Remember if we had something like tau and sigmas and all of those. And we saw them as a set permutation. So all these elements in there, we permute them. There's some permutation of them and that becomes an element in the group. And if I can do that, I can show that conjugation is actually just setting up these permutations of the elements in my set. So can I view a group action? So what are we going to have then is the following. I can have this permutation sigma of G mapping G onto itself. That means sigma G can be viewed as this conjugation G binary operation A, binary operation G inverse. I hope you can see in the corner there. So that means that I can view every element in this group as a permutation of the set that makes up that group. So is this so? Well there are two properties that must be obeyed. Two properties that must be obeyed. I must have the fact that if I have G1 and I have G2 elements of G and I have A as an element of A. That's for all these elements. I must have the following. That if I have G1 dot G2 dot A, that must equal this binary operation between these two. So if I compose these two, dot A. So that must hold. The second one that must hold is that E dot A, that must equal A back again. And E dot A, how do we write E dot A? Well that will be E according to our definition, the way that we have to find it. There would be A and E inverse. Well the inverse of E is just E. So that will just be E and A and E and no matter which way you would do that, that would be A, binary operation E and that would just be A. So the second property of group actions is obeyed. Let's see if we can get this first one to go. So we are told that G2 is an element of G and therefore, and we have A as an element of A. So G dot A we can write as this. So here on the left hand side we have G and we have here G2 composed with A, composed with G2 inverse. I'm just leaving out these binary operations. And more so, this is now a new element and the action of that on that element would be G1, G2, A, G2 inverse, G1 inverse. So I'm just the group action of this on this thing. I put G1 in its inverse on both sides. G1 is an element there so I can use this definition that I've set up. So what can I do here on the right hand side? What happens on the right hand side? What happens if I take this element and I act on A? So on this side I have G1, G2, I'm just leaving out the binary operation, composed with A, composed with G1, G2's inverse. So that on the action, action on A, action on A means it's the first one, it's whatever this is and it's inverse. So whatever this is and it's inverse. And we saw before, remember, the inverse of that would just be G1, G2 on this side and A. And remember that's G2 inverse, G1 inverse, we looked at that before. And lo and behold, the two are equal to each other. Left hand side equals right hand side. So I can see group action here, this conjugation just as this fact. It's this mapping of G2G so I can see every element just as a permutation, every element in here is a permutation of this. That is an extremely powerful thing. One more thing is just if, well I don't know if it's really interesting, but what if G is abelian? If G is abelian, what does that mean? Well G1 composed with G2 equals G2 composed with G1 for all G1, G2 elements of G. So that's not, you know, that's not. So what we're going to have here is that we have G1 and it's action on A. Well we'll just make a G's action on A, it will be G, A, G inverse. But remember now if we look at that, these are all elements of my original set. So I can just write that as G, G inverse A and that's just A. So I have the fact that G's action on A just gives me A back. And if you think about this, this is just the identity permutation. Because if I act on A and just get A back for all the A's that I could possibly have, I'm just going to get it back when I act on that. And that permutation is just going to be exactly the same as all the elements. So if I were to map all of A here at the top, so I have A1, A2, A3 all the way. And I act with G on A, I just get A. If I act on A2, I just get A. I just rewrite this over and over and over again and that's the identity permutation. So if I have an Abellion group, it just means that all the group actions are actually just the identity permutation. So have a look at this, I think it's quite easy to understand that we can see this conjugation in a new light. We can see conjugation in a new light and that is that we can create here from the elements that make the set itself to see them as permutations of that exact same set.