 So now eventually we get to the orbit stabilizer syndrome. There's a lot to write and number one I want to say sometimes I've written a few things and number two it is not the easiest thing in the world to understand. And the proofs really, you really need to concentrate. So I want to keep it together. First of all I just want to talk to you about a new definition of something called an index. An index of the stabilizer, remember that's a subgroup of G. So the index of a stabilizer in G we write it as G and then the colon G dot A. And that just means the number of left cosets I can form. Remember this is a subgroup of G. So if this is G I can form these cosets. And the number of these inclusive of this is the index of the stabilizer in G. And remember if I have a set A and I have the orbit. Here's the orbit of A and in this orbit of A there's going to be a number of elements. And I can also just express the number of elements that are in the orbit. So then we just write it with this absolute value sign, the number of elements in an orbit. So I have the number of cosets of G formed by this stabilizer of G. And I have the number of elements inside of the orbit of A. And what the orbit stabilizer theorem is saying is that these number of cosets equals the number of elements in the orbit of A. So the index of the stabilizer with respect to A and G then is equal to the number of elements in the orbit of A. So carefully remember there's all elements in the same coset. So now I have various elements in this coset of mine. What we are trying to say of this is true. So if the number here, so just to make it easy to understand, so I've got 1, 2, 3, 4 here. And I've got 1, 2, 3, 4 here. So this 4 is supposed to equal 4 here. That means everyone in here has to go to the same element. No matter which element I take here they must go to the same one. And if I take an arbitrary element in one of the other cosets it will have to go to one of the other ones. So they all going to map to the same element. So all elements in the same coset acting on A, element of A results in the same element in this orbit. So they've all got to act on A and give us the same element in the orbit of A. That's the only way that the number of cosets here is going to equal the number of elements in the orbit of A. And we've got to prove this. So first of all let's see that all of these in the same coset is going to map to the very same element in the orbit. So I've got this, taking this G bar and I'm suggesting that it's an element of this stabilizer of A. The stabilizer of the group with respect to this element A and A. So I can form this left coset. So I'm going to take an element G and G, compose it with a binary operation G bar. And I'm running through all the G bars in that. So it's just an arbitrary one but that is my left coset. Now if I look at this, remember this is a subgroup of that. So the identity element will be in there and the identity element composed with is just going to give me... One of these is going to be the identity element composed with G, G composed with the identity element is just going to be G. So G is going to be in this, G is going to be in this. So it's going to act on A and that just gives me A because that's how we defined this. Now let's take an arbitrary other element in there and let's not just do this one. So G O, G bar, that's an arbitrary element inside of this coset and I'm going to act on A. By the first property of group actions, this will be G dot, G dot A, G bar. But remember this thing is part of the stabilizer. This is part of the stabilizer. So this is just A. So this is G dot A, exactly the same one. It's exactly the same one. So no matter which one I take here, it is going to map to this V same element in the orbit of A. So we've got that. Now I just need to show that if I take an arbitrary element, I'm going to call it G star and it's in a different coset. And I'm going to have to prove that it will then not go to the same as one out in there. So let's assume to the contrary that it does map to the exact same thing. So I'm going to say this G which is in here acting on A is going to be exactly the same as G star acting on A. So I'm saying that both of them acting on A will give me the same exact element. So that's to the contrary and let's see if we can run into a contradiction. So what I've done for you here is left compose with the inverse of G both sides. So acting on this element, acting on the same element. By the first property of group actions, I can rewrite it like this. It's the binary operation between those two, the binary operation between those two. G inverse composed of G, that gives me the identity element. So I have that A equals this G inverse. You have to look at this on a big screen to see I don't have a big enough board. So that equals this G inverse composed of G star dot A. But hang on a minute. That is just the definition of the stabilizer. Because if I have an element acting on A and it gives me A back, that means it must be an element of the stabilizer. That's how we define the stabilizer. So that means it must equal this being this arbitrary element in the stabilizer. So we have G inverse composed of G and let us go to equal that. It is in this element of the stabilizer. And if I left compose with G on both sides, what do we have left? What do we have left here? That's this identity element. So I have, I should say G star there, G inverse, G star, G inverse or G star. So I have the fact that G star equals GOG. That means it must be an element of this. So if I take another one out there to the contrary suggesting that they go to the same, it actually means no, it never came from that one. It actually was just one of those. So my assumption to the contrary that if they are different, they match to the same shows, no, no, no. If that one is different, it is going to go to a different one. And that was just arbitrary. So if all of these, which I've shown on this side, they're going to map to the exact same one. All of these, they're going to go there. And all of these means they're going to go there. And I have my orbit stabilizer theorem. Difficult to get, you probably have to read your textbook very well. Listen to your lecture, watch these videos again. But in the end, if you watch it long enough, you'll see that makes absolute sense. And that's a very deep thing that we've got here. That the number of cosets formed by the stabilizer is just going to give me the number of elements, the index of that. So the number of cosets that gives me the number of elements in the orbit of this A element of A. All mapping to the same, all mapping to the same, all mapping to the same. So the number there has got to equal the number there. Orbit stabilizer theorem.