 So now we come in this video to the class equation. Now the class equation is very difficult by the course by the way in which we write it. And I always find it difficult because you have to think of something inside of something inside of something inside of something inside of something and you've got the help on that side. But the concept itself is very easy. If you have a group let's have a group. The first part of the group might just be the center of the group. So that's all the elements with that commute. And that center, if I choose any element in the center and I make a conjugacy class of that, of course that element is going to be the only one in that conjugacy class. So I can have all the elements in this little, now all the elements that are in the center they're going to make their own little conjugacy class. Now I'm left with all the elements outside of that and if I choose one of them, one of them I can make another conjugacy class and I can make another conjugacy class and another one. So if we talk about finite groups, if we talk about finite groups we say that the order of the group or the number of elements in that group I can do that as the sum of the following. So that is going to be the order of the center so that each of these plus the order of each of these conjugacy classes. So how many are in there? Remember we showed that these are all this joint. So the number in there, the number in there, the number in there, the number in there and if it's all finite, if I add all of those that must just equal all the elements that were in the group. So that's, I mean that's fairly simple and that just makes a lot of sense. It's the way in which we write this that is horrendous. We say that it's the order of the center plus the sum of the index of all the centralizes of the element in the group. So the index, the index of the centralize of the element in the group, in the group of the group, the index. Remember the index we wrote it like this from orbit. So where do we start? Let's just start, so I'll show you how to deconstruct this awful thing because that is just a fancy way of saying all of that. So I start with g and I say g equals the center of that center of the group or the elements there and then all these conjugacy classes by some other element g1 in there, the conjugacy class of g1, the conjugacy class of g2 until all the way to n. How many of these there are anyway? So if I were just to look at the order of g, if I were just to look at the order of g that would be the order of the center of g, as we see there, each of the conjugacy classes and then I'm going to have the sum of vehicles 1 to n, all of these n of the order of each of these gis. So I'm going to take how many elements are in that conjugacy class, how many in that conjugacy class, how many in that conjugacy class. But remember these conjugacy classes, remember we can just see them as the orbit of A, so the orbit of gi in this instance. But let's take to the orbit of A, remember we said that the order of the orbit of A we can write as the index of the stabilizer of A in G. So the index of the stabilizer of A in G in G and that's how we wrote it, this index format. Remember from that video. So and just to remind you of what this stabilizer of A in G is, the stabilizer remember that is where the group action on the elements, now remember this is even more complex because A is just our set that makes up the group but we just end up with the same thing. So instead of the orbit of A we're talking about the orbit of all these gis, so that is just going to be the index of the stabilizer of gi in G, this index of this in G. Now I just want to remind you of this stabilizer or at least stabilize this this index of this stabilizer and the way that we wrote that is remember we are dealing with group action but this is conjugacy so I run through all my gis and I act on it and I conjugate it like that and I end up with just with a gi. Remember that, now think back of that video again, this is nothing other than the centralizer of the gi in G, this. So I can replace this there with this, this is nothing other. So look at those old videos, look at your notes and just see that these things are as we defined all of these and that is why we can bring the centralizer of gi in G, the centralizer of gi in G and the index in G that is the index in G. So that is nothing other than just counting up all of these elements, counting up all of these elements. So this would have been a lovely way to write it but in the books you might see it written as this which is this complicated thing because this centralizer can be written as this, this is nothing other than coming into the index which is nothing other than the order of the orbit of gi and that is nothing other, the orbit is nothing other than that and remember the conjugacy class is nothing other than the orbit. So the order of the conjugacy class equals the order of the orbit of that element. So you just have to all the definitions we've had before and realizing you know what that this is, oh I've seen this before that is the centralizer of gi in G or you know as we had of A in G. You just got to realize or just got to recognize you know what all of these are for and if you work it back you get to this very difficult looking class equation but I think it's more important just to remember that if we have the conjugacy classes that make up the center and the order of all the other conjugacy classes we add that all together we get the order of the finite group and it's got to be finite.