 Welcome back to our lecture series Math 4230 abstract Algebra 2 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lectures one through three in our lecture series, we introduce the idea of group actions. So that is, we have a set for which some group G is acting upon it. This is often referred to as a G set, okay? Where X could be any set, it might have some algebraic structure, it might not. All that we're assuming right now is that there is a group acting upon that set X. And in those lectures, we presented many, many examples and some properties of group actions. In this lecture four, we're going to talk about the so-called class equation, which when you first look at the class equation, it actually feels like a somewhat obvious observation. But believe it or not, this obvious observation can actually have some powerful consequences, especially in the case of P groups, which we'll talk about in the second half of lecture for the next video here. But we'll introduce the class equation in this video right here and look at some examples. So let me remind you about some notation we introduced previously when we discussed group actions. So there's the idea of a stable set. So if you have an element G that belongs to the group and the set X sub G, capital X representing the G set here, this is the collection of all elements X inside of X, such that G dot X is equal to X. So this is the stable set. These are all the things that were fixed by the element G itself. Related to this on the other side, there's the so-called stabilizer. There's lots of different notations, of course, for the stabilizer. The one that we're mostly using is this set G sub X here. This is the set of all elements G inside of the group such that G dot X is equal to X. So the important thing here is X sub G is a subset of X, and G sub X is actually a subgroup of G, which kind of makes sense. Since X is a set that may or may not have any structure besides that affected the G set, then X sub G is just a subset. But the subsets of G we don't really care about. Really, we care about subgroups. We had these stable sets, these stabilizers, sometimes called the isotropy subgroups. What we want to do is add to this collection of symbols here. And so we're going to consider now the symbol X sub G. So notice what's going on here, that X sub little G, this is a subset of X, and then the subscript then tells us how we construct that. So for a little element who gets fixed by the little element, the individual element X right there, then capital G sub X here, right? In that situation, we're taking a subset, a subgroup, in fact, of G that's determined, that's fixed with regard to this element X right here. Now we're going to take capital X sub capital G. This is actually very similar to this set right here. We're going to take the subset of X, right? So X sub G is the collection of all elements of X. So this will be a subset of X right here. But what's different now is that we want to find all those elements X that are fixed by G, but in fact, they're fixed by every G whatsoever. So now the capital G is suggesting here that the little G is allowed to vary, right? In this notation, X sub G here, the little G is fixed, but the capital X is allowed to vary. You can take on any element of X so long as it's fixed here. So now we allow X and G to vary. And so this gives us all of the elements in X that are fixed by everything. These are things that are trivialized by the group action. No group element does anything to them whatsoever. So in fact, this right here is the intersection of all of these sets, XG, as G is allowed to range over the elements of the group. So this would be the collection of all of these, it's the intersection of all the stable sets. It's a very important set when one studies group actions. Another way of characterizing is the following. XG here is then the union of all of the orbits sub X that have this format. Each orbit is just a singleton, because after all, each element of the group just sends X back to itself, so its orbit is going to be trivial. So this is, this stable set is all of the stable sets together. I should say it's all the, it's all, it's the intersection of the stable sets, it's the union of all these trivial orbits. This is the collection of all the trivial elements of the group action, of the G set here. And so since the orbits of a group action form a partition, because after all, the action does cause an equivalence relation on the set. So X is a union of all of these, all of these cells here. So X, you know, you're going to have maybe a couple elements that are trivial. X2, move it up, in which case we get some like X, K, right? And then you're going to get a bunch of orbits. You have some orbit here of like X, K1, another orbit here, X, K2, or something, you know, just keep on going. X is the union of all of these things. Now the trivial orbits we can put together as this set X sub G, like so. And then the other non-trivial orbits, we just keep them as they are. Because X is a union of all those things, we get the so-called class equation where a G set can be partitioned into its orbits, where you take all of the, you take all of the trivial orbits, we're just going to glue them together for simplicity's sakes. And then you have these non-trivial orbits right here. It's important to mention that with a G set, the orbits are essentially the G subsets. That is, the orbits are kind of like subgroups in that the orbits of X are going to be those subsets which are closed under the group action. If you take an orbit X here and you times it by any element of the sub, of the group, right, you act on it, you're just going to get back the same orbit, like things might get scrambled up, but you're going to get back the same orbit. So the orbits of a G set kind of are like the subgroups that we think about it in an algebraic category sense. And which case then, of course, you can union, these will be sort of like the smallest cosets, the smallest subgroups, so to speak. You can take unions of orbits that will also give you G subsets. And one important one, of course, to take the union of all the trivial ones, right? This is the trivial subset of X there. And so you get this very important equation. It's often called the class equation for which the cardinality of X here is going to be the cardinality. We'll call it the order, I guess, the order of the trivial stable set there. And then the union, the sum of the orders of all of these orbits here. Now when it comes to group actions, there's one group action that's very important compared to any others. And this is the conjugation one. I mean, there's a lot of important group actions, but the group action of conjugation is very, very important. And why is that? Well, the group action of conjugation is a group acting upon itself in an algebraic manner. And so any information we can learn about the G set actually says something about the group itself. So this is a group action that actually tells us information about the group itself. We learn things algebraically by studying a group as a group action. And so oftentimes, when we talk about the conjugation action, we can inherit sort of special terminology, special notation to describe the symbols we saw on the previous screen just for the group action. So in particular, the orbits of the group action with respect to conjugation are called the conjugation classes. A notation that's often used in this situation is that if you want to take the conjugation class of x, an element x here, this is often denoted as x to the G. So this would be the collection of all elements of the form G inverse x G, where x is fixed here. And so G is allowed to vary here. Now technically, I'm writing this as a right conjugation. We really should be working with left conjugation because we made the decision to do left actions in this lecture series. But the thing is the action, the left action by conjugation and the right action by conjugation give you the exact same partition of constancy classes. It doesn't really matter because if you are right conjugating by an element G, then you're left conjugating actually by its inverse. So you really get the same thing. No big deal. We're not going to be too worried about that. So this notation here suggests what you're doing here is you're taking the element and you raise it to the G. Why the superscript? Well, this is a very common thing because this notation can get a little bit cumbersome at times. So oftentimes in group theory, we use exponents where you take x to the G here, and this represents G inverse x G, for which if you want to be somewhat, you know, conventional with what we're doing here, you can put that on the on the right. In practice, though, when people write x to the G, they mean the right conjugate. If you want the left conjugate, that should be G to the G. The superscript should be on the left. Again, that's how it's typically done in the literature. This is a pain to type up in latex. So it's not used very common. So I think honestly, that's one of the reasons why the right conjugation Jews more commonly in the literature. But despite our agreement to do left actions, the right conjugation action, like I said, is really not any different. So it's not a big deal. So you might see notation like this used by me, x to the G to represent conjugates. If you mess up and switch it from the right to the left, not a big deal, it won't make much of a difference in this situation. So that's what this notation here means x to the G. The capital G means that you're allowing the exponent to vary, thus the conjugates vary, but who's getting conjugates left fixed. And this gives you your conjugation action. All right, the conjugation, excuse me, the conjugation classes with respect to the conjugation action. Knowing the congisly classes of a group is very important when you study that non-Abelian group. They're trivial for Abelian groups, of course. The isotropy subgroups or the stabilizers will commonly be referred to as C of x. Some people like to call this C sub x, right? Now this, of course, agrees with the notation we had before of G sub x, right? But again, like I said, this is often referred to as C of x. Because again, since the group that's acting on the set x are actually the same thing, you get a little bit confused when you see things like G sub x versus x of G, because the G's and the x's are the same thing for conjugation. So we introduce that new symbol. So again, typically it's called C sub x, or C of x right here. It's the centralizer. So we're looking for all elements of the group for which they commute with the element x. This is always a subgroup, and this is a subgroup that always contains, of course, the cyclic subgroup generated by x. Because x will commute with itself, and in fact, every power of x will commute with itself, right? And so the centralizer always contains the cyclic subgroup generated by x. It will also always contain the center of the group, because those commute with everything. And that gives us the centralizer subgroup. It's a very important group, a subgroup when we're studying non-Abelian groups. Now speaking of this subgroup, ZG, the center, it's important to consider the center when you're talking about the conjugation action. So again, the center is the set of all elements that commute with everything in the group. With regard to this discussion of stable sets, the center is this set x sub G, right? x of G, which we saw on the previous slide. This is the collection of all elements x inside the set that are trivialized by the action. So these are all the elements G.Z, which are equal to Z. But the group action here is conjugation. So we're looking for all those elements G, Z, G inverse, which are equal to Z, which is a course equivalent to saying that GZ equals ZG. Like so. These are things that commute with everything. So this stable set is in fact the center. Now the class equation, when we translate it to the language of congenital classes, and honestly, when people talk about the class equation, this is usually what they're meaning. They're talking about the congenital class equations, because the more general formula 1421, which we saw on the previous slide, you might call that the orbit equation, right? But the congenital class equation, the congenital class equation tells you that any group is going to have the form. The order of the group is going to be the order of its center plus the indices of each of the congenital classes. Let me unravel that for a second. Okay. On the previous slide, we saw that a G set X, its order, which now that we have the conjugation action here, the X set is itself the group G. So the cardinality there is going to equal the order of the center, which for the previous equation, that is we take the collection of the size of all of the elements which are trivialized by the action, that's the center in the situation, which is a normal subgroup. Then we take the sum of all the sizes of orbits. Well, we saw previously by the fundamental counting principle that the size of an orbit is actually the index of its isotropy subgroup, which in the context of conjugation, the isotropy subgroups are the centralizers of the element itself. That is the stabilizers with respect to conjugation are the centralizers. And so putting all that together, we get this right here. And so what this tells us is that the order of a group is equal to the order of its center plus the collection of indices of centralizers, where these Xs, these XIs range over the non-trivial congenital classes. And for a finite group, this observation is very, very important. And we can use this to study various things. So for example, since the congenital class has cardinality equal to its index, so if we take the congenital class of X here, this is going to equal G dot the centralizer of X, which for a finite group, this is the order of G divided by the order of its centralizer. And so this is sort of like a Lagrange theorem-like result. The size of a congenital class must divide the order of the group in a finite setting. Let's look at two examples of this. Again, it only makes sense to really look at non-Abelian groups because this observation is trivial for Abelian groups because in that case, you just get that G is equal to its center. It doesn't really tell you much at all. So let's look at the smallest non-Abelian group possible. Let's take S3. The congenital classes of S3 turn out to be the identity, the three cycles, and the two cycles. This is actually a general principle that in Sn, the congenital classes are exactly the cycle types. So if we want to talk about the congenital classes of S4, we should look at the cycle types. You could have one, you could have a two cycle, you could have a three cycle, you could have a two-two cycle, and that's it. There's going to be four congenital classes in S4, and it comes from the different cycle types. I don't want to list all of them because there's 24 elements in S4. We see that same thing for S3. We get three cycle types, and those are the three congenital classes. Also in Sn, for S3 and above, S4, etc., the center is actually trivial in that situation. So we get that the symmetric group, again, for S3 and larger, it's a centerless group. The center is trivial in that situation. For S2, of course, it's abelian, so that's not true in that case. So if you look at these together, so the center is just this one. This is the only trivial congenital class for permutations of three letters or more. And so when we look at the class equation, we're going to get the order of S3 is going to be the order of its center. Let me write S3 there. Plus you're going to get the size of that first congenital class, this one right here, and then you're going to get the size of the next congenital class, which you see right here. So basically what we're saying here is if you add together the sizes of these congenital classes, you have a congenital size 1, 2, and 3. And so you're going to get that order of S3 is 6 because it's 1 plus 2 plus 3. And in each and every case, these have to be divisors of the order, 3 divides 6, 2 divides 6, 1 divides 6. The class equation guarantees this. Now for 6, that's not too surprising. 6 is a perfect number. It is literally the sum of its divisors, 1 plus 2 plus 3. So that might seem like, oh, you got me there. You had a perfect number. Give me 28. We can do the same thing again. Well, okay, let's try D4. The congenital classes of D4 are the following. 1 and R2 are central elements. Their collection together is ZD4. The other elements you have R and R cubed are conjugates. S and R squared S are conjugates. And RS and R cubed S are conjugates. So we get these congenital classes. Let's throw the center together like so. And so the class equation tells us that the order of D4 is going to be the order of the center plus the sizes of the three other congenital classes. So this one, this one and this one. So you get that 8 is 2 plus 2 plus 2 plus 2 for which we see that those have to all be divisors of 8. So in particular, they're all 2s. So 2 divides 8. The individual congenital classes are also 1. So those singleton ones, right? So the order of a congenital class always divides the order of the group. And then the sum of all the congenital classes has to add up to the order of the group. So what we're telling you here is that the group has to be a sum of its divisors. The order of the group has to be a sum of its divisors, which again, that doesn't seem much because you know, it should take 1 plus 1 plus 1, et cetera, right? And to get up to your group order there because 1 divides everything. But that's just an abelian group. If all of your congenital classes are trivial, you get an abelian group. For non-belian groups, it gets a little bit more sticky. Like how can I form congenital classes? The congenital classes have to divide the order of the group. And we have to add them all up to give you the order of the group itself. So in some respect, congenital classes kind of behave like subgroups, at least from a Lagrange theorem point of view.