 Let X and G be elements inside of a group G. Then we're going to denote the element little G as a superscript on the left of X. That's going to be defined to be GXG inverse. Now from a text latex type situation, this is kind of an awkward little symbol here because superscript usually should go on the right. This actually looks like a carat G and then there's space X, right? It's kind of weird. Now you have to be careful if there's anything over here that this little carat's going to want to attach to that. So you have to be very careful that there actually is a legitimate space. So if you had some other symbol like say H over here, you might have to put like a backslash with a blank to make sure it doesn't attach to the thing right here. It's a really annoying thing to do in latex. Superscript and subscripts on the left are not well used and actually it's I think the main reason why the group theorists don't like this notation here. But nonetheless, this element superscript GX is defined to be GXG inverse. This is called the con... This is a conjugate of X with respect to G, okay? And honestly, this is what one calls a left conjugate, which I'll talk some more about that in just a second. If we take the set of all conjugates inside of G all conjugates of X, this is denoted as capital G sub X and this is called the consciously class of X for which the reason why we call it a consciously class is that well turns out conjugation forms an equivalence relationship. So if X say equals, so if two things are conjugates of each other, what does that actually mean? So if Y is a conjugate of X, that means there exists some element G, so it's that GXG inverse is equal to Y. This is what we mean by X twiddle Y right here. There exists, and that's the wrong E. There exists some little G inside of the group so this happens. So X and Y are conjugates of each other. This is an equivalence relationship. It's reflective, reflexive, it's symmetric, and it is transitive, okay? So this gives you these consciously classes. Turns out you could define conjugation a little bit differently. One, and this is actually, in my opinion, the more common way of defining it, you could define X to the G as actually G inverse XG like so, and then the corresponding consciously class of X you would actually denote as X to the capital G, and so this is gonna be the set of all elements X to the G where G ranges over G like so. And so in my opinion, this is the notation that's used most often. Now it could be because in terms of latex, it's a whole lot easier to denote this just right in here, X care of G or something like that. But it turns out there actually is a big conversation about the left convention, which we see right here. So this idea, this is known as the left convention, and you kind of see that with how we denote it. The little G is on the left there convention. And over here in blue, we get the so-called right convention. And it turns out prior to this video, if you've been following along with this lecture series, we've seen this already before, all this left and right. So you have like left cosets versus right cosets, right? Which one do we talk about? Do we do right cosets or left cosets? Do we do left translation versus right translation? Which one should we do? And so when it comes to these conjugates, this left and right convention comes into play. Now these differences of conventions give slightly different interpretation of things like cosets and conjugates and translation, but it develops all the same theory, right? Like for example, when it comes to these conciously classes, if you take the left conciously classes, this is actually the same thing as the right conciously class. Conjugation is, the conjugates, whose conjugate to X is independent whether you're doing left conjugation or right conjugation. And so some of these small little things make no difference whatsoever, right? The difference between left conjugation and right conjugation is analogous to this difference between left cosets and right cosets and left translation versus right translation, I said that. There's no difference between the theory of left cosets and the theory of right cosets. Like for example, if you take the number of left cosets, this is equal to the theory of right cosets. I go the other way around, like so. It makes no difference which one you use and that's what we define to be the index of the group, right? Because it doesn't make a difference which one you use over the other. So how does one actually make a decision then? So in this lecture series, we're gonna use the left convention as the dominant convention. I say that because there are some situations where we have to talk about left things and right things at the same time so we can't ignore it entirely. So we're gonna use the left convention as these notations are gonna be derived from the fact that we write functions on the left. So in the United States, the most common way of writing a function is like f of X where the function's on the left and the input is on the right. And other places in the world, it's actually quite common to write the function on the right. So you might write something instead as like X of F like that or probably you'd see more often X and superscript F right there. So that's kind of like what we're seeing right here that the element G is kind of acting like a function on the input element X. And so that's the main difference of those things, right? But be aware that though there are many times, there are many, I should say there are many times where left and right conventions must interact. So we can't just ignore them in general or we can pick one convention but we have to be aware of both. Like for example, normal subgroups are those subgroups for which left cosets are equal to right cosets. So there's something special about normal subgroups cause it's where like things come together if I dare make a political analogy in this video right here, you know, normal cosets are those bipartisan, so excuse me, normal subgroups are those bipartisan subgroups that's where left and right come together and there's no disagreement, which is again kind of ironic because the United States that behavior is not what I would call normal. I know that's a horrible pun, but there it is for you. So most of the, multi-interactions will be the study of group actions. That is when it comes like this left and right, this is a big deal when one talks about group actions which is not something we're gonna worry about in algebra one. This is something we'll get more into in abstract algebra two, math 42, 30. Okay, so you should be aware of both conventions. That's the thing I really want you to get away from this conversation here. And so some other things to mention here. I just also wanna mention that the study of constancy class is getting back to the idea here of conjugation here. The study of constancy class is an important part of group theory. For example, let's see, I doodled all over my page so I need to come down for a little bit. So if we were to look at the element superscript gx right here, what happens when that's equal to x itself, right? Well, that would mean that gxg inverse is equal to x, which if you multiply on the right by x, you actually get that gx equals xg, right? Okay, so an element is equal to its own conjugate when you commute with the conjugator, okay? And so when you see something like gx here is equal to just x itself, this would suggest that x commutes with everything, which in group theory we'd say that x is central, right? Central because it belongs to the center of the group. This is noted zg. This is the set of all elements z inside the group such that zg equals gz for all g inside of g, right? So when, so you're central exactly when your congenital class is yourself, right? And it turns out this center, this center group, it can be proven, I'm not gonna do it right here, that the center is always a normal subgroup. So this is a pretty important group and the congenital class being trivial actually means the element is central. Another important element, right? To talk about is what we call a commutator. So let's say we take something like g comma x right here. This is defined to be gx, g inverse x inverse. Now you have to be careful because with the right convention, when you see the commutator gx, this is actually defined to be g inverse x inverse gx. So slightly different definition, but it doesn't make much of a difference here. You'll notice that this commutator, this looks like gx times x inverse. And the use of commutators also is a very important tool to, the use of commutators is important tool also to measure commutivity, right? Because if a commutator turned out to be, if a commutator turned out to say be the identity, right? That would suggest that gx, g inverse is the identity, which if you multiply on the right hand side by x inverse, by x and g, excuse me, you end up with the statement that xg equals gx. So a commutator being trivial is, again, it's equivalent to being a commuter. That is your, these two elements can be with each other. That's why it's called a commutator. An important group in group theory, a subgroup, it's called g prime. This is known as the commutator subgroup. This is gonna be the subgroup generated by all of the commutators, right? Or g and x range over the elements of group right here. So this is the subgroup generated by commutators. There could be things in the subgroup that are not commutators, but they're gonna be products of commutators and things like that. It turns out that the commutator subgroup is also a normal subgroup of g. And these two subgroups measure how abelian something is. Like if you have an abelian group, the trivial, the center of an abelian group is gonna be everything. And the commutator subgroup is gonna be nothing, right? The identity because everything would commute. And so again, I'm really trying to emphasize here that these conjugates are a pretty big deal when it comes to group theory, right? Coming back to a previous theorem, we said, oh no, it's still messy. But you'll see right here what I just erased, right? About the conjugate. You can talk about the conjugates of a subgroup. And so one equivalent form to be in a normal subgroup is that a subgroup is normal if and only if it's closed under conjugation. So we say that a subgroup is normal if and only if for all g inside of g and n inside of n, we have that g n, g inverse is inside of n. So a subgroup will be closed under multiplication. It'll be closed under the identity. It'll be closed under inverses. But the normal subgroup has this extra operation of conjugation that normal subgroups are closed under conjugates. And that's what makes them a little bit extra special. Let's look at an example of such a thing, okay? So in S3, the three conjugates, it has three conjugates, you have the identity, you have the three cycles, one, two, three and one, three, two. And you have the transpositions, one, two, one, three and two, three. And I'll leave it up to you to check and verify these conditions right here. But if you look at all the possible conjugates of like the element one, two, right? There's six possible conjugates. You go over the elements of the group here. You're gonna see that two of the conjugates turn out to be one, two. Basically, again, I'll let you compute that. Two of the conjugates turn out to be one, three and two of the conjugates turn out to be two, three. In the dihedral group, D4, right? The conjugacy classes are gonna be the identity. You're gonna get R2. This is because R2 here is central. So is the identity, identity is always central. R and R3 are conjugates of each other. S and R squared S are conjugates and R, S, R cubed S are conjugates of each other. In the alternating group A4, the consciously classes turn out to be the identity. You have the two, two cycles. They form a consciously class. And one thing I wanna mention is if you put these things together, the identity plus the two, two cycles, this actually gives you A4, the climb four group. Now, you'll notice here that the climb four group is actually a normal subgroup of A4 because the climb four group is actually a union of consciously classes. That means a V4 will be closed under conjugation inside of A4, so therefore it is a normal subgroup. Normal subgroups are those subgroups which can be partitioned using consciously classes. The other two consciously classes are the three cycles. There's eight three cycles in A4, but they're broken up into two groups. You have one, two, three, one, three, four, one, four, two, and two, four, three. Then there's their inverses. The inverse of one, two, three is one, three, two. The inverse of one, three, four is one, four, three. The inverse of one, four, two is one, two, four. And then the inverses of two, four, three is two, three, four. This is a very interesting property about consciously classes. If you take the inverse set of a consciously class, you always get back in a consciously class. Now, that's not so exciting for these sets right here because the inverse set is itself, right? But when it comes to A4, these two sets right here are actually, these two sets, this one and this one, they're actually different sets and they're inverses of each other. That always happens if you have some consciously class, say GX right here, and you take its inverse, that is to say this will just be, this will just be the consciously class of the inverse elements, pretty cool property. And we'll talk some more about conjugates, of course, in the future, but this introduces you to the idea of a conjugate. For Abelian groups, we don't really care about conjugates because conjugates and Abelian groups are always singletons because every element is central. But for non-Abelian groups, studying the consciously class is a very important measurement of the group.