 Welcome to our lecture series Math 4230, Abstract Algebra 2 for students at Southern Utah Universities. This is the first lecture in our series, and if we haven't met before, my name is Dr. Andrew Misildine. Like I said, this is Abstract Algebra 2. If you're not quite familiar with many notions like groups and rings from Abstract Algebra 1, I would recommend you go towards the previous lectures in the series Math 4220. In our first semester of Abstract Algebra, for which this is the first lecture of really semester two in Abstract Algebra. In the first semester, our series focused primarily almost entirely on groups. We did talk a little bit about rings, mind you. But honestly, our discussion of rings was really to just lead to the idea of a group ring. So why are groups so important that we focused almost an entire semester on it? Well, I mean, honestly, if it was just to me, I would say because they're cool. They're awesome. What else do we have to say? But the viewer might need a little bit more convincing than just they're dope or something like that. So really, the importance of group theory comes from the fact that groups are ubiquitous in mathematics and in science even. I mean, they're everywhere, which of course, this is a result of how they interact. That is, this is how groups interact with other objects. That is to say groups are important because they act on other sets. So our very first unit in this lecture series, particularly lecture one here, is going to talk about the idea of a group acting on a set or so-called group action. So this first video is going to introduce us to some important vocabulary we should be aware of with discussion of group actions. So we're going to have a group G in consideration here, but we're also going to have a set X. For which this set X, we don't suppose necessarily any algebraic structure on it. It's just a set. I mean, there could be more to X than that, but for the moment B, X is just a set and G is just a group. We define a so-called left group action, or when it's clear from context, we'll just call this a group action. But as there is a possibility of a right group action, which we'll define in just a moment, a left group action from the group G on to X is in fact a map. It's a function of the form G cross X towards X. So the domain is the Cartesian product of the group G and the set X. And then the co-domain is just the set X right here. Now, we will often denote the image of the ordered pair G comma X using a symbol like the following G dot X. And there's a lot of different notions that people use when it comes to describing a left action. There's no universal symbol that everyone uses. In our lecture series, we'll primarily just use the symbol G dot X to denote the image of G comma X for a group action. But some people will use like a little circle, like kind of like function composition. Some people use like a little triangle G triangle X for which the triangle points towards the element being acted upon and it points from the group element which is acting entirely. Another common one is you use a superscript on the left. So superscript G X. This isn't a very common one, I think mostly just because it's extremely tedious to typeset that symbol in latex. It's possible, but it's really annoying. And honestly, probably the most common notation people use for a left group action is just the juxtapositioning of G and X. So they're next to each other with no further symbol whatsoever. And in my professional literature, that's how I would write it as well. In this very first lecture, I'm going to stick with the little dot here because the group action itself kind of acts like multiplication, but we don't want to confuse this with the group multiplication. So until we have a little bit more maturity, a little bit more experience on group actions, I do want a solid symbol to help distinguish between what is multiplication of the group and what's the action of the group on the set. This can get particularly confusing when the set we're acting on is the group itself. So again, we're going to use that symbol to help us out here. So getting back to the definition, besides that aside there, a left group action is a map from G cross X into X denoted as G dot X. And it satisfies two axioms, kind of like when we defined a group. It's not just good enough to have a binary operation. There have to be axioms associated to that binary operation. Now the first one, we're going to call it the identity axiom. You'll recall that a group, by definition, has an identity element. The identity axiom for group action tells us how does the identity element of the group interact with this action? So if you take any element X inside of the set X, then the identity of the group, we'll typically call that E, the identity acts on the element X by doing nothing. The identity element does not change the elements you're given. And that kind of makes sense. I mean, when it comes to group multiplication, the identity doesn't do anything. So with regard to group actions, the identity also has that property there that the identity acts like the identity function. And then the next axiom, there's a lot of different names that you could give to it. We'll call it compatibility. Some people call it homogeneity. Some people call it associativity, which that can get a little bit dangerous, but it does kind of act like an associativity axiom. But let's call it compatibility because we have the, for the group, its multiplication is already in play here. But now we're introducing this new function that kind of looks like a binary operation, doesn't it? You take two things for the input and that gives you a single output that kind of feels like a binary operation. In fact, the symbol we're using to denote the group action resembles multiplication. But of course, with our group multiplication for G here, we'll use juxtaposition there. So there won't be any confusion about whether we're multiplying the group or we're acting with the elements there. So again, one should be very cautious, but a group action itself is not a binary operation because the two sets in the domain are not necessarily the same thing. We could have it be the case that G and X are in fact the same set, but in general, that's not going to be the case. And so it's not a binary operation in the normal sense that we don't just take two elements of a set and produce a third element of the set. So one needs to be careful when we talk about combining these things that we actually get something that makes sense. So first of all, it makes sense to talk about the element G dot H dot X because if you look at just H dot X right here, the group action says that if you take an element of the group and an element of the set and you combine them together, they give you an element of the set. So H dot X belongs to capital X there. Therefore, H dot X as an element of capital X is something that G could act upon. It could act upon this element right here. So what we see on the left-hand side is that we have that H acts on X and then G acts on whatever H did to X there. So G acts on the element H dot X. This makes sense in terms of the composition of functions with respect to this action. In contrast, we should be very cautious with things like the following G dot X dot Y where here G is an element of G and X and Y are elements of the set X. We have no meaning what X dot Y is. This would be complete nonsense. So we want to be cautious about things like that. We can't just have arbitrary combinations because the G's and the X's are different potentially. But what compatibility does for us is that those left-hand side makes sense. You act by H then you act by G. Compatibility says that this is the same thing as acting by the product G times H. The product of G and H acting on X is the same thing as H acting on X and then G acting on that. And so the two axioms together give us a so-called group action. Now, when it came to defining a group, we actually took on three axioms. One was very similar to this identity axiom, which we call the identity axiom. The associativity axiom is very similar to the compatibility we have on the screen right here because compatibility has to do with, if you have three things, you have a G, an H and an X, how do you do with it? What do you do with these things? Well, in some regard, you can associate this way and in some regard, you can associate this way. So again, compatibility looks like an associativity axiom, although associativity really only makes sense for groups, right? A binary operation, I should say. Clearly, there's non-group binary operations that are associated. That's the definition of a semi-group. But we don't have an axiom for group actions. We don't take on an axiom about inversion. What do inverses do with a group action? We'll actually be able to prove that later on. We don't assume an axiom because we can actually prove things about them later on, but I get ahead of myself. So just again, in summary, we have this group. We say that it acts on X here and X is then, if it's equipped with a group action, it's often called a G set. So X is the G set because the group G acts upon X. And so I want to take a quick moment and talk in general about the idea of a left action. This is not going to be an official definition. We need to worry too much about going forward here, but in general, a left action in an algebra is a map of the form A cross B and it maps to B. So same thing like we saw before, where loosely speaking, I'm going to say that A and B are so-called algebras. That is, they're sets with some type of algebraic structure to it. There's some operations there. We can call multiplication, subtraction, addition. It doesn't really matter, but there's some operations in play which have some properties. Those could be associativity, commutativity, distributive laws, what have you. We're not going to necessarily nail them all down. But these two algebras don't necessarily have to be the same category. Like in this situation of a group action, G is a group. It has an associative binary operation with identity inverses. That's a category of algebras, groups, right? A set is essentially an empty algebra. That is, there's no algebra structure attached to it, but you could think of it as an algebra in sort of like a null sense, all right? But in linear algebra, for example, we often deal with the idea of scalar multiplication. With scalar multiplication, you have your vector space v. You act on it by a field. Oftentimes, this is the field of real numbers or complex numbers, but if you have a vector space, you could take any field whatsoever. And this produces a vector. So if you take a scalar times a vector, this produces a vector. This is an example of a left action. Of course, in vector spaces, things are commutative. So left versus right, that distinction doesn't really matter. Again, I'll get into that in just one moment. But in linear algebra, this idea of scalar multiplication, it's not a binary operation. We don't take two vectors multiplied together to get a vector. We take something different, a scalar and a vector, for which without scalar multiplication, a vector space is really just an abelian group. And when it comes to what do we multiply by? Well, you take a field, which is a very, very well-developed ring, lots of stuff going on there. We can also talk about the idea of an R module, which is kind of like a vector space, but we no longer require the scalars be a field. It could be an arbitrary ring. These are all things we'll talk about later on in this lecture series, future lectures. Stay tuned for those. And so there's a lot of rich theory in this idea of actions. Algebraic sets act upon other algebraic sets, including just plain old sets that have an empty algebraic structure, like a set. Of course, in this chapter, we're going to focus on the idea of just group actions. But we'll often just call it an action for short, because there's no other confusion. But this idea of an algebraic set acting on something else is very widespread in abstract algebra. Group actions, of course, is a good introduction to that. All right, now finally, I keep on talking about right group actions, right group actions. Will it be a right group action by analog? You would take some type of map X cross G, and this produces X like so. And in which case, you have the similar axioms, excuse me, for the action. So we require that X dot E is equal to X. So the identity basically doesn't do anything. We have to be a little bit more careful on the compatibility axiom, because in that situation, we're looking at X dot H, like so, dot G, like so. What do we want that to be? Acting by H, then by G. Well, we want that to be the same thing as acting by HG, like so. In which case, that's what we look like. You just got to flip everything around. It's basically the same type of stuff. Similar notations. We can use a dot notation for a right action. We could use a circle notation for a right action. Like I mentioned earlier, there's this little triangle that people sometimes use. The triangle always points towards the set and points away from the group element. It's not pointed the other direction. This triangle notation is very popular when you actually have a left action and a right action simultaneously, sometimes called a by action, because there's two actions going on. So sometimes you use the direction to emphasize that we go into the left or the right. We aren't really going to worry about that in this lecture series. A juxtaposition is another popular notation, but actually in the literature, I think the most common notation is to use a superscript, which is very easy to put into LaTeX. X to the G there. For our purposes, we're just going to focus on left actions. If there's really no advantage to it, it's just sort of like in the road of life. If we have run across a fork, do we take the left road or we take the right road? We're just choosing to take the left all the time. This is a convention that showed up in our previous lecture series about left cosets versus right cosets. It really doesn't make much of a difference in the most part. Just be consistent left versus right. But like I said, with that by action, sometimes there are situations we have the left and right interacting. And so in that situation, it's important that we do have as an option, but for us, we're going to stick with left actions. Check out the next video or two to see some properties and examples of group actions.