 In the previous video, we introduced the notion of a group action, more specifically a left group action. In this video, I want to introduce to us some important examples of group actions, because I claim that group actions are everywhere in mathematics. Let me point out some group actions that we've seen for a long time now, but maybe we just didn't attach the right vocabulary to it. So our first example comes from linear algebra. Take our group to be the general linear group of N by N matrices on the real numbers. And take as our set the vector space of real vectors within entries. So X right here is indimensional Euclidean space, and then G is the general linear group of N by N matrices. Then G acts upon, G acts upon X by just simply using matrix multiplication as we multiply on the left of these vectors by a matrix. So in particular, A dot X, so this is the action, how does the matrix A act on the vector X? Well, this will just be matrix multiplication times the vector by the matrix. This gives you something back in RN. This will be a vector within entries. So that gives us a set, does it satisfy the axioms, right? Well, the first axiom was the identity axiom. If you times by the identity matrix, well, you get back the vector X. So the identity action is trivial. What about compatibility? If I act by G and then by A, what happens there? Well, that's the same thing as multiplying by the matrix A times B. So for matrix multiplication, the compatibility axiom for the action comes immediately from just the associativity of matrix multiplication. And so matrices act upon vectors, and this is therefore called the matrix action. Another example, very similar to the one we just saw here, is permutations. This time take G to be the group S in, that is the symmetric group on in letters, and then take the set that we're gonna act upon B, those in letters, right? So we take one, two, three, four, all the way up to N. How does a permutation act upon the set X? Well, if sigma is some permutation in SN, and X is some letter inside the set X here, then sigma dot X we define to be just the image of X via the permutation sigma, right? Honestly, what we're doing here is that if you take the general linear group, or if you take the symmetric group, basically we're taking some collection of functions that forms a group, and then we're setting the set X to be the domain of the group. Now, that is the domain of the functions in the group. And now, in order for multiplication to be defined here, we, you know, because the operation, I should say, the operation is defined as composition. For permutations for the matrices we talked about a moment ago, the domain and co-domains are the same thing. That's why we can compose them together and get a group option, a group operation here. And so, in that setting where our elements are these functions with the same domain and co-domain, you can always form a group action by setting the set X to be that domain of these things. And therefore, the group action is then just function evaluation, evaluate the function at the element X, all right? We proved it for the general linear group. Let's just clarify, make sure it's good for permutations as well. If we take the identity permutation, this is the identity function. If you evaluate the identity function X, you get back X, doesn't matter what X is. So the identity permutation acts trivially on any element X. What about compatibility? If I have two permutations, sigma and tau, and they act upon X, tau will act upon it first. Well, what does tau do to X? It gives you the image of X via tau. So this is some other number in capital X there. It's a number, what is sigma gonna do to that number? Well, you just plug tau of X inside of sigma. So this is now function composition, sigma of tau of X. So using, of course, composition notation, we write this as sigma of tau evaluated at X. But sigma of tau here, the composition of the functions is what permutation multiplication was defined to be. So this is equal to sigma of tau acting upon X, right? So compatibility is very straightforward. This is known as the permutation action. And like I said beforehand, as long as we have these functions that act upon the set, I should say if the function's image, a domain and co-domain, be more specific, there is X, and we have a group, well, basically they have to be permutations. We can define an action using evaluation of the permutations. The general linear group, you can think of a special case of this going on right here because the general linear group is only some of the permutations you could do on vectors. But of course, it's one that is relevant for linear algebra there. So I'm kind of putting the card in front of the horse here. I wanna generalize the examples we just seen previously. Let's take any group action whatsoever. So X is a G set for some group G. Then take H to be a subgroup of that group, call it H. Now, since X is a G set, there's some left group action of G onto X. So we have some map, G cross X to X. If we restrict the domain, we can restrict the first factor of the domain just to be H. So by restriction, this gives us a function. We just shrink the domain. The output was always an X. This makes an H set. Well, why is that? Well, the identity, which belongs to H here, will do nothing when you act by it because it didn't do anything with G, so it won't do it when you throw away some stuff that's not the identity. And then compatibility will also be inherited. So one can naturally restrict a group action from a group to a subgroup, and you then get a new group action. Now, X was originally a G set, but by restricting the group G down to H, X also becomes an H set. So any G set is also an H set for any subgroup of the group G here. So for example, we had the permutation action just a moment ago, SN. One important subgroup of SN is DN, the dihedral group. It's the symmetries of the regular in-gon. DN naturally acts upon the set one, two, three, four, up to N, where we can think of these as the labeled vertices of the regular in-gon, as DN is a subgroup of SN. This means that X, it's an SN set, but it's also a DN set. And by restricting to the subgroup, we're kind of thinking of the set X a little bit differently. So when we think of X as an SN set, it's just a bunch of letters, we're scrambling them up. But when we think of X as a DN set, then we're really thinking of it as the regular in-gon for which this as DN is the symmetries of the regular in-gon, that's the action we apply the symmetries to get those. And so by Cayley's theorem, every subgroup is isomorphic, excuse me, every group is isomorphic to a subgroup the symmetric group SN. So by Cayley's theorem, it turns out that every group action essentially is just the permutation action up to relabeling. We identify the group with some subgroup of SN and we identify the set X with the elements one, two, three, four up to N. So even the general linear group acting upon a vector space, this is a special type of permutation action where in that case our set X is infinite. So we have an infinite symmetric group that's possible. But Cayley's theorem also can be used to go the other way around. Every group can be embedded inside of the general linear group. After all the symmetric group can be identified with permutation matrices in the general linear group. So if we can embed groups inside of SN and SN can be embedded inside the general linear group, every group can be identified with a subgroup of the general linear group. And the general linear group as it acts on a vector space there, we can in that situation identify basically every group action also with the matrix action. Of course, if we look at smaller sets of vectors but your main takeaway here is that essentially every group action is a permutation action. The matrix example is a little bit more murky because of our set X is finite. What do you do with that? Don't worry, every action is essentially a permutation action up to relabeling. And that's a big consequence of representation theory. We're not gonna really utilize too much of that in this lecture series but I do want you to be aware of that. I wanna do a few more examples but these next examples, we're gonna not leave X so arbitrary anymore. We actually want it to have some type of algebraic structure, right? In fact, X for some of these examples will be a group it'll actually be the group itself. And that's our first example here. A group G actually can act upon itself. So we have that X is equal to G and there's a very, very natural action of G on itself. And it's in fact just left to multiplication. So if X and G are both elements of the group G then G can act on X by multiplication, okay? And so it's then very immediate for the group actions that the action axioms, I set aside that backwards, sorry. It's immediate from the group axioms that the action axioms are satisfied. If G is the identity, then you multiply by the identity and you'll just get back X. Compatibility is just associativity at the situation. So if a group acts on itself by left multiplication this is referred to as the left regular action. Similarly, you could define a right group action from G onto itself by right multiplication and this would then give us the right regular action. This is actually one of the most important, believe it or not, group actions, a group acting on itself. And in fact, in some degree, this regular action represents a lot of generic actions. I'm not gonna provide the details of that just yet but you can believe me that the regular action, left or right, is one of those fundamental of all group actions, a group acting on self by multiplication. Now we actually addressed this action previously in our proof of Cayley's theorem we proved, like I mentioned just a moment ago, that every group is isomorphic to a permutation group. What we did is we actually permuted the group itself by left multiplication. We were utilizing the left regular action. And in fact, in a future video, we will strengthen Cayley's theorem to really prove the statement I said a moment ago that every group action is a permutation action. I'll provide you all of the details to that in a future video here. So using left multiplication, we're basically identifying group elements with its left multiplication. So we identify with these permutations. Another very important example of a group acting upon itself, this time not by left or right multiplication, a group also can act on itself by conjugation. So if we have x, or excuse me, g.x, we can then get the conjugate to be the image here. So gx upon x by conjugation. You have g on the left, g inverse on the right. Yet be careful with the ordering here. If the inverse is on the right and g is on the left, this gives you a left action. If the inverse was on the left and the g is on the right, that actually gives you a right action. So you do get these two conjugation actions, but one's left, one's right. For the most part, it doesn't really matter, but if we wanna be consistent with left actions, we should have g inverse on the right, all right? So it's conjugation and action, right? It produces an element because since g is a group, g inverse is well-defined, g times x times g inverse, that's a product inside of g. It's an element of g, no worries about that. The identity axis, if you conjugate by the identity, you're gonna get ex, e inverse, e inverse, of course, is just the identity. Anything times the identity is just that element. So the identity times x times identity is just x. So the identity acts trivially for conjugation. Compatibility is a little bit more challenging, but conjugation is beautiful in this regard that it works exactly how we want it to. So let's look at the details of that. So if we have x acting on x, excuse me, h acting on x and g acting on that, well then let's first apply the action of h. It's conjugation. So h dot x becomes hx, h inverse, then g acts upon that. So we're gonna get g times hx, h inverse times g inverse. So then we wanna factor this thing. So h, excuse me, g and h, we can redo parentheses to end up with that. But what about on the right-hand side, right? So we have gh here, we have an x, then we have an h inverse, g inverse. Now notice that we have h inverse and g inverse. By the Shusok principle, this is the same thing as gh inverse. And therefore, this is the conjugate of x by the product gh. So compatibility is satisfied. This gives us the conjugation action. Very, very important group action. So one final one for this video here, again, this is a fundamental action. In some degree, this next group action represents a whole lot of other group actions, in particular, transitive group actions. But I haven't defined what that is and I'm not going to at this moment. So let's take our set x to be g again. Excuse me, our group is gonna be g this time. Our set is going to be this time x, x is g mod h, where h is just any subgroup, any subgroup whatsoever. In particular, h does not have to be a normal subgroup because this set g mod h is defined to be the set of left cosets of g. I don't claim that g mod h is a group. Of course, if h is a normal subgroup, then this would be a group. But for a group action, the set x does not have to be a group. But I am saying that the set x is algebraic and that it was created from groups, even though it doesn't have a well-defined multiplication, at least not with coset multiplication. We can still define a group action on cosets right here. So if you take an arbitrary coset like xh right here, how does g act upon it? Well, it acts upon it by multiplication. You multiply the representative by, you represent the representative x by g. And that's what we do. Well, is this a group action, right? Well, if you act by the identity, you're gonna end up with ex times h. Well, that's just xh right there. Compatibility, right? g dot h dot xh here. Well, again, just by associativity here, you're gonna get g dot hxh, like so, which is then going to become gh times xh. So yeah, that's fine. Is it well-defined though? Like if you use a different representative, is that gonna make a difference there? So like let's say xh is equal to yh. Let's say that's given, does gxh equal y, excuse me, gyh? That's a very important question here. Now from the first one, we know that if xh equals yh, that tells us that xy inverse is belonging to h, like so, okay? And so then if you times that by the following, and I think actually, I mean, this statement's true, but to make the argument a little bit easier, I want it to look like actually x inverse y of h. Again, this is the whole left versus right convention. What I said was before was correct, but I wanna use the left convention. So again, this statement's also true here. So then this, so x inverse y belongs to h. This will also be true that x inverse, g inverse gy belongs to h because I just slipped the identity in there. And so you have this x, g inverse, excuse me, x inverse, inverse, you have x inverse, g inverse times gy, this belongs to h. And then the Schuchlach principle, again, you get gx inverse gy. And therefore, yeah, gx and gy do represent the same cosets. So this is in fact a well-defined action. This is commonly referred to as the coset action. And it's a very, very important action. And so I provided in this video some good examples of group actions. And some of these are very fundamental, the matrix action, the coset action, the restricted action, conjugation, regular permutation. These are some of the most important group actions there are.