 So, we have been studying quotient spaces, you may suddenly say why this group action is on, group actions give you a big supply of quotient spaces, that is the motivation of introducing group action circuit level, it will be used in many other ways later on, group actions occur in a natural way in all branches of mathematics, it is an essential part of modern geometry, it may be used as a technical tool in the study of certain symmetry of mathematical objects, the symmetry of a certain structure is defined by its group actions. So, we cannot go on exposing that part of it, but let us start with what is the definition, what are the basic concepts with the group action. Some of it you might have seen while studying groups themselves, start with a set X and a group G, by a left action of G on X, we mean some kind of a binary operation not exactly, G cross X to X, a saturated function, mu of GX, we shall write shortly as GX, as if it is a multiplication in G. This notation follows the special case when X is also G and mu is the multiplication in G, G cross G to G. So, we are writing the same notation here, but you can, if you different actions may be there, then you will have to write it in different notation, that is what we have to do. So, this is simplified notation, if there are more of them, you cannot simplify all of them to the same thing, that is one we have to remember. So, this mu has two important properties, it is associative, G of X and followed by action of H is action of HG of X. So, HG of X is as G bracket X, it is identity hypothesis, namely if is the identity element of G, then the action of E on any X, namely EX, which is the same thing as mu of EX, that must be always X itself. So, action itself is identity action, that is all. Once a map mu like this satisfies these two, we will call it a action, because we are writing it on the left side and this, this composition etcetera is defined from left side with a left action. Exactly same way, you could have defined a right action also, I will not bother to define that here, it is restricted forward. Now there are a few inbuilt structures here, take X belong to X, we denote G power X, G power X denotes what? All elements of the group G which keep the X fixed, G of X is X. See you think of action of G in X as a motion, it is moving the particles, elements of X are particles, they are moving and that motion is given by G, that is the dynamic way of thinking about the group action. Similarly, G lower X is set of all GX, where G varies over G. So, this is called orbit, G upper X is called isotropic subgroup of G at X. G upper X depends upon X, if you take X, it will be different for X, of course. This is isotropic subgroup of X and G little X, lower X is called the orbit of X. We introduce an equivalence relation in X as follows, X and X over X prime are equivalent if there is a G which will bring X to X prime. Obviously, G inverse will bring X prime to X and identity takes X to X and if G takes X to X prime and H takes X prime to say X double prime then HG would have taken X to X double prime. So, this is a equivalence relation. So, just like we had three different ways of looking at the quotient map, that is precisely what is happening here. So, all orbits are obviously disjoint and for each X there is an orbit. Therefore, X is the union of all these orbits under the G action. When you have equivalence relation that just means that two elements are in the same class, same orbit. If you look at all the orbits as a space Y, then you have a surjective mapping from X to that Y. Here I am writing Y as X slash G on the left because it is left action. If it is a right action, I would write it as X slash G on the other side. So, this map is nothing but X going to its orbit, its equivalence class. So, all the three pictures are here of a quotient map. So, you have all they got a quotient set here, X to X by G. One can if you find I told you right action also. In fact, every left action can be converted into right action by defining XG XG as G inverse of X. G X should define the law of associative will be going to rubber. So, G inverse of X, then it is really check that you know this becomes right action and if you have right action by the same formula, you can make it left action. Therefore, roughly speaking, there is no need to study right actions once you have studied left actions or vice versa. So, depending upon your left your right you can choose whichever one you like. Sometimes there are both the actions. One action and other action may be quite different. That is why we need to go these concepts. Here is another remark for each G in G, the assignment X goes to GX is a bijection because G is a group, you see. So, every element is invertible. What is the inverse of X going to GX? It is X going to G inverse X. So, let us write this left action by LG. Now, LG is a map from X to X and I won't say it is a permutation of X because it is a bijection. So, what you get is for each G, the assignment G going to LG defines a map from G into the permutation group, permutation group sigma X. This itself is a group homomorphism because of the associativity. LG composite LH is L of GH. That is what you get associativity. So, this is a group homomorphism. So, given an action, I have a group homomorphism from G into the group of all permutations of X. Conversely, suppose you have such a group homomorphism, they call it as capital L. Then you can define mu of GX as L little G of X means LG operative on X. Then this will become a left action and if you do again LG of this one will be LG the same L. So, there is a 1-1 correspondence between actions and group homomorphisms from G into permutation group of X. Here are a few more terminology. An action is set with transitive if for each pair X, Y belong to X, there exists a G such that G of X is Y. So, any two elements are related by an action of G. The element G may depend upon the elements X and Y. Any two are related. It is the same thing as saying, look at this one. One single orbit, the entire X will be one single orbit. Start with any point, you can go to any other point by an action of G. So, there will be just one single orbit here. It is the same thing as the action is transitive. Now, the action is called effective respectively faithful. The first let us look at effective. If GX is X for every X implies X must be identical. What is the meaning of this? What is the meaning of this? That every non-trivial element defines a non-trivial permutation. So, this is some people call it faithful also. Just means that the corresponding homomorphism from G to sigma G, say sigma X is an injection. It is a monomorphism. Okay. So, this equal to saying the corresponding homomorphism is injective. We say the action is fixed point free. Sometimes merely just free action, whenever GX is X for some X implies G's identity. So, this is very, very strong. In a sense that if you have a non-trivial element, then it will not fix any element. All the members are moved. Okay. GX is X for some X, even if it is fixed is one X, then it must be identity. Identity of course X is everything. So, such thing is called free action. Okay. Some people fix a point free, just free action also. Free doesn't mean that it is, you know, it doesn't cost anything. It costs very high. There are a few other concepts like this, which I will use much, much later, but right now I will just introduce them. Take a group G, which is acting on a set X. Suppose you have homomorphism from another group H to G. Then through this homomorphism, we can make H action on X. By the formula H of X is take rho H that is an element of G, take that action rho H of X. A generic name, this is called restricted action of G to again, this is a typo here, to H. This name is borrowed from the special case when rho is the inclusion of homomorphism of the subgroup H. Suppose H is a subgroup and rho is an inclusion, then there is a, this name restricted action makes sense. The same thing has been generalized to any group homomorphism, whether rho is injective or not. On the other hand, you can have a different point of view. Suppose you have a homomorphism alpha from G to another group K, then we want to have what is called an extension here. For that you have to construct a space X, enlarge space X as follows. So this X alpha X together with alpha is X alpha. This is a set on which now K will act. But how this X alpha define? For that you have to wait here. Let us look at this, how this done? Look at K cross X as a set and on this set, you take the action of G via the first slot K. See there is a homomorphism from G to K. So these are group homomorphisms. You can think of G acting on K by this formula namely G K. So I have to take some action on this side namely K times alpha G inverse. Alpha G inverse is an element of K. So I am multiplying them inside K, K into alpha G inverse. On this side take just the action of G on X. Essentially, if you do not write this alpha at all, suppose it is inclusion map, it is K G inverse G X. Like G and G inverse are cancelling out. You are introducing that G, G G inverse G. If you combine them, it will be no action at all. That is the kind of thing that we are thinking about. So let X alpha denote the orbit space of K cross X under this action. So this is going to be a quotient space of X cross K cross X under this action and let K X denote the orbit of the point K X. So these are the equivalence classes here. For example, if I write alpha G inverse on this side and G on this side, it will represent the same element. So that is my X alpha. This alpha will already include what is K because alpha is homomorphism from G to K. Now you can define the action of K on X alpha from the left slot. Every equivalence class, the first slot is from K, K prime comma X. K times that, you just take K K prime times K K prime comma X the class of that. So we first created room by enlarging X, created some room for K to X by just taking K cross K cross X. Then you can take a left action, but we do not want the whole of K cross X. We have to take a quotient of that. So it is a matter of verification to see that this indeed defines an action. It is all straight forward. There is nothing to verify here. We refer to this action as extension of GA action to an action of K. Note that the set on which K acts is not the same as X. We have extended it, same as X on which G acts. So this is somewhat larger and larger fact. Let us take some examples now. The simplest example of a group action is the natural action of G on itself. G acting on G where G is a group. The orbit space will then consist of just one single element because the action of G on G itself is transitive. If we have G and H, which element will bring G to H? H G inverse operating upon G will be H. That is all. If H is subgroup, then we can take the so-called restricted action of H. So H will act on the entire G, say on the left or on the right, whichever way you want. He is called restricted action through the homomorphism row from H to G of the action number one, the example one. The orbit space here are nothing but the left cosets or the right cosets according to which action you have taken. So this is what you study in group theory, right? Under the inclusion of homomorphism, row from H to G. Now if we extend, see first you have G acting on G itself, G to G. You restricted it to an action of H because H is a subgroup through wire. Same row, if you extend it, you will not get back G. You have to be careful. Okay, you will get something archer. So it is interesting to check what you get. Now that is all about general group action on a set. You know in group theory you study this one very deep, go up to all silos, theorems and so on. So many interesting results can be obtained by just studying this one. But now we want to go to topology. Suppose now X is a topological space, then we are not satisfied by just permutations as actions. But the homomorphism must be inside homeomorphism group, not permutation group. The set of homeomorphism is a subgroup of set of all permutations, right? So that is the extra condition that we need when X is at topological space, which is the same thing as saying that the action from G cross X to X is now continuous. In what way put G, take G just a discrete topology. X has its own topology. Under that mu must be from, must be a map from G cross X to X, along with the two hypotheses of associativity and identity that we have learned. That is called a topological group action. In other words, as I have told you already, the homomorphism G counter G takes values inside the homeomorphism of X. This is a subgroup of all permutations, self-homomorphism. Again, you have the same quotient set. Now you give a quotient topology to this. This becomes a orbit space, not just orbit set. What is the topology? Remember, a subset of, subset U of X by is open if and only Q inverse of U is open. In X, the set of orbits becomes topological space is called orbit space. In group theory, group actions are most useful study properties of groups themselves. Here, we shall use them to study properties of the quotient space. Starting with pretending as if we know everything about X, what can you say about X by G? So that will depend upon the kind of actions that we have. Accordingly, we will introduce a few more concepts here, definitions. An action is called even if for every point in X, you have a neighborhood U of X in X such that if you translate U by G, G U, it will never intersect U. G U intersection U is empty for all G not equal to E. So, this is much stronger than the fixed point free action. This could not be defined for arbitrary sets because there is no neighborhoods or no apology there and so on. So now, we have a stronger notion of fixed point free. G U intersection U is empty for all G not equal to E. Here, G U denotes the set of all elements which look like G X where X runs over U. G is fixed. We shall call such a neighborhood U of X an even neighborhood. It is easy to see that an even action is fixed point free. That is what I told you. Converse is not true in general. But of course, if we assume G is finite and X is also dog. Now, you start doing topology. So, this is an easy exercise. I will have to test an exercise. It is easily checked that restriction and extension of an even action are even. Even a topological space clearly the entire group of homomorphisms acts on spaces. We get more interesting actions by taking subgroups. What kind of subgroups you take? Only that is all. You see the action of homomorphisms on a topological space you do not have to define it is there. They are homomorphisms. So, which sub space you take, which subgroup you take that will define the symmetry of X. So, let us stop here and next time we will see examples.