 So, the next thing I want to say is generate. Let us take our dictionary. What is a dictionary? Dictionary if you bring a dictionary it has all possible words. The words are made out of some fundamental objects they are the alphabets. How many alphabets you need? 26 alphabets you need. You can concatenate these 26 alphabets in whatever form you want and you get a word. Some words will be meaningful, some words will be meaningless, but you can concatenate. Concatenation is like a group operation. Does still dictionary form a group? Closure property is there, empty space I can treat it to be like identity, inverse descendants. I cannot find another word when I combine with a space it will or combine with some other word it will give you a space. It is not possible right. So, a dictionary the generators are 26 alphabets, but definitely it does not form a group. The same argument we can start looking at generators for sets which form a group. For example, this example we said to I can say that the fundamental operation is pi by 2 which I can call it as a generator. I can call that as a generator. What do I mean by calling it as a generator? I can take the generator some number of times and you should generate all the elements which are there. So, if you want to get r pi I take r pi by 2 twice just like an alphabet 26 alphabets I can create a word ok. Here I have only one alphabet which is r pi by 2 with just r pi by 2 any number of times written because if I had two different fundamental operation I can do many things. I just have one fundamental operation I can do this many times, but there is also some limit this pi by 2 up to 360 degrees only I can go right. So, r 3 pi by 2 will be. So, in that sense for this particular group which is the symmetry for a square I can say that r pi by 2 is a generator of a group ok. And since this generator also satisfies as identity ok, sometimes we say order of the element r pi by 2 is 4 that 4 is because of this power ok. So, let us take some simple example like cyclic groups ok. So, let me give you some simple examples. So, whatever I did for a square I can do it for any polygon right. The rotation will be can you tell me what will be for a pentagon or a hexagon? Let us take a hexagon regular hexagon I am not drawn it very well take it to be a regular hexagon. What are the rotation operations? Someone 2 pi by 6 there it should have been 2 pi by 4 technically right. This one which I written I could have said that there is a since it is on order 4, 4 vertices are there I could write that it is a generator is 2 pi by 4 which is nothing but pi by 2 here in this case. Here it is 2 pi by 6 in general n gun if you take what will that be? 2 pi by n for an n sided regular polygon generator will be 2 pi by n that will be the generator. What is the order of that generator? Order of that element if you take r to the 2 pi over n n times it will be identity and this power is what is important. So, cyclic groups a generator let me call it as a whose order is n. So, I mean by this that is a to the n is e that is the meaning of this. So, what will be the group elements? Cyclic group elements of order n will be identity a, a is the generator all powers of it. So, order of the group is also order of the generator and is it abelian or not abelian? It is abelian. So, cyclic group is and because I am looking at cyclic group of order n sometimes the notation is cyclic group with a subscript n. So, this group will be called C subscript definition of a cyclic group is just that it is given by one generator and the order of the generator which is the order of that element defines the order of the group. You can permute it and start looking at any arbitrary element does not matter because you can do that here also you can take a square to be you know you can redefine things, but that should not really matter you call it as a cyclic group if there is one generator and it is an abelian. So, just like alphabets let me give a group with two generators let us start with let us do some examples. So, let us look at a group with two generators let me call it as a and a b order of a is 2 what do I mean by that? His squared is identity order of b is 2 identity you want to construct just like using 26 alphabets all words I want to construct all words using these two generators and powers. Let us start off. So, first is identity anyway a you can have a b in generally if it is different order you would have got to a squared a cube and so on here it is a squared is identity we do not need to do that, but you can have one more what is that one more or two more a b and b a right. In fact, you can even show that a b is equal to b a how do you show that somebody you have to first draw the multiplication table and see what happens ok. So, let us do that also. In fact, that group which is generated by a and b it is called the Klein group ok that is why I am saying a b will be b a inverse of a squared is identity a is self inverse b is self inverse. I am giving you some data I am telling you form a group generated by two different generators a and b how many elements will be there in that group. Technically the set will have this right with the generator you have a and all powers of a, but I have given you additional condition that the order of a is 2 order of b is also 2 which means I cannot have in that set a squared and b squared I can have an a b and then I could also have a a b I could also have a b a what else can you have. So, how many powers you can go if the order of that a is only 2 and order was n then you can go up till a n minus 1 if order is 2 you can only go e and a right that is why I do not have all powers of a will come to that variation I have given only the specific case. So, this also implies a inverse is a this also implies b inverse is b. So, with this set I do not think I can have anything more than this can you have anything more a b squared is a a b to the power of any power even power is does not matter odd power will get back to the same set. So, I cannot have anything more than this I agree I agree it cannot be, but I am saying if I say the group is generated by just like the 26 alphabets creates all words. So, it is not a single generator you can have many generators and you form the set no that is only for cyclic group which has one generator, but if you have more generators closure property tells you that combination of two generators should also be an element of the set generators are a and b in this case I am not able to hear you. Generators are basic fundamental objects by which you can construct all the elements of the group. All the elements we have to make sure that this forms a set which is satisfying all the axioms of the group properties that is not required that is not what is meant by that. The general statement is if you have a and b are generators any element g i of the group should be some power of n and b per m this is all this. So, let me write it if a and b are generators of g then g i should be some power of n and some power of m. If I on top of it I say that a squared is identity and b squared is identity then the group set can have e a a b b it is not clear whether a b a is independent or new, but right now as a set I can write this and then make sure whether a b is b a. a b a will also be allowed good point I think. So, what you are saying is that I could also have a b a. In principle if there is no condition on a b with b a then this can I think go on like this. So, this is possible, but let me also add one more condition a b equal to b a ok. So, let me put that condition suppose I put this condition then it truncates here I agree with it. So, let us put that condition. So, I am going to take a squared equal to e b squared equal to e and a b equal to b a then generate the group if I say then you know that otherwise you know it can go in the I agree with it inverse of yeah. So, let us do that inverse of a b is what b inverse is b a n. So, that is also going to be a b by this property ok. So, everything is taken care of and this forms a group whose generators are a and b, a and b are of order 2 and a b is same as b a is what I have given as a condition a b a b inverse is b inverse yeah this is fine right. What is the confusion on this? This one. No, but any operation when you do it as operators even in quantum mechanics these properties we do not violate. If you do 2 operations by any group operation if you want to do the inverse of it then you have to reverse it with the inverse that is always true not only for matters those are the linear algebra properties we do not want to violate linear vector space ok good. So, let us write the multiplication table here for this group which I wanted and this is what I call it as a Klein group order of this group is again 4 what are the subgroups here anyway I leave it you to write this what are the subgroups I think I have to stop a is one subgroup H 1 H 2 is e b is one subgroup can you also have e a b this is a way to get some training on what is a subgroup how to write the multiplication table and so on. So, I have just put in the multiplication table for you for the Klein group, but you can yourself do it and check it out ok this will be there on the website. Let me stop here and on Monday I will continue with conjugation conjugacy class and so on. It will in general it would have involved, but if I give this condition in the beginning itself yeah inverse of this. So, because a b inverse is a b that is why