 In this video in the next I want to talk to you about group actions now group actions are not always the easiest thing to understand because when you read a textbook or when you When you when there's a lecture that sort of goes Quickly over some concepts, and you can't really grasp what is going on here And I want to just use an example So when you use an example because I think it's important to understand what group actions are So the idea behind group actions is that we have this this the first this first video is just going to be about something That's called faithful faithful group actions And then I'll just show you some examples in another video about unfaithful Non-faithful group action. So let's just stick to faithful group actions. I don't mean don't worry what it means It's the concept of whatever group action is and I'm going to have this group G And it's going to have some set and let's just call that set G just to make it easy in a binary operation and then I want another set A and For you know, we call it call it a target set doesn't matter what this is just a set So I just have a set of elements in the group action What that is is the is the action of the elements of the set on the set And what that does is it binds properly together the fact that each of the elements in this set Can be a representation is a representation of one of the permutations of the set and we've seen that before So that's all that group actions was trying to do it is just to show that there's this connection between elements of a set in a group and the permutations of another set and we've seen this before and that was The set A Let's have a set a being one two three So just three elements is not the numbers one two three just representing three elements So three elements meaning I'm going to have six permutations and they are the permutations and we gave them names before so We named that the e that is just the the identity permutation takes one two one two two and three two three and This just swaps one and two so that was tau one two We called it this swapped one and three so there was tau one three this swapped two and three So that was tau two three and we called this one sigma and we've applied sigma twice. We called that sigma squared So we've seen that and our group G was made up of these six elements and each of them was a representation of another sets permutations and we also showed that if we do a composition of these of these We'll get a new one a new permutation and that permutation would have been the same permutation that we've gotten if we did a Binary operation between these two that would have given us another one of these It's a group says closure and that would have been exactly the same as you know The permutation is the composition of two of these permutations and we showed that so we have we know already that this is there's this intuitive Connection between permutations and and elements in a group and that's all the group actions are saying that there's some action Of the group element on a set So there's a group element and it has an action on a set and it gives me that you know That's that's kind of what we're trying to say here So what we're going to do is we take we take the the Cartesian product of G and a and yeah I refer to the s the set of G that one The group so so if I if so that would just be this set of all elements g comma a such that G is an element of G set the set that makes up the group and We have and we have a is an element of a so this whole all these and if we looked at This instance example that we have here. So we'll have e comma one e comma two e comma three So that would be you know all the ease Running just that's one of the G's running through all the a's and then tau running through all the a's tau two And what I've done here is I've just made this tau one tau two tau three sigma and sigma squared So don't worry about that. Not seeing the same here. So let's just make this tau one and tau two and tau three and That is that you know That that really is that and all we know all we're going to do now is we're going to have this mapping Let's just call it f and all we're going to do is just map all of these G comma a and we're going to map them to one of the elements of a and You know, this is exactly what we do here and instead of writing g comma a we usually write a g dot a Just a notation and books just to confuse you So this would be this would be akin to a dot one and this would be e dot two and this would be say tau Two dot three and what's that going to equal well that equals one that equals two and this one equals three And so for this tau one, which is there. What does it do to one? It takes one, you know It takes one and it changes it to two So that's going to be two. What does it do to two? Well, it takes two to one And it takes three to six six three two three So for tau two, which was this one it takes one To three it takes two to two Where are we taught to it takes one two three stays two stays with two and Three goes back and three goes to one if we look at our three here one stays with one two goes to three and Three goes to two for sigma one one goes to one goes to two Two goes to three and three goes to one and here one goes to three and Two goes to one and we have the fact that three goes to two There we go. So that's all that is happening I'm taking the Cartesian product one of these and I map it onto one of the set elements in set A I Can also do this see this function sigma g don't worry about this is not the sigma It's not the same as all of these sigmas and what that can do is I can map g dot a to a So this this group action of g on a maps back to a so I'm actually just taking one of these and I'm mapping it back to that and We can really show by the fact that we can get this inverse That equals a and that's be the same as sigma g Sigma g inverse of a And by this sigma g dot a by that I mean g dot a So, you know by this I mean g dot a it's just another way of writing this and then the inverse of that G inverse of that and that would be the same as this way around showing that this is actually a bijection So that this mapping is actually just just a bijection which we needed to be What we want to know what we want to say there are two properties two axioms two properties that must be obeyed if I take G to dot a that is going to be you know one of these With one of these and that's going to give me another one of these and then Which will now become one of these and I take it with that so I do this G1 dot g2 dot ga that That would be the same thing as getting the Which is just g1 g2? Dot a six This different ways to write that this was this is property of them number one that must be so that if I To were to take the and that's what we just had when we when we first looked at this if I took the binary operation between any of these two I You know and Look at what it does to one of the elements that would be the same as if I did this first I do this get another element plug that three into whatever it was there get that and I'm going to get the same Answers on both sides. I think I want to clear the board and actually show you By one of those that this is actually to true and then just very simply that e dot a must be a So don't get confused by writing it like this It's just the same thing as writing it writing it like that But I think you can understand intuitively here that this group action this group action on another set Just think we think of it in these terms The difficult part is just conceptualizing the fact that we have a bijection here And I've not shown you that these two are actually the same and this is called a two-sided inverse and in set here If you have a two-sided inverse it proves that this mapping here is actually a bijection It's just actually a bijection because what what we're mapping here with Sigma is we just mapping a to a And if we're just mapping a to a what are we doing? We're just creating permutations So all I'm saying here is that we have these permutations and the next video I'll just show you we just end up with Cayley serum here actually because you know, we'll have one two three will have one two three will have One two three, you know, they'll all be there's permutations of each other these things fit together beautifully for now I just understand this concept of the fact of the action of a group on another set as this representation of This the elements in this group is permutations of this faithful meaning. There's there is a bijection there And we'll talk about I'll show you that you can also get non-faithful groups in the next video I just want to clean the board. Let's just take any arbitrary ones of those that will just Take some and just show that this first property does in fact hold for this So I haven't you know, I haven't proven any of this for you I haven't shown proofs in this video because I just want you to understand really what a group action is It's just connecting these two to each other and you might get confused just by using this different Different notation the notation is just another way to write What is actually just happening behind the scene? So let me clean the board and we'll do we'll do one of these as an example So let's let's like g1. Let's have it here g1 Let's just make that tau 1 and g2 will make sigma and we'll see what happens there So we're going to take the tau 1 of the sigma of a Remember that that would be nothing just saying that that I'm just take you know Just wondering what what does sigma do what does sigma do to two and there we go. So sigma dot a So where's sigma and we we've forgotten to say what a let's make a equal to two So we have here. We have tau one dot the sigma of two So what is the sigma of two? Yes sigma? Sigma of two that is three So that is the tau one of three Tower one dot three. So where's tau one dot three? Well, that's just equal to three and Let's see if we have this tau one binary operation with sigma If we have those and we apply it to you know that to a so what does tau one and compose the sigma? So tau one where we have where's tau one. Let's just think this through. So tau one Is what did we make tau one is tau one two? So that was tau one two. So what does that tau one two means it takes? one to two and two to one and these three on its own and sigma Sigma was going to be What do we sigma sigma sigma sigma sigma one goes to two So one goes to two We're sigma sigma one goes to two one goes to two One goes to two two goes to three And three goes all the way back to one. So that's what we have. So with these two combined. I actually have one two Composed with one two three And I've got to do that Hey So let's see what happens here one goes to two one goes to two and two goes to one So one just actually stays on its own Two goes to three So if I do two two goes to three and three just stays with the cells so two goes to three So I'm left just with two three. So which one of these is two three that was tau three So we have tau three two and If we have tau three Dot two that equals to one be all three Exactly the same thing. So our property does hold so this is really a representation This is a group action. This is an example because this property holds of a group action on a set This property holds and of course if we do this one, you know, it's always just going to always is going to have that So that is that that is the faithful group action On a set of which this is always a beautiful example But this is all this is the only thing that's happening behind the scenes forget for forget for a minute just all these you know Conceptual proof or these proofs that you have to go through rather rather just understand What this is this connection between the elements of a group and the perm each of those are being a permutation of a Set and that is a group action on a set