 In this video I want to start looking at some more interesting aspects of group actions and we're going to start looking at the orbit stabilizer theorem and there are two words in there orbit and stabilizer so let's start looking at the orbit, the orbit of a set. So we're going to imagine that we have two things we're going to have a group G it's going to have a set I like writing that is G set these days and some group action and we're going to have a set A just a set of elements and what we're going to take is we're going to take a specific A which is an element of A and we're going to allow all the elements in this set to act upon this A so that's going to give us this thing called the orbit of A the orbit of A and that is a set G dot A such that G is an element of G set so what you can think of you take all the elements of G all of them one after the other you acted on this one specific A and that will give you a new set and that set is called the orbit of A first thing that we'll notice is really that A is an element of the orbit of A why is that so well we know that the identity element of the set G set is one of the elements in the set and if we act on A we are just going to get A so we know that A is an element A is really an element of the set so that is one thing we can look at just to see what goes inside of this set the next thing we want to look at is if we take this set all the G's acting on A and we take any one any arbitrary element of that we've seen A is already in there but we take any arbitrary element of that let's take G dot A so that G dot A is an element of the orbit of A as simple as that we just take any arbitrary one of them what we are suggesting is that the set that we get from doing this from taking one of these elements as our new element as our new element so we just take this as a new element of that orbit that we're going to get exactly the same orbit back we're going to get exactly the same we're going to get exactly the same orbit back and how do we how do we know that well just imagine that we take G dot A the what we are what we say the definition is so now we're going to say that GA that's an element let's call this of GA so GA is an element of the orbit of GA and let what can we do let's take G star this time and we act on G dot A so what we are saying is this is exactly the same as our initial definition of the orbit of a set we take all the elements G so this is going to be a new set and this time we just call the G star just to make sure that we're not talking necessarily about the same G is an element of G set in other words we are going to run through again all the elements of G set the set from my and we're going to act upon this one element arbitrary element that came came originally from the orbit of A so I should actually just say that it is part of the orbit of A let's just do that the orbit of A and we're going to call this the orbit of GA because I'm taking a new element and remember that element is just another element of my set here and I'm doing this whole orbit thing again now by the properties of group of action I can rewrite this as G star and composed with G that's the binary operation of that acting on A that's what I can do but by the properties by by of of a group this is just another element of G this is just another element of G so this writing this is nothing other than just rewriting this because I'm taking all the elements of G and I'm running all of them and I'm acting on a all of them and I'm acting on a because this is just you know under the column of G in my Cayley stable of group operation some by Kelly serum I'm just running through all of those again so I'm running through all of those acting on a running through all of G set acting on a and this is exactly so this is OGA is this the set such that this this is all elements of G set so I'm running through all of them again and what am I left with I'm left with the orbit of a because that is exactly the same thing so those are exactly the same thing so I can take from the start taken element of a all the elements and G set act on that that gives me a new set and I'm saying this specific set which is very unique because it comes from a is I'm going to create a set and I can take any one of those as a new element I'm gonna take then a new it as a new element and I'm going to let G all the set all the elements G act on all of them again and I'm going to land up with exactly the same orbit I'm going to end up exactly the same orbit so that's very interesting the other thing that we really need to note is the fact that that this orbit of a is a partition it's a partition on a so it takes a and it partitions it well what it might do is it might just be equal to a this is the trivial one but let's imagine that it's not trivial and that there's another specific orbit and we're going to take an element B element of a so that we create the orbit of B remember that's going to be G dot B and we run through all the elements of G set we run through all of them and what we are suggesting is that we can have these two be this joint so that if I have this element here and it is it's not part of that you know if we can show that then it is this joint so that we have this that OA the intersection with OB that that is an empty set they are this joint how would we prove that how would we prove that now let's just to the contrary have this G dot a and we set that equal to G dot B and we're going to make this G star let's make this G star so that we know there's not necessarily that we start off with the same element as this we run through we run through all of them so let's say they are not this joint so that you know that these two I can find an element in the orbit of a and that's going to be equal to some element in the G in the orbit of B imagine then that that would mean that they are not this joint but let's act on the left hand side by G stars inverse G dot a and it's completely legitimate to do and I'm going to G star dot B by the properties you know what we can do so this is going to be G stars inverse composed with G star and that acts on B we know from the properties of the group that that would be the identity element so we just have B on this side and what do we have on this side well we're just running through all the elements by the Cayley's table for G we're just running through a whole one by Cayley's by Cayley's theorem that means I'm running through all of them and what do I have I'm running through one by Cayley's by Cayley's theorem that means I'm running through all of them and what do I have I have nothing other than G dot a here G dot a element of the orbit of a so now B is an element of the orbit of a so if this was so then it means it is originally part of this orbit of a so originally part of this orbit of a showing that it if it is not part of the orbit of a then we have to disjoint then they are disjoint showing clearly that that if I if I do create an orbit say it is part of it is one of the partitions of my set a so in summary a couple of beautiful things we know the definition of an orbit we take a specific one we know that a itself will be an element of that orbit we also know that if I take any one of the elements in that orbit and I try to create a new orbit through that it's just going to be exactly the same as the original orbit and that these orbits they really partition they partition a set they partition the set in these disjoint in these disjoint sets so that is the orbit in the next video we're going to look at what the stabilizer is stabilizer and that's for the next video