 In the previous videos for this lecture one on group actions we introduced the notion of a group action and provided some important examples. We're now in a position where we actually want to prove a very important first result about group actions is that group actions naturally place a partition on side of the g-set x. And because there's a partition associated to that is also an equivalence relation. So what is that? So imagine that x is some g-set where g of course is a group. We define a relation on x where we say two elements x and y are related to each other if there exists some element g in the group such that g acts upon x and gives you y. So if y is the image of x with respect to some element acting upon it then we say x is related to y. We claim that this relation is in fact an equivalence relation and this is called g equivalence on x there. The equivalence classes of this relation are called the orbits sometimes called the g orbits of x in case g is ambiguous in the situation. It's commonly referred to as o sub x. So it's the orbit that contains the element x like so. So if we can prove that little twiddle here is an equivalence relation then it has associated a partition that partition we call the orbits of the set x. Let's prove that it's an equivalence relation. So there's three things we have to check. We have to prove that our relation is reflexive symmetric and transitive. We'll start with reflexive here. So take any element inside of the g set x. Well if you act on it by the identity you get back x by the identity axiom of a group action and therefore this tells us that x is related to itself so the relation is in fact reflexive. Let's now check for symmetry. Let's take two elements inside the g set x and y such that x is related to y well by definition of our relationship we have that there exists some group element g so that g acts upon x and produces y. We then want to reverse this process. So what I'm going to do is I'm actually going since g is in fact a group g inverse belongs to g and it can act upon any element of x in particular g inverse can act upon y. Well what happens there well by assumption y can be so to speak factored as g acting on x like so so g inverse or g inverse dot y is g inverse dot g dot x notice you have two actions happening right here two elements acting that's to say well by the compatibility axiom we can factor this instead you know sort of a reassociated as g inverse g dot x which of course g inverse times g is the identity and then the identity acts on x so you get x so notice here that if there's an element that sends x to y from the group then there's its inverse will send y back to x now you you'll recall when we defined a group action we only took on two axioms what it says what does the identity do and what do products do now groups have a third axiom but inverses but for group actions we don't need an inverse axiom because in fact we can we can infer from the identity and compatibility axioms what inverses have to do so if one group element does something the inverse will do the opposite it'll send everything back to where g sent it and particular this then proves that our relationship is symmetric so next we have to check transitivity so let's imagine we have three elements that belong to capital x we'll call them x y and z let's suppose that x is related to y and y is related to z so that tells us there exist two elements of the group we'll call them g and h so that h acts upon x to produce y and g acts upon y to produce z so then we're going to claim that the product of g gh will send x to z and this follows immediately from the compatibility axiom since we have the product gh acting on x there using compatibility this becomes g acting upon h acting upon x well h acting upon x gives you y g acting upon y gives you z and therefore we get that x is related to z thus giving us the transitivity axiom and so in particular we've now shown that this relationship is an equivalence relation it gives a partition on the g set x and we call the equivalence classes the orbits are the g orbits again if we have to specify so when you look at this proof here I want you to pay close attention to where the axioms of a group were used and where the axioms of a group action were used to prove all of the three axioms of an equivalence relation these axioms wonderfully interplay with each other and it just shows you the rich theory we have when it comes to groups and particularly down group actions so let's look at one example and compute some orbits of a group action I'm going to take a permutation action I'm going to take the set one two three four five so s five of course acts upon that set but any sub group of s five also acts upon that set so in particular I'm going to take the subgroup generated by the the three cycle one two three and the two cycle four five this is also the product of the two sets there's the cyclic subgroup generated by one two three there's the cyclic subgroup generated by four five so this group is really isomorphic to z six it's a cyclic group of order six but nonetheless take this group here how does it act upon the set one two three four five what are the orbits let's consider the orbit of the letter one where can one go well if you take the number one and you multiply or you act on it by one two three that'll send you to two so one and two are in the same orbit if you have the letter one and you act on it by one three two it'll send one to three and so that's an option if you take the element four five it'll actually leave one fix it sends it to itself one two three and then four five it'll send one to two if you take one three two and four five it'll send one to three and then I skip the idea but the identity will send one to itself so the element one can be sent to itself it can be sent to two it can be sent to three that gives you the equivalence class the orbit containing one so the orbit containing one is one two three which is also the orbits containing two and three and then lastly what about four where can four go well the identity since four to itself one two three since four to itself one three two since four to itself four five will send four to five as the name suggests one two three and four five will send four to five and then one three two and four five will send four to five and that's it so four goes to itself it can go to five and that gives you the orbits for four and the orbits for five and so that brings us to to the end of lecture one for math 4230, our introduction to group algebus. Thanks for watching. Throughout any of the videos in this lecture or through any videos in this entire series, if you have any questions, feel free to post them in the comments below and I will gladly answer them at my soonest convenience. So thanks for watching. If you learned anything, give this video a like. And if you wanna see more videos like this in the future, please subscribe to the channel. Thanks everyone. See you next time.