 So in this video I just want us to gain some deeper insight into cosets. Remember if H is a subgroup of another larger group G and A is an element of G then we have AH that's the left coset of H under A and HA is on the right hand side is the right coset of H under A. Now Kayleigh's tables really help us to get a deeper insight into what is going on here. For if this is the whole of G that I have here and the whole of G that I have there H is a subgroup of the whole G and I can write this in any order in any order the elements so I'm going to take all of the elements in H in a set that makes up the group H and I'm going to bunch them all in the beginning and then all the rest there's nothing wrong with the order I can put that in every order so I have my binary operation here and I'd have all the elements of the binary operation there and if H if A is an element of G it doesn't say anything about it being an element of H it might be it might not be remember that so I've got these two scenarios here A is not an element of H and remember how Kayleigh's tables how they work so I'll have all the elements of H and this will be A binary operation with H and here it'll be H binary operation with A and the way that these work remember we've shown with Kayleigh's theorem you know these will be unique and they will be it'll be exhaustive in other words all the elements in H you know all of these elements of these binary operations they will all occur there and they will all occur there and especially if you look at this it's easy to understand because this is just a mini Kayleigh's table you know for a single group that all these rows and all these columns are unique so they'll all fit in there so remember on this side I have A not an element of H and this side I have A being an element of H it says nothing about A being in there it might be and it might be in there and it might not be in there so what insights can we gain from the fact that it is there well if A is there it really implies here that that it's also an element A is also an element of AH the left coset and A is also an element of HA the right coset why is this so because remember I define these as this and I define this one as this but H is an element of H H is a group that implies that E must also be an element of that group so if this H is E which it might be the identity element the identity element the identity element on this side I'm left with A and I'm left with A so A would then also be an element of this left coset and right coset because the identity element must be there and you can see that insight that we gain here we can see that that insight so this is now going to be an element of as I've written here of the left and right cosets and from this from from from the closure problem from the Cayley's theorem we know it must be in you know we know it must be inside one of those because it is exhaustive all the elements must be all the elements must be there therefore A must also be one of those and A must also be one of these and it's easy to prove that more interesting though if A is not an element of H you'll note that this left coset this right coset is not going to be in the left and right coset which are there you can see that visually but how do we prove that how do we prove that if A is not an element of H that implies that really the left coset its its intersection with H is the empty set and the right coset's intersection with H is also the empty set how do we prove that how do we prove that now let's prove that by contradiction and we first we're going to consider this left coset A H and let's say let let this not be let the intersection so the intersection the intersection is not the empty set and we're saying that A is not an element A is not an element of H so what do we do it's the we're just considering the left here so we say A the left that is an element of the left coset and let's just choose something let's make it H star whatever let's say that's an element of H and I said that this is not the intersection is not empty since these are arbitrary values I'm going to say well if the intersection there must be something that is equal between this these two so let's do that it's equal to H star and I can just rewrite this and get A on its own so I'm going to say A equals H star binary operation with H inverse now this is an element of H this is an element of H seeing that H is an element H is an element of H H is a group that implies that it's inverse by the inverse property must also be an element of H so I have the binary operation between two elements in a set that in a group so that product that binary operation there the product with a binary operation that must also be in other words this binary operation with H inverse that must be an element of H but I've just shown that this equals to A therefore A must be an element of H and that's a contradiction because my my first assumption is that it wasn't so that is a contradiction so we know that that is true that must be an empty set and the same is going to be go for this I'm going to under I'm going to assume that the intersection is not an empty set it's not an empty set and I'm going to have HE and that is an L that is going to be HE that is an element of of HE my right coset again I'm going to choose this star as an element of H I'm my base assumption was it's not the empty set so we can set them equal they arbitrary and we've chosen them just such that these two are equal to each other because they fall in that Venn diagram where it's not an empty set and again I can have that A equals H star binary up oops now it's going to be this way around remember left so it's inverse binary operation H star that is an element of H that must be an element of H by this same property so that's an element of H so A is an element of H because that equals that and that's also a contradiction because of my base in this assumption I said here that A is not an element of H that was my base my base assumption so this is a this is a contradiction and therefore this holds as well and you can clearly see that when we look at this picture here the the intersection between these is going to be the empty set so you can see this is a very nice proof Cayley's theorem will help us there and on this side by contradiction very easy to show these but if that is not so then A then then A cannot be A cannot appear there in or at least the intersection between these two if A is not an element of H that that intersection between those two will give us give us the empty set because remember I'm now seeing just a set that makes part of the group and this is a coset in other words it is a set I'm just looking at the elements in a set so a very nice proof and a nice deeper intuition of really what is going on when we are talking about you know we are talking about a coset