 Good, so we're still on this very mysterious little journey somewhere with these cosets and it's coming quite soon I don't want to spoil the surprise One little step further remember that we had Nice little visual representation. We have our group G here We have our group G here. We have a subset of that group. We call it H Okay, such that H is a subgroup of G and We had instances where A was definitely One of the elements of H and A was not an element of H. Let's choose that. So that was A. We generated this whole Left coset that was A. We generated this right This whole right coset Now the question would ask if we were to take one of these elements in here Let's let's just consider the left cosets here. If I were to take one of these Remember it's A and you know, this could be one of them obviously will be the identity element and Let the others be H sub 1, H sub 2, H sub 3, H sub 4 You could name them like that, whatever you choose. What if I were to take any one of them and I Take that as a new element and I generate its coset I generate a new coset of H on that. So instead of if I have a left coset that being This such that H is an element of H. I now say I'm going to take one of these and generate its coset So what I've got here is this new element and it's going to be one of those elements in there And I want to generate it's it I want to use it to generate this coset of H. So instead of using that I'm using this So it's this with the whole of H. I've just got to be slightly careful here I've just got to be slightly careful here in as much as Remember the one would be A with H sub 1, 1 with A with H sub 2, 1 with A with H Binary operation H, H sub 3, etc. etc. So once I have one of them this one is now fixed and This still refers to the all of them. So let's make this, you know, just something different such as This is an element of H and this is my new coset The theorem is that we're trying to state here is this is still nothing other than the left coset Still nothing other than the left coset and that looks a bit strange, you know Is that so because say for instance that would be here That would be this element here and if I'm going to generate that coset. So that's that one It's a bit small apologies for that you'll have to increase the scheme size, you know These are not the same and Cayley's theorem showed but what we're stating specifically here is the order in which these things go Will not be the same because remember they cannot be by Cayley's theorem So the order will be different around from this one But exactly the same elements that are there will be in that coset and is that possible to prove? Yes, remember we can use associativity. So let's just have a look at this little one. So that is going to be this I Can do that and now I'm concerned with what this is this says I'm taking an H which is in there this little H and I'm creating the left coset with all of them So this is actually this part here. This is just actually this So it would be one of these in here and for the right cosets one of those in there And that's just the whole of H all over again. So this is nothing other than just the whole of H So what I'm doing here is A with the whole of H such as H is an element of H and what am I left with? The left coset of H as simple as that and Visually and we don't have to reprove this because we did it before when we said a was an element of H Now I'm saying that I've chosen a specific H one of them here It is inside of that. You know, it might as well have been this is this exactly the same argument as we had a an element of H and I'm doing its binary operation with the whole lot of them So for the left and for the right cosets and you know, I'm just left with the whole of H And I'm just left with the whole of H. So that's just the whole of H Nothing other than that. So that's a little surprising fact and if This the same would go for here if we had this If that was our element in what we would have here is you know H To star all of them with this specific one with that specific one And these two are going to be exactly the same same Same argument and I think visually it makes a lot of sense that that is so and we've actually done all the proofs You know so that we can just reference those proofs and just Stating stating all of that the importance is just to realize that this is now some H in here with the whole set Some H in there with the With the whole set so with a whole subgroup then the elements that make up The set of the subgroup So quite quite easy to see in a bit of a surprise But another just something we need in the pocket to get to our end journey with these cosets