 In this video I want to talk about uniqueness of elements in a group, because in the previous video we talked about how we can denote the identity as an E or as a one or something like that. But that's kind of putting the cart in front of the horses, isn't it? Well how do we know there is the identity? Why isn't it like AN identity? In English we use these articles to kind of specify like plurality a little bit right. How do we know there is only one identity? When you look at the axioms of a group it just says there exists an element that has such and such property. It doesn't say that that element is unique. It turns out that for a group the identity element is unique. That is there exists only one element that has the property that EG equals GE which equals G for all elements of the group. The identity element is unique. Now when one wants to prove that something is unique, typically what you do is you take two elements with that property and argue they actually have to be the same element. And so there's kind of two ways you could do that. You could do a proof by contradiction to be like for the sake of contradiction take elements E prime and E double prime. And if there are identities they have to have the following property. G times E prime has to be G and also E prime times G has to be G. I do both cases because in a group we don't assume that it's commutative. And then we take another element E double prime and assume that G times E double prime is equal to G which is the same thing as E double prime times G. So we take two candidates of the identity so that we have these two candidates. We have two candidates for the identity. And so we have two elements that are going to act like an identity inside of this group. How do we know there's only one of them? Well one argument could just be that for the sake of contradiction suppose there's two of them and get a contradiction. I'm going to take a slightly different perspective here. I'm just going to take two of them. I'm not actually supposing they're distinct but I'm going to show that they're in fact one and equal. Because their identities this element G that we're playing around here is arbitrary. This property holds for any G. In particular E prime is an identity for E double prime and E double prime is an identity for E prime. So if you take this element and expand upon it a little bit right E prime times E double prime. Well if we think of E prime is an identity for E double prime that means this should equal E double prime. But conversely if we think that E double prime is acting as an identity on E prime this gives you E prime. And so you can see then connecting them here that E prime is equal to E double prime thus finishing the proof that two things are actually one of the same thing. If you did a proof by contradiction right now you would have got a contradiction to our original assumption that they were actually distinct. And so because I'm just contradicting the original assumption I don't really think there's a need for a proof by contradiction but just a thing is still a valid argument. So we actually do get that there's one and only one identity inside of the group. We can do a similar thing for inverses that when you look at the inverse axiom it just says that for any element there exists a inverse but how do we know there's only one inverse? There could be multiple inverses right? Well the group axioms help us figure this thing out here. And so I want you to be clear that when you look going back to this proof of the identity here all we used was that we had a two-sided identity right? And so I want to make sort of like a comment here. We didn't even use the full blown axioms of groups here. So if there exists a two-sided identity this is true for any type of object any type of algebraic object. If you have a two-sided identity it is unique. Now if you have only one-sided identities things can get a little bit more funky which is why we required in the axiom that the identity would be two-sided. But let's get back to the idea of inverses. If G is any element in the group then the inverse of G is unique. So to prove this we're going to take two candidates for an inverse. So we're going to say G prime and G double prime are both inverses for G. What does it mean to be an inverse? Well for the sake of G prime it means that G times G prime and G prime times G is equal to the identity which we'll call it E right here. And what does it mean for G double prime to be an inverse? It means that G times D double prime and D G double prime times G is equal to E. And so what we have here by assumption is two candidates, two candidates for inverses for the inverse of G which like I said earlier we will call the inverse G inverse. But at the moment there could be multiple ones right? And so what we're going to do is we're going to consider the product the product of all three of these elements together. So if G is an inverse or I should say if G prime and G double prime are inverses G they should act like inverses of that. So let's consider I'm actually going to start off with this point right here. Take G prime G and times it by G double prime right? Well if G double prime is the inverse of G that means G sorry I said that I mean that's true but I want to say if G prime is the inverse of G that means G prime times G is the identity. And the identity times anything will give you back the element. So this object right here is equal to D G double prime. But conversely because we're in a group we can re-associate I can redo the parentheses and get G prime times G G double prime right here. And so in this situation if you take G times G double prime since they're inverses you get G prime E which is equal to G prime. And so you see that these things are actually one in the same thing inverses are unique. Now this time we were using of course the inverse axiom to show existence here because to be unique unique means that you have at most one but you also have at least one. The inverse axiom gives us that they're unique. To use the inverse axiom you have to have the identity axiom because the inverse axiom references the identity so if no identity exists the inverse axiom becomes null and void. But you can also see here that the associativity axiom was necessary. So to prove the uniqueness of inverses we used all three group axioms associativity identity and inverses. Now I want to show you why is it so important that the inverses be unique? Why is it so important that the identities be unique? I'm going to show you two very quick applications of the inverse element being unique. So we're going to show what's commonly referred to as the shoe sock principle. The shoe sock principle why do I call it that? So if we have a group G so G is a group and elements G and H are elements in the group we often will denote groups with capital roman letters and then the elements of the group will be lowercase roman letters. Typically we like to use capital G for generic groups lowercase G for elements and set group. That's sort of convention we do. So what the shoe sock principle has to do with is if we take the inverse of a product so we take the inverse of G times H. This is actually equal to H inverse G inverse. Notice how the order gets swapped around when you take the inverse. In general groups do not demand any commutation on the operation here. The multiplication is not necessarily commutative so the order matters. And so when you take the inverse of a product you reverse the order and the principle is the following. In the morning when you get ready for the day you first put on your socks and then you put on your shoes. So that's how you get ready for the day. On the other hand when that's when you put them on but when you take them off the roles get reversed. You first take off your shoes and then you take off your socks. So the inverse operation switches the order of these procedures around and that's what we're trying to claim right here. And this proof is basically just the uniqueness of the inverses. What we're going to do is the following. We are going to take G times H and I'm going to multiply it by H inverse G inverse like so. So we're going to multiply these things together and I'm going to prove that this is the identity. Well because the group has an associative operation we can redo parentheses so we get H inverse times G inverse G times H like so. But G inverse and G they're inverses of each other so their product will give you the identity right. So we get the identity EH right there. Well EH times anything. EH is just going to give you the element back. It's just the identity. So you get H inverse times H which as those are inverses you get back here the E. So I want to give you where associativity was key in this discussion. Obviously for inverses they must exist and you must have identities but the associativity axiom was necessary so that we could redo parentheses at our leisure right. So what we have now shown is that H inverse G inverse when you times it by GH gives you the identity. But wait a second it should be that GH inverse times is by multiplied by GH to give you dot dot dot dot dot. This should be the identity. Since inverses are unique the only way that H inverse G inverse could act like an identity is because it is the identity. If your element walks like an identity and quacks like an identity then it has to be an identity because they're unique. In this situation if your element walks like an inverse and quacks like an inverse it must be an inverse because inverses are unique. The only way this element could give you the identity is it was the inverse but look at it it worked. It's an inverse and the other's direction works as well right. So you see that I illustrated doing which one did I do. Here in the proof I wrote up here GH times H inverse G inverse right. I did the other details. The thing is the two arguments are basically the same. You don't necessarily have to do both. I typically would say the other one is similar. So the uniqueness of the inverses shows us the shoe sock principle. Let's look at one other corollary of the uniqueness of inverses here. So in any group if you take an element G the double inverse is the original element again. This again follows by inverse uniqueness right. If you take G times G inverse that gives you the identity because G inverse is the inverse of G same thing on the other direction. But wait a second, wait a second, wait a second. If I take G inverse the only thing that should multiply by G inverse to give you the identity would be the inverse of G inverse right. By construction G inverse that's an element of the group. It should have an inverse element. But as G is already doing that then uniqueness implies that G inverse inverse actually is the element again. So the simple statement that inverses are unique can be a powerful tool to prove more powerful theorems, propositions, corollaries about properties of groups.