 In our previous video, we proved what we're calling theorem 914, that if two groups, G and H or isomorphic, then there are some properties that must be the same about the two groups. That the two groups will have to have the same order. If one's abelian, the other one's abelian. If one's cyclic, the other one's cyclic. If one has a subgroup of order n, the other one has to have a subgroup of order n. These element, these properties right here are often referred to in the literature as invariance. These are things that do not change. They do not vary as you go from one group to the other because the two groups are isomorphic. Because after all, when two groups are isomorphic, we're not saying they're equal as sets. There could be some cosmetic differences. We saw previously that the group Z4, with respect to addition, it has its elements 0, 1, 2, 3. This is the same group as Z4. That is, we're looking at fourth roots of unity. So you get 1, i, negative 1 and negative i. Clearly there's some important differences here, right? The names of the elements, the labels themselves are different. The name of the group is different. We call one of the operations addition and the other operation multiplication. These are all cosmetic differences. This is sort of a linguistic difference between it. This is just how we describe the group. But the algebra of the two is going to be the same. And that's where this idea of invariance comes into play. An invariant is something that doesn't change as you go from one group to the other. So like both of these groups are abelian. And so that's something that's true when you go from one group to the other. These are both groups of order four that stays the same. They're both groups that are cyclic. They both have a unique subgroup of order two. Those are things that stay the same in variance. Now the reason why invariance is so important is what happens if you want to show that two groups are not isomorphic. Showing that two groups is isomorphic just means here's the isomorphism for the world to see. You just have to give someone a bijection between the two groups that's homomorphic and then it becomes an isomorphism. It preserves the operations between the two. But how do you show something that's not isomorphic? You don't show the world an isomorphism. Like, look, my hands are empty when I try to display the isomorphism. Well, that doesn't make any sense. That's sort of like the same argument about Bigfoot, right? Is Bigfoot running around the California mountains or anything like that? Well, I can't find him. Does that prove he doesn't exist? Maybe Bigfoot's really good at hiding. I have seen numerous movies about Sasquatches and Yeti's to explain why man hasn't found them but why they've existed the whole time. Admittedly, most of those were comedies and fictions and things like that. But nonetheless, is Bigfoot a good hider? And that's why we can't find them? Or can we not find them because he doesn't exist? That's that's the concern we're in right now. If we can't find an isomorphism, is that because it doesn't exist? Or does that mean we're bad at finding them? Which could very well be the case. I mean, at the time of this recording, the Riemann conjecture is still an outstanding conjecture, one of the seven millennial problems of 2000 worth a million dollars if you can solve it or disprove it, right? And the Riemann, the Riemann hypothesis has to do with the existence of nontrivial roots to the so-called zeta function. I won't go into all the details of that. But you know, the best mathematicians in the world have worked on this problem. Yet no one can find nontrivial solutions. Does that mean they don't exist? Or does it mean they're just really, you know, very, very well concealed, right? That these are sort of the problems one has what one struggles with when you try to prove non existence. So the usual strategy one takes to prove non existence is actually proved by contradiction. We can't prove that it doesn't exist if we can't find it. So what we're gonna do is we're gonna assume that it's true. We will assume that it exists. And then we find a contradiction, a logical contradiction that would follow from the existence. And so if we can get a contradiction, then we must then we didn't have to take the opposite of what we assumed to be true. That is, we'd have to then assume what was what we have to then we would then know the assumption is false. So it doesn't exist. They're not isomorphic. And this is where invariance comes into play. Because if there was an isomorphism between two functions, that isomorphism must preserve all of the invariance. If there's an invariant that disagrees between the two groups, then that means they're not isomorphic. There's no isomorphism that could exist because otherwise it would have preserved those things. So let me show you some examples. Let's look at the dihedral group D4 versus D5. So the dihedral group D4, this is the symmetry group of the regular square. Well, I guess every square is regular. And D5 will be the symmetry group of the regular pentagon. Okay, these two groups are not isomorphic. How do I know that? Look at their orders. D4 is a group of order eight, two times four. And D5 is a group of order 10, two times five. We proved this previously. Now, like I said, in the previous in the previous video, we proved theorem 914. And so the order of a group is an invariant. Two groups are isomorphic. If they're isomorphic, that means their orders are the same. But D4 and D5 are they don't have the same order. So there cannot be an isomorphism between them because otherwise that would prove that eight equals 10, which is contradiction. Therefore, because they disagree on an invariant, it means that D4 is not isomorphic to D5. And there's something particularly special about the numbers four and five right here, we actually get a general principle from this, that the dihedral groups dn and dm, they're only isomorphic if and only if the parameters in and m are the same. So if you take the symmetry group of different regular gons and in gons, that is they have the different number of vertices, then those won't be the same groups. So dn is not equal, is not congruent to dm when in and m differ. All right. Now order is a very nice invariant. If two groups have different orders, they can't be isomorphic. But there are groups with the same order that can still be non isomorphic. Take S3, the symmetric group on three letters, and take Z6, the cyclic group of order six, the cyclic group as this label has order six, S3, its order is going to be three factorial, which is equal to six. Both of these groups are order six. What does that tell us about isomorphism? It actually tells us nothing. It tells us that they could be isomorphic or they could be not isomorphic. We don't have enough information. We have to look for a different invariant if we want to show that they're not isomorphic. Now the invariant to use here is the commutivity principle. Now Z6 as a cyclic group is a bilion, but S3 on the other hand is non-abilian. If we take, for example, the permutations one two and one two three, notice what happens here is one goes to two, two goes to one, so one is fixed, two goes to three, and then three will go to one which goes back to two. So one two times one two three is two three. But on the other hand, if I take one two three times one two, you get one goes to two, two goes to three, right? That's already enough evidence right there. Three goes to one, and then two goes to one which one goes to two so it's fixed. So notice you get two different transpositions. This is a non-abilian group, S3, but Z6 is a bilion. But like we said earlier, if the two groups were isomorphic, if one was a bilion, then the other would be a bilion. This is a violation of one of these invariants. Therefore, Z3 is not isomorphic to Z6. You could also make an argument about cyclic. Z6 is cyclic, but S3 is not cyclic. So the two groups cannot be isomorphic to each other. There's a different, they disagree on invariants. Now I want you to be clear that this list of four invariants is not comprehensive. These are all four conditions that must be the same for isomorphic groups, but there are other conditions. If all four of those conditions are the same between the two groups, they could still be non-isomorphic. And so it's to show that two groups are non-isomorphic, it's important to kind of develop a good understanding of these invariants. So let's look at another example, right? Let's consider the cyclic group Z8 and Z12. These groups are both cyclic, they're both non-isomorphic, they're not isomorphic, but they're not isomorphic because they have different orders, right? Z8 is order 8, Z12 is order 12, those are not the same integer. Therefore, those two groups are non-isomorphic under addition. But what if we look at the associated groups under modular multiplication? If we look at Z8 star and Z12 star, okay, those are both groups of order four, right? Let's just be, let's be clear, Z8 star, Zn star will contain all of those integers relatively prime to the list. So Z8 star will look like 1357 and Z12 star will look like 15711, like a slurpee, in which case, those are both groups of order four, that there's a potential that they could be isomorphic. And in fact, I'm going to define a bijection between them. I'm going to send one to one, three mod 8 to five mod 12, I'm going to send five mod 8 to seven mod 12, and I'm going to send seven mod 8 to 11 mod 12. So that is clearly a well-defined bijection because I'm telling you which congruence class goes to which one. And it's clearly one to one and onto because there's just four to four, okay? So we have a bijection, is it, is it homomorphic? That one's a little bit harder, because in this situation, I just, when it comes to an isomorphism, it's easier to show it's isomorphic when there's like a formula that this goes to that formulaically, is that even a word, by some formula, right? This formula isn't really a formula at all. It's just, I'm saying this goes to here, this goes to here. I told you one by one where each of the elements goes. So proving that this is homomorphic is much more challenging, and for a very large group, this would be a very, very challenging task. Now, since these are both groups of order four, it's not so hard to do. So what I'm going to do to prove, to prove that this, this, this function phi is actually homomorphic, is I want you to compare the Cayley tables of the two groups. So if you take z8 star, its elements are 1, 3, 5, 7, 1, 3, 5, 7, right? If you look at all the possible products, 1 times 1 is 1, 1 times 3 is 3, 1 times 5 is 5, right? 5 times 3 is 7, 5 times 5 is 1, okay? So look at that, that's the, that's the Cayley table for that. Do the same thing for z12 star, right? 5 times 5 is 1, 7 times 11 is 5, things like that. And so remember that the identity, that the map here phi, since 1 to 1, 3 to 5, 5 to 7 and 7 to 11, I've re-labeled everything. And so let's see what happens if I start labeling these things. So for the first one, let's label every single one we see in this picture. So you'll notice that the one, when you're working mod 8, is the identity. Looking at the first row, you see the exact same label, right? So one is the identity of this group. And notice how you have the ones across the diagonal here. This suggests that every element is its own inverse. So if you take an element times itself, you get back the identity. That's kind of interesting. If I come over to z12 and label all of the ones, because remember 1 goes to 1 here, I label all the ones, you will see, you will see the exact same picture. The ones are located all in the same spots. Interesting, all right? Let's see the labeling 3 goes to 5. If I label all of the 3s, right here, this is going to be the second column in row here. If I label all the 3s, see their locations on the grid. 3 is identified with 5. If you look at all the 5s, notice they're the exact same spot. Okay, what about 5 goes to 7? If I list all the 5s now, right? 5 is associated to 7. If I list all the 7s, you can see, again, they're all in the exact same spots. 7 goes to 11. If I label all of the 7s, which in the middle I guess I could have just left it blank because that's an indicator, right? You can see that according to the color codes here, they're all the same spot, right? And therefore, the Cayley tables are the same. These two groups have the same Cayley table. And in fact, earlier in this series, we played around with this idea about two tables, two groups have the same Cayley table up to relabeling. That was sort of a precursor to this idea of isomorphism, for which if two groups have the same Cayley table, they necessarily have to be isomorphic to each other. Now, you have to be careful. The Cayley table itself is not the best tool to use, per se, because one, for a group of a million, right? That's a big Cayley table. It's a lot of data to process. And also, if you list things in different orders, so if I had listed like 1, 7, 5, 11, the colors wouldn't have matched up the same way. So if you're doing the Cayley table, you got to list them in the same order, according to this bijection right here, which isn't too hard to do. If we list the elements in the same order, the same rows and same columns, based upon this bijection, then you'll see this same Cayley table emerging right here. And so in the literature, we would say here that the Cayley table determines the group. What I mean by that is the following. If you give me the Cayley table of the group, then I can determine the group up to isomorphism. Up to isomorphism, there's only one group that has this Cayley table. And that group is the Klein-4 group, namely Z2 cross Z2. And so there's lots of different ways of representing this group. We could represent this as, well, we could do a mod 8 multiplication, mod 12 multiplication. We could think of this as direct product, 00101011. We could describe it that way under addition. Another one that's kind of fun to do is the Klein-4 group. You could think of it as just the group 1abc, like here, for which the multiplication is then given as 1abc, 1abc. All right, so multiplication by 1 is pretty easy. Multiplication by a, you're going to get 1. a times b is c, a times c is b, b times a is c, b times b is 1, b times c is a, c times a is going to be b, b times c is a, and c times c is 1. You see the same exact structure right here? We could do all of the same labelings. We could look at the identity, look at where the identity is along the diagonal. We could do the next element, call it yellows, or it's a right here. You could find all of the a's, exact same spot. We could then do my next element, that's the b's, that's going to be blue. See the blues right here, just like above. We could have all of the reds, which are going to be the c's right there. Up to relabeling, all of these groups are the same and the Cayley table determines them. This is the Klein-4 group. That's why I say it's so as if it's unique. It's the Klein-4 group, because uptie someorphism, these are the only, these are the only group that has this Cayley table. And so when I say that, the Cayley table determines the group. This is, when we say that such and such determines the group, what we're saying is there's a list of invariants that if all of these invariants match up, that determines the group. So if two groups have the same Cayley table, which is an invariant, then they will be the same group. Another way of determining the group is if a group is cyclic, if two groups are both cyclic and have the same order, then that determines the group, right? Being cyclic in the order determines the group. It's a sufficient list of invariants, because as we started this whole video, going back a little bit, as we started this whole video on this discussion of invariants, which we had right here, these are all necessary conditions for two groups to be isomorphic. Necessary conditions means they have to be true if they're isomorphic, but if they all are true, they might not. They could still be non-isomorphic. This idea of such and such determines the group. That means that you have a sufficient list of invariants, which really is a powerful tool that you can show that two groups are isomorphic without even coming up with the function itself because you have a sufficient list. At the time of this video, the only criterion I'm going to give for determining the group is one, the Cayley table determines the group, and then second, if two groups are both cyclic and have the same order, that also will determine them up to isomorphism. Later in this chapter, we will provide necessary and sufficient conditions to determine abelian groups. That is, the fundamental theorem of finitely generated abelian groups will give us conditions to determine when two groups are isomorphic or not. Two abelian groups are isomorphic.