n a n and tau n a n, you have to show sigma tau is an a n. Good. Sigma is an a n doesn't mean that it equals 2m. It means that it can be written as the product of 2m transpositions and tau is the same thing and then blah, blah, blah. Couple of additional comments. If you have sigma as the product of a certain number of transpositions and tau as the product of a certain number of transpositions and you form the product sigma tau, the resulting permutation is going to be the product of the number that you use for sigma plus the number that you use for tau. You're not multiplying the number of permutations together, some of you missed that point. Similarly, I saw this on a couple of papers, it was a little bit unfortunate. When you write sigma as a product of transpositions, or it doesn't matter what you call it, if you write it as something like a1, a2, and then a1, a3, or something like that, and then a1, an, let's say, some of you wrote something that makes completely no sense. Equals, so this is meaningless. In fact, does a1 inverse mean, folks, these are just numbers, one, six, two, three, or something like that. So this notation is absolutely meaningless. Of course, what you're looking to do is put the inverse sign out here, that makes perfect sense. Then you're looking at this transposition and its inverse, and the inverse of the transposition is itself because it's a transposition. But boy, any sort of notation like that just makes no sense here. All right, next in question seven, here was the breakdown of points. Let's see, part A was worth a half. Parts B through E, one each. And then part F was worth a half for a total of five points. See any comments here? No, other than for question F, find a positive integer so that sigma inverse is sigma to the t. Well, because sigma cubed is the identity, sigma inverse is sigma squared, that's no big deal. What some of you said is that sigma to the fifth is sigma inverse. It is, that's OK. So maybe you were thinking geometrically. So there were actually a couple of different answers that were given there, both of which were correct. Two was the one I was sort of anticipating and looking for, but five is a perfectly good answer to question seven F. All right, let's see. The last page, page four, was worth 15 points total. I broke up in question eight. Part A was worth one. B was worth one and a half. C was worth one and a half and D was worth one. There's a total of five on that one. No issues there, you just pounded it out and didn't see any issues coming up. Question nine, both parts were worth one and a half. B, one and a half. What some of you did in part B, which is where you can compute the order, let's see. Using an easy computation, why? Because the two permutations, the two cycles are disjoint. That's why you can use the LCM game, the least common multiple game. If you didn't use the word disjoint somewhere, then you haven't given me appropriate justification as to why things should be true there. So be careful. The only reason you can use the least common multiple in part B, and it doesn't apply in part A, is because the two cycles here are disjoint. And let's see, in the true, false, the only thing that I saw come up more than once is in question E, some of you thought that that was false. Well, I mean, I know why that was true, too, because there was a similar question on the practice exam where the word cyclic didn't appear. So G1 and G2 are groups that have the same order than their isomorphic is not true. But if they're cyclic groups with the same order than they are isomorphic, so that should be true. All right. So in the end, it becomes obvious that when I sit down and write up an exam, I don't really worry about how many points total are on there. This one happened to be worth 47. That's fine. Usually the total number's around 50 or so, but that doesn't necessarily have to be the case. Now, remember the way I assign grades in here. If you're up over 90% when it's all said and done with homework and quizzes and exams and this and the other thing, then you're guaranteed to some sort of a, might be a minus or something like that. That's OK. And if you're up over 80%, you're guaranteed to some sort of b, et cetera. I don't give letter grades on individual exams, but I'll at least give you an idea of how you did as a group, as a group, as a collection of students, just to give you some clue as to where you sit compared to your buddies. And it turns out that as a group, you did extremely, extremely well. So it turns out that the average, the mean on this exam was 39.3, which is right at 83.5%. That's really, really nice. And just for your info, let's see, I've sort of cut the lines here at 90, 80, 70, but again, I'm not implying that one's an a, one's a b, or anything like that. Just 42 and above, there were 1, 2, 3, 4, 5, 6, 7. There were 9 of you. That's really super. 38 to 41 and a half. There were 1, 2, 3, there were 6 of you. 33 and a half to 37. There were 1, 2, 3, 4, 5, another 6 of you. And below 3 of you. And that's the total number of students in here. So again, there's no letter grades implied here. It's just roughly here is how you did as a group. All right, two comments, because this is the first exam that you've gotten back from me. Comment one is there's nothing in concrete here. If you have any questions on what I did, why I marked things off, or why you lost points somewhere, if you can't read my writing or something like that, please feel free to come and see me. I encourage you to come in and ask questions on the specifics. I guess I'd suggest first take your exam, compare it to the solution sheet, and see if you can figure out on your own why it is that maybe you didn't get full credit for a question, or why something sort of went south. And the second comment is there's no statute of limitations here. Because remember, and I'll certainly talk about this in more detail near the end of the semester, the way exams work in here is you get exam one, then you get exam two whenever that happens, end of October, beginning of November, something like that. Then exam three will happen during the final exam period. It's a 75 minute exam that'll cover the last chunk of the course. But then also you have this opportunity to take the sort of optional final if you want to replace your exam one or your exam two score. And for example, when you're perhaps preparing for that optional final if you decide to take it, if you see something on this exam one that you don't understand or you're not sure why, come in then. I don't care. That's fine. If you deserve points back, I want to give them to you, folks. I'm not trying to play gatekeeper here. Anyway, take them home. And if you have questions, come and see me or send me an email and we can make an appointment. But I was really pleased with this. It's clear that most, if not all of you are working really hard at this stuff. Someone asked, as we were leaving the exam, the exam looked a lot like the practice exam. Do you always do that? I mean, it did look a lot like the practice exam, but those are the important ideas. And if you've got them down, I'm obviously not copying and pasting. I'm throwing the word cyclic in or not cyclic, or I took the permutation in one of them. So you're doing the same flavor of problems, but you're not necessarily doing identical problems. I don't know when exam two comes around or exam three I'll certainly be passing out practice exams. Will they look as similar? Maybe. I mean, they don't have to, but it's not uncommon for that to happen. So there's merit to preparing using the practice exams. Any questions or comments or remarks? OK, very nice. Very, very nice. So we move on. Let me remind you where we were a week ago tonight before the exam. So recap. Is this? Well, we talked about a new way to build groups. The idea is, if you start with a couple of groups, or in fact maybe a whole bunch of groups, that you can somehow hammer them together to get a new group, it's easy to do. You just form what you learn to do back in Math 2.15 and discrete math. You simply form the set Cartesian product of the sets. And you say, oh, but yeah, these are more than sets. They happen to be groups. So can we put a group structure on the resulting Cartesian product? And the answer is yes. In fact, the operation is pretty easy. You just do everything what we say component-wise. If you have a couple of pairs and you want to combine them, you just combine them in each component. And that makes sense because each of the components is assumed to be a group. And so what we did was we looked at the following sort of interesting question. If you know something about each of the pieces that are used to make up this product, whether it's just two pieces or more than that, can you say something about the resulting product in terms of the individual pieces? And the answer is sometimes yes and sometimes no. For example, if all of the pieces that you use to hammer together are each a billion, then the results are going to be a billion, it turns out. But it might be the case that if you hammer together some cyclic groups, the result might be cyclic or might not be cyclic. And that's what we showed at the end of last Monday. So we formed what we called the direct product or the Cartesian product of groups, G1, direct product, G2. Sometimes there's only two of them, but there might be more of them, G sub t, any number of them. Maybe you're hammering six groups together or something like that. And what we saw is you can do this process, folks, in any situation you want, whether the groups are billion or not a billion or finite or not, it doesn't matter. Just hammer them all together. But what we're focusing on for now, focus on for now, is what happens if each of the groups that you use, G sub y, happens to be a billion, ah, happens to be cyclic. And what we observed a week ago, observed this, is that the result might be cyclic or it might not be. Maybe the result is cyclic, the result is cyclic, but maybe not, but maybe not. And the examples were, if we computed z2, direct product with z3, well, there's no issue in trying to figure out how many elements are in the direct product. This is just a statement from Math 215 about sets. The number of elements in this group is six. Question is, is it a cyclic group? The answer turned out to be yes. We actually sat down and found a generator for this group. So it turned out this is cyclic. We just showed that by pounding out a generator. In fact, 1, 1 is a generator. Not the only one, but it is one. But on the other hand, sort of a non-example, example, if I look at the direct product z2 cross z2, this is not cyclic. So sometimes if you take two cyclic groups and you hammer them together, the result is a cyclic group. And sometimes when you take two cyclic groups and you form their direct product, the result is not a cyclic group. And what we're going to do tonight is identify when it's the case that if you take the direct product or you form the direct product of two cyclic groups, that the result is cyclic. And from that, we're not going to prove all of the details of this big result. But from that, what we're going to do is write down, surprisingly, all of the abelian groups that have a specified number of elements in it from this result.  So if I hand you the cyclic group having that many elements in it, and I hand you the cyclic group having that many elements in it, then it turns out that this is cyclic, and I'm going to put in parentheses before I get to the punch line, and therefore necessarily isomorphic to z sub r times s, if and only if the g, c, d, r and s is 1. Turns out the only time if you take the direct product of two cyclic groups that the result is a cyclic group is when the two numbers that you have handed me are what we call a number theory business, relatively prime. They may not be prime numbers individually, but the point is that there's no number bigger than one device above them. So look, if I hand you two and three, those are relatively prime, and so the result turns out to be cyclic. On the other hand, two and two are not relatively prime, so the result is not cyclic. I'm not going to prove this proposition for you, but the intuition is, yeah, if you're going to try to find the order of this particular element in the group, it turns out if the two numbers that you've handed me are relatively prime, the behavior here is almost similar to the behavior that happened when we were looking at orders of permutations that are written as disjoint cycles. You sort of keep computing, keep computing, and you're trying to figure out when it is the case that you get back to the identity element. Well, for permutations, the identity element looks like I. And one of them sort of worked, but then that one was in the order wound up having to be the least common multiple. It turns out here, if you hand me this element, if the two numbers that you've handed me are relatively prime, in other words, if their greatest common divisor is one, then necessarily their least common multiple is the product. So the least common multiple of these two numbers turns out to be six, and you can show that that implies that the order of this element is six. And as soon as you've got an element of order six inside a group of six elements, then necessarily it's a generator. So that's intuitively what's going on here. But again, I don't want to get into the details of the proof. It says, though, that you have a pretty good idea of when it's the case that you can take a group like this and sort of replace it by or trade it in for something like this. So if somebody hands you z2 cross z3 and somebody else hands you z6, those two groups are essentially the same. Of course, they're not written the same, but they're isomorphic, meaning what? That if you relabeled all the elements of z2 cross z3 appropriately, and you relabeled all the elements of z6 appropriately, that the two would look indistinguishable. The same beast. On the other hand, if I hand you z2 cross z2, well, it turned out z2 cross z2 looked indistinguishable from this group that we've been calling v for the last three weeks. It's the group that has the property. It's got four elements. It's got an identity element, of course. And all of the non-identity elements have the property that when you combine them with themselves, you get the identity. In other words, a squared is the identity region. But those you can't somehow relabel or reconstruct so that one looks like the other. z4 and z2 cross z2 are different beasts. But z6 and z3 cross z2 are essentially identical because they're basically. So the question is going to be this. If we keep going, can we somehow, rather than stopping at just the direct product of two cyclic groups, if we somehow hammer together three cyclic groups or four cyclic groups or 18 cyclic groups or something like that, can we devise a way that we can somehow move around or trade one in for the other and say, well, this one, if you were to write out the group table, looks like this one. Or this one, when you write out the group table, doesn't look like this one. And the answer turns out to be yes. And it all boils down to this particular remark. Whenever you hand me the direct product of two groups, if the two numbers are relatively prime, then you can trade it in for one group, the cyclic group. If the two numbers are not relatively prime, then you can't trade it in for that cyclic group. And the point is it turns out that you can play that game regardless of whether you've taken just two groups or whether those two groups somehow appear in the middle of maybe the direct product of three groups or four groups or more. So it turns out this extends to more than just two groups and just the direct product of two groups. Bring a better pen next time. Here's one, maybe this will work. Direct product, a little better, two groups. And let me just explain what that means by presenting an example here. Example, if I look at the group Z2 cross Z3 cross Z4, so it makes sense to form the direct product of these three groups, let's see how many elements are in here. That's easy, six times four, there's 24 elements in here. Question, if I hand you another group with 24 elements like Z6 cross Z4, there's a group with 24 elements. Is that group isomorphic to that group? Well, here's another group with 24 elements. Is this group isomorphic to that one? Or is this group isomorphic to that one? Here's another one, how about Z8 cross Z3? Is this one isomorphic to that one? Here's another one, how about Z2 cross Z2 cross Z3? So what I've done here, folks, is I've written out a bunch of different groups, each of which has 24 elements in it. Heck, I'll give you another one. That has 24 elements in it. And the question we can start asking is, look, here are some naturally occurring groups. They're groups that I can form just by taking some cyclic groups or direct products of cyclic groups or something like that. And the question will be, can we identify which groups on this list, if any, are isomorphic to each other? In other words, can we identify whatever might be viewed as a repeat on this group or on this list? Well, here's the idea. Two and three are relatively prime. So this piece is, and here's the symbol. I don't know if I've used it this semester, but I'm going to start using it more and more. When we talk about two groups being isomorphic instead of having to write the word out all the time, I'll use this symbol, it's an equal sign with a little squiggle on top. This group is isomorphic to Z6. So what this comment means is, if you have a chunk that you can trade in for a cyclic piece, then you can drag along any additional direct products that might be hanging on. So that, because this piece is isomorphic to Z6, this whole thing is isomorphic to Z6 cross Z4. So in fact, these two are isomorphic to each other. Yes. Isomorphic. I need to write down one more here, folks. I forgot this, I had this. I want to write down Z4 cross Z6 as well. All right, let's see. Question is Z6 cross Z4 isomorphic to Z24. No. Reason? The GCD of 6 and 4 is 2. Because the greatest common divisor of these two numbers is not 1, you can't trade that in for this. So this one is not the same as that, which then necessarily means it's not the same as that. How about this one and that one? I don't know. How about this one and that one? Do you like that? Yeah, because the GCD of 8 and 3 is 1. So these two are the same. It's in blue here. So these two are isomorphic by using the same result. But look, folks, we determined this wasn't the same as that. So therefore, this can't be the same as that. Because isomorphism is what we call an equivalence relation. If you have two groups or isomorphic and you have another one that's isomorphic to this one, then these would have to be the same. And because we've established that this isn't the same as that, but this is the same as that. It's just like equality. All right, let's see. How about this one? Z2 cross Z2. Oh, boy. Well, how about this piece? Z2 cross Z3, that looks like Z6. So at least I can trade this in for Z2 cross Z2 cross Z6. But I don't see that up here anywhere. So I don't see how this can be isomorphic to anything else that's on the list. And I say, well, wait a minute, how about this one? No, that's no good. Because this is Z2 cross Z2 cross Z6. This isn't Z2 cross Z2 because the GCD of 2 and 2 and 2. In other words, I can't trade in that piece for this one. So this one sits by itself. Not that there aren't other groups that you could write down that this would be isomorphic to. For instance, this is isomorphic to Z2 cross Z2 cross Z6. But I simply haven't listed that one out here. How about this one? Z4 cross Z6. Yeah, it turns out the following is true, folks. Regardless of whether the GCD is 1 or not 1 or anything, if you take ZR cross ZS or ZS cross ZR, it doesn't matter which order you do things in. So the process of forming direct products is independent of the order you do them in. So let me slide that proposition in here. Proposition ZR cross ZS. And I'm going to use this notation, this squiggly line notation. This is totally standard, folks. Is isomorphic to ZS cross ZR always? Always. So it turns out that the order that you trade things in for doesn't really matter. How about, let's see, Z4 cross Z6. Oh, yeah, that's already up there. So this one is already here. That's isomorphic to this one, which in turn, then, is isomorphic to that one. So what have we got here? We've got what we'll call equivalence classes of isomorphism. This particular group happens to be isomorphic to this one, which happens to be isomorphic to this one. So if somebody asks you to write down a billion groups having 24 elements, folks, it's unreasonable to think that you should write down this one and this one and this one because they represent the same structure. They're isomorphic. So all we're going to do is ask you when we're asking you to write down groups that have the a billion property together with a certain number of elements, we're going to ask you to write down just one of those groups. And we'll phrase that as uptysomorphism. So uptysomorphism here is one group. You might say, well, should I call it that or should I call it that or should I call it that? And the answer is what we typically agree to do is call it whatever you get so that if you can take one of the numbers that appears here and break it up into a product of two relatively prime numbers, you go ahead and do that. We could call this Z6 cross Z4. It's a perfectly good name for this group. But because 6 can be written as 2 times 3 and 2 and 3 are relatively prime, we typically like to trade it in for that. All right, we've got this one. Let's see, we've got Z24 or Z8 cross Z3. Of course I could call this Z3 cross Z8. There's a lot of things. So here's another one. Here's one that can't be isomorphic to anything else in the list, right? Because this is not a billion and everything else is a billion. So here's this process. We can construct a lot of a billion groups using cyclic groups. But here's a good question. If somebody hands you an abelian group and tells you how many elements are in it, do you know which group they're talking about? Definitely not. If somebody says I'm thinking of an abelian group that has 24 elements, question? Megan, question? Please. John? Why it's not? Okay, this chunk of Z2 cross or this chunk of it we can trade in for Z6. So it might be isomorphic to this, but the problem is we're left over with Z2 cross Z2 which we can't trade in for Z4 because these are not relatively prime, two and two. So this chunk can't be traded in for that. Similarly, these three together can't be traded in for that because the twos are not relatively prime to each other. So think the only way you can move around in these diagrams is if you've got two numbers that are relatively prime. So I can trade this in for that but I can't trade that in for that. Now, is that clear things up a little bit? Let's see, so this we can trade in for Z2, Z3, Z4. That's what isomorphic remains. Well, all I'm worried about is that piece there. So the two, Z2 cross Z3 is Z6. I mean two and three are relatively prime, so I can trade. And then the point is if you've got another piece that you're dragging along, you just drag along the same piece. All right, so where from there? Oh yeah, here's where from there. So yeah, think of the question this way. Question, if you're told that you have your friend, this is a better way to think of it, is thinking of an Abelian group, bless you. Thinking of an Abelian group and your friend tells you how many elements are in it and you are told how many elements are in it. In other words, the order of the group. Do you know the group? I have somebody says, all right, I'm thinking of an Abelian group and it's got 24 elements. Do you know which group your friend is thinking of and the answer is no. Because your friend might be thinking of Z2 cross Z3 cross Z4 or your friend might be thinking of Z24 or your friend might be thinking of Z2 cross Z2 cross Z3. I'm sorry, I didn't say enough Z2. Z2 cross Z2 cross Z2 cross Z3, that's got 24 elements. So the answer is in general, no. In general, no. And the example is if the number of elements is 24, if N equals 24, then you don't know because I've given you a lot of examples of Abelian groups. So let me take one step back just to give you some context. The question is how much do you need to know about a group in order to know what the group is? And when we say know what the group is, all right, the group Z2 cross Z3 and the group Z6 are treated as the same group here. They may be written slightly differently but because they're isomorphic because they have the same structure because there's a way of relabeling them to describe the same structure, we don't really wanna call those different. So the first question to ask is, is knowing the number of elements in the group enough to tell you what group it is? And the answer, of course, is definitely no. In general, if somebody says I'm thinking of a group with six elements, do you know what the group is? No. It might be Z6 or it might be S3. There's two different groups with the same number of elements. Of course, you might look out if somebody says I'm thinking of a group with five elements in it, do you know what the group is? The answer to that question is yes, I know what the group is because we proved that if you have a group that has a prime number of elements then necessarily the group is cyclic. So somebody says I'm thinking of a group with five elements, you know exactly what group they're thinking of, Z5, because that's the only possibility. If I ask you to, I don't know, if I tell you I'm thinking of a group with six elements, forget it. You don't know if it's S3. If I'm asking you to think of a group with 24 elements, then it's big, forget it. I mean, there's lots of different groups up here. So the answer in general is no. Just knowing the number of elements in the group doesn't tell you what the group is. But here's a more reasonable question.  So now I've got two sort of pieces of information about the group I'm thinking of. A, that it's a billion and B, how many elements are in it? Is that information alone enough to tell you what group you're looking at? And the answer turns out to still be no. Here's an example. But here's the incredibly nice thing about the adjective abelian. If you just focus your attention on abelian groups, which means admittedly folks that you're leaving out a lot of things. But if you just focus on the groups that happen to be abelian, there's sort of a miraculous thing that happens and we're not gonna prove it in here. The miraculous thing that happens is the following. I know how to construct. I know how to explicitly, concretely write down a lot of abelian groups. Take cyclic groups and form their direct products. So think, I've got this sort of building block process which allows me to produce a lot of abelian groups. The miracle is, in fact, that gives you all the possible finite abelian groups. The process of taking cyclic groups, those are special. And forming direct products of those, that's a very concrete process. In fact, in the end, not only gives you a lot of possible abelian groups, it gives you, up to isomorphism, all the possible finite abelian groups. And that statement, and again, it's not really beyond the scope of this course, but it would take us a full class period to run through the proof and I don't wanna spend the time doing that. Other than just have you understand the, I don't know if the miraculousness of the result is, turns out, turns out, the following is true. Any finite abelian group is isomorphic to to the direct product of cyclic groups. Now, this is usually referred to as the fundamental theorem of abelian groups, theorem of, I'm gonna call it finite abelian groups. And we're, this is the third time this semester that my notation and statement is gonna be on the surface sloppy, but the interpretation will mean exactly what you would hope. Folks, I'm not saying that if you hand me an abelian group, it's necessarily the direct product of more, of two or more groups. For instance, if I hand you the abelian group Z7, it's not the direct product of cyclic groups. It's already a cyclic group. It's just like when we said, if you have a permutation, it can be written as a product of cycles. It might be a cycle already, or if you have an integer bigger than two, that it's a product of prime numbers, except for if it's a prime number already here. If you have an abelian group, if we say it's the isomorphic to the direct product of cyclic groups, it might be a cyclic group. So I could put this in, but I'm just trying to get you sort of speaking mathematics ease here. It should technically say any finite abelian group is either already a cyclic group or is isomorphic to the direct product of two or more cyclic groups, but that just gets too cumbersome, right? So it's a miracle. It says that these very basic abelian groups, the cyclic ones, together with this very concrete construction, direct product, allows you to describe all possible abelian groups. Now there's some significant extensions to this. We can talk more generally about what are called finitely generated abelian groups, rather than just finite abelian groups. When we talk about finitely generated abelian groups, that allows us to include the integers and direct products of the integers and these are bands and those sorts of things. So there is a more general statement that we could make that, again, is not beyond the scope of this course, but would take us too much time to get there. And then what happens beyond what are called finitely generated abelian groups? Well, then there's other abelian groups, infinitive abelian groups, like ones that you gave on the exam, maybe the rationals with addition or the complex numbers, non-zero complex numbers under multiplication or something like that. Those might not be isomorphic cyclic groups. In fact, the structure of abelian groups in general is really intricate and really interesting. In fact, that's what Professor Rangaswamy does for a living. He studies the structure of abelian groups and you can talk about certain properties of some that are shared by some and not by others. All right. So look, if somebody says, I'm thinking of an abelian group and it has a certain number of elements in it, 24 elements or 120 elements or 50 elements, do you necessarily know what group they're thinking of? No. But what this allows you to do is say, all right, you've got an abelian group, therefore it's the direct product of cyclic groups by this fundamental theorem. Can I write down what the cyclic groups are? Can I write down what the possibilities are? And the answer is yes, because we can just systematically go through this process and say, hey, how are or how is it possible to take a direct product of cyclic groups and get 24 elements in it? Well, it looks like there's a lot of different ways, but we can do it in a systematic way and I'll show you how to do that. So example, let's see example. Find all abelian groups having, let's do, 36 elements. And I'm going to put in parentheses, uptie some morphism. Don't list two isomorphic copies of the same group. So in fact, what we're asking is the same question that we looked at some examples of, but instead of asking it for groups having 24 elements, we're going to look at a question for groups having abelian groups having 72 elements. Well, I can think of one right off the bat, Z sub 72. The cyclic group of 72 elements is an abelian group having 72 elements. So when it's all said and done, at least that group's going to come up, but hey, right off the top of my head, I can think of a lot more, Z8 cross Z9 or Z2 cross Z2, oh, I'm sorry, 36. Okay, let me do that again. I can think of one that has 36 elements. Z sub 36. I have 72 in my notes, but this will be a little cleaner example. So Z sub 36 works. Z sub 4 cross Z sub 9 works. Z sub 2 cross Z sub 2 cross Z sub 9 works. Z sub 6 cross Z sub 6 works. I mean, so I've written down a lot of examples. The miracle, again, is by simply writing down all those possible direct products of cyclic groups, the fundamental theorem says I'm guaranteed to have written down all the possible abelian groups that they all have to arise as a direct product of cyclic groups. So that's an incredibly nice simplification that we can make. Now the only extra step in the process is we have to come up with a systematic way of listing all these out so that we've made sure we've captured all of them and that we did any more than once. And the systematic approach typically is this, folks, take the number that you're given, write it in its prime factorization, so 36 is 4 times 9, there's its prime factorization, and then simply ask how many ways are there of taking 2 squared times 3 squared and writing it in chunks that are somehow, well, I won't say relatively prime, that are either identical or distinct as far as the factorization goes. For example, I can write 36 as, well, I can write it as 2 times, yeah, okay, so 2 times 2 times 9, I can also write it as 2 times 2 times 3 times 3 and those two I'm going to view as different because, well, 3 times 3 and 9, well, 3 times 3 is obviously 9, but if I had Z3 cross Z3, I can't trade that in for Z9 because these aren't relatively prime.  In the end, folks, there's lots of other ways to do it. I mean, I can write it as 6 times 6 or I can write it as 18 times 2, but in the end, there are only 4 factorizations of the number where you've written things either as primes or prime powers. So I've got 9 or 3 times 3, I've got 2, I've got 4 or 2 times 2 and so in the end, here are all 4 of them, so here they are. Z2 cross Z2 cross Z9 corresponds to this first factorization. Z2 cross Z2 cross Z3 cross Z3 Z4 cross Z9 and Z4 cross Z3 cross Z3 and here are all of them. So what happened to Z6 cross Z6? Well, the claim is that I've already got it captured somewhere on there. Well, yeah, let's see. This is the same as what? Z2 cross Z3 I can trade this piece in and Z6 is Z2 cross Z3 I can trade that piece in but let's see the order that I write these things down is irrelevant. So Z6 cross Z6 is actually already on the list and it's isomorphic to that one. Z2 cross... I'm just making some up and asking, alright, here is one. Here's a direct product of a billion groups. It's got 36 elements. You've claimed that this is all of them. You better make sure this one's already on the list. It is, let's see. Z18 is isomorphic to what? Z2 cross Z9. It's not isomorphic to... Let's see how it's going to do this. It's not isomorphic to Z3 cross Z6 because 3 and 6 aren't relatively prime but it is isomorphic to Z2 cross Z9 and so let's see. Is that one already on the list? Yep, right there. Well, how about Z sub 36? There's going to be a group of 36 elements in it. Is it on the list? Where is it? 4 and 9. So in the end, all you need to do is take whatever integer you're given, write it in all its possible prime power factorizations where the prime powers are viewed as different if you write 9s 3 times 3 and our 4 is 2 times 2, etc. Write it out in all possible configurations you can and then simply write down direct products of cyclic groups based on those factorizations. What's unfortunate is there's no nice formula that says if you give me the number of groups that there are, I'm sorry, if you tell me how many elements you want to be in the group like 36 here that you can somehow predict how many of being groups there are having that order. I mean, there is a formula for it but it's nasty enough that it's not worth trying to memorize or use or something like that. So, look, there's four groups, four being groups that uptystomorphism having 36 elements. There's only one of being group uptystomorphism having 37 elements. There turns out to be only one of being group uptystomorphism having 38 elements because 38 is 2 times 19 and then there's only one factorization of that. It's 2 times 19 and 2 and 19 are relatively prime. So, let's see what would be next. 39, oh yeah, 39 is 3 times, there's only one of being group uptystomorphism having 39 elements. There's a lot more having 40 elements because there's a lot of ways of writing down 40 in a prime factorization. Yeah, David, a question? Okay, here you go. So, unfortunately, there's no, oh, the number 36 means that I'm looking for four groups or something but what you see after doing enough examples of these and those are included in the homework is that, all right, I see the pattern. I sort of have to break down each of the prime powers and all its appropriate factorizations and go from there. All right, that is good, good, good. So, let me give you, yeah, let me give you a little bit of an overview and we'll start on the next topic tonight because we've still got 15 minutes left. What we're gonna do is put, this is chapter 11, put chapter 11 to bed. This was the fundamental theorem of finite and being groups and how to describe all the being groups having a certain number of elements in it. We're gonna skip chapter 12 for now. We might pick up some of those ideas later on in the semester but what we're gonna do is start essentially a new idea. This is chapter 13, it's called homomorphisms. This is actually a new section. Intuitively, homomorphisms are the analog to groups of what linear transformations are to vector spaces. Homomorphisms are ways of getting from one group to another. All right, so you're saying, well, you mean we're just looking at functions? Yeah, we are looking at functions but not just any sorts of functions. The intuition here is the same as the intuition was for linear transformations in vector spaces. You know, linear transformation between two vector spaces and nothing more than a function between the two vector spaces but it's a function that somehow understands that what's going on is not just two sets being connected but two vector spaces being connected. In other words, it's a function that says whatever the operations of the vector space are over here are somehow preserved by the function corresponding to the operations of the vector space over here. So linear transformation is what? T of v plus w is T of v plus T of w. In other words, the transformation respects the one operation called plus and it also respects the scalar operation. Well, a homomorphism is nothing more than a function between two groups that respects or preserves or keeps intact the underlying binary operation. And what we're going to do is look at examples of homomorphisms between groups, understand what they might represent maybe geometrically and then look at various ideas or various quantities associated with these special sorts of functions and it turns out those special quantities like certain subgroups that get kicked out of these things will play an important role so that eventually by the end of this new section not only will we have some sort of idea of what goes on with functions between groups but we'll actually come up with a bit of producing yet another group. Not just subgroups or factor groups. No, I'm sorry, not just subgroups or direct product groups or this that. And the other thing we're going to eventually write down what are called factor groups and that'll take us a little while to develop. So here's the idea. I'll call one of them G1 and G prime and you know what folks, I'm going to revert back to our technically correct notation. Our groups. So the binary operation on G and the binary operation on G prime might be completely different beasts than a function. And we usually use Greek letters here. Phi from G to G prime is called a homomorphism. Homo meaning same and morphism meaning structure. In case, well let's see, given the little introduction I just gave you the hope would be that you'd be able to actually write down what it means. It's a function that somehow preserves the binary operation. In other words, if I take two elements of G let's call them x and y. So take two elements in the domain. Domain is just a set. But hey, it's a set with more. It's a set with a binary operation. What I'm going to ask you to do is perform the binary operation on these two elements. So you get another element of G and then run it through the function so that what gets spit out is something over in G prime. I don't know what it is. Well folks, here's another way of producing something in G prime. Take each of these two elements individually. Don't worry about somehow combining them in the context of the binary operation yet. Take them individually. Run each of them individually through the function. Well that's going to give you two elements over here. And then combine them over in G prime. So that's two ways of somehow taking these elements and hooking them up with the given function and the requirement is in order for G to be called a homomorphism that somehow the function has to behave correctly regardless of which pair of elements you start with. For all x comma y in G. That's what it means for a function to be a homomorphism. Structure preserving. That whatever the binary operation is if you decide to combine the two things first inside the domain group G and then move the result over into the group G prime you get something in G prime. Another way of attacking the problem is take the two things that you're handed in the input group run each of them individually over to the new group G prime and then combine them over there. If you get the same answer both ways then we call the function a homomorphism. Let's write down some examples. Example. Let's write down the easiest one first. Yeah. Here's a homomorphism. Define phi from this group. So by default I've told you what the operation is because it's the integers. There's only one operation. To some other group in this case the two groups are going to be the same. There's certainly nothing that precludes G and G prime from being the same or star and star prime from being the same as follows by setting phi of any little z equal to 4 times z. And let's show that phi is a homomorphism. I'll abbreviate that HOM homomorphism. Well it's certainly a function from the integers to the integers that's no big deal. If you take an integer and you multiply it by 4 you get another integer. The question is, is it the case that multiplying by 4 preserves whatever the group operation is in the underlying group? And again, the operation here is plus and the operation here is plus. Same group. So it showed it is. Well what do we have to do? To do this, pick any two x and y in the input group. We have to show or prove that phi of x plus y plus because that's the operation of the group equals what you get if you were to do the operation to the two of them individually. Now here folks, this plus is being interpreted as happening inside the domain. This plus is being interpreted as happening inside the range. So we have to show that. Well let's see. Well, phi of x plus y, by definition, this function does what? It multiplies by 4. It's 4 times x plus y. But phi of x plus phi of y is 4x plus 4y. Of course this is just 4x plus 4y. So the left-hand side equals the right-hand side. Check. So phi is the homomorphism. Supplying by 4 gives a homomorphism. Let's do a non-example just to keep you honest. Non-example, define, let's call it something else. How about Greek letter psi from z to z? Psi of little z equals 4 plus z. So it looks like I'm doing the same sort of thing as before. It certainly is a function from the integers to the integers. You take an integer and you add 4. You certainly get another integer. But let's see. If I do psi of x plus y, x plus y is what? Is 4 plus x plus y? On the other hand, if I do psi of x plus psi of y, that's 4 plus x, and that's 4 plus y, which gives 8 plus x plus y. Are these two equal? Technically, what should I do at this stage? It doesn't look like these are going to be the same. Better write down a specific counter example. So example, if x equals, it doesn't matter what you pick. How about x equals y equals zero? Just make them both zero. Then psi of zero plus zero. Psi of zero plus zero is psi of zero, which is 4. But psi of zero plus psi of zero is 4 plus 4, which is 8. And so psi is not a homework. I've written down a specific counter example by giving you two integers x and y. They happen to both be zero. That's fine. And I've shown that when I run those two integers through psi of x plus y, I get something different than if I had to run them through psi of x plus psi of y. So I'm just trying to clean up and drive home again this same sort of idea. If you're going to convince me something's not true, write down a specific counter example and convince me that's not. Here's the specific counter example. You want to write down a different counter example? Sure, there are many. In fact, most anything will work. Questions? Comments? All right, let's try one more example. This one's going to be interesting. These homomorphisms, well, in the situation that I wrote down, the one that worked was a homomorphism from a group to itself. Here is a significantly more interesting one. For example, I'm going to write down a function. Let's call it tau from Sn to Z2. So what I want you to do is start with any n bigger than or equal to 3. I don't care which one you pick. Pick n equals 7 or n equals 4, n equals on or on. I don't really care. And what I'm about to do is write down a function from Sn to Z2. And here is what the function does. Tau of, let's see, what am I looking to put in here? A permutation. So the notation looks sort of strange. So here tau denotes a function from Sn to Z2. So it's waiting for you to input something in Sn. That's what the things in Sn are typically denoted as. I don't want to use a tau here because I don't want to use it twice. Equal to, there's two possibilities.  Piecewise defined function. If you input an even permutation, then I want 0 to come out, and if you put in an odd permutation, I want 1 to come out. So at least it's a function from here to here. Why is the function well-defined? Well, it's this big theorem that I apologize for not proving for you. This is a big theorem. It says, if you hand me a permutation, there might be many different ways of writing it as a product of transpositions. But either, I don't care how you do it, either each way that you do it is always going to involve an even number of transpositions, or it's always going to involve an odd number of transpositions, but it's never going to be both. So there can't be any argument about which of these two numbers is supposed to come out. So the well-definedness of this particular function is actually a relatively deep result. But we're using that as known. It's really going on under the scenes. Okay, that's fine. Now the claim is going to be that this particular function is a homomorphism. So we'll prove that tau is a homomorphism. And the reason is, well, you have to show me that regardless of what two permutations you hand me, sigma one and sigma two, that tau of sigma one. Okay, well, let's see. So pick any two permutations, any two elements of Sn of Sn. What do you want to call them, X and Y? Yeah, if you want, that's fine. But typically we use Greek letters to denote elements of Sn. How about tau and sigma? Well, I've already used tau for something else. How about sigma and kappa? That's fine. Let's call one of them sigma one and the other one sigma two. There's two elements of Sn. Now, in this particular case, it turns out that there are few enough possible outputs that we can go ahead and pound this thing out just by brute force. So there's really only four possibilities. There are what? Let's see. Sigma one even and sigma two even or sigma one is even and sigma two is odd or sigma one is odd and sigma two is even or finally that they're both odd and sigma two is odd. So in this particular situation, if you hand me two permutations because all you're interested in is whether or not they are even and odd that's all you need to define what the function does. Let's just look at what happens in each of the situations. So if I compute in this case, let's say in case one here this is good. So let's call this case one, case two, case three and case four. So really all we have to do is check that in each of the four situations that the equation that we need holds true. And let's see if it does. Well, in the first situation, what is tau of sigma one, sigma two? Let's see. If sigma one is even and sigma two is even then what is sigma one, sigma two? Is it even or odd? It's even because the product of two evens is even. That's what you're proving in the exam. So because this is even, this gives zero. On the other hand, what's tau of sigma one? Well, let's see what are we supposed to do? We're supposed to do the operation as whatever it is inside the target group and the target or the codomain or the range group is z two. So when I do the operation to the two things that come out what I'm asking that in this particular case is add in z two. Well, let's see. Sigma one is even, so tau of it is zero. Sigma two is even, so tau of it is zero. And the question is, does zero equal zero plus zero? Yes, it does. So it worked out fine. Not bad. Case two and case three and case four, I'll just knock out these. These will be totally quick. We'll get out of here in about a minute or so. Case two, tau of sigma one, sigma two. Well, wait a minute. If sigma one is even but sigma two is odd, folks, if I do the product of an even permutation with an odd permutation what do I get? It's odd because I've got an even number added to an odd number and an even plus an odd is an odd. So this is odd. So tau of it is one. Well, let's see. On the other hand, if I do tau of sigma one plus tau of sigma two, let's see. If sigma one is even, that's zero. If sigma two is odd, that's one. I get zero plus one, which is one. That worked out good too. Case three, I won't go through it similar to case two. Case four is the interesting one. What happens if I do tau of sigma one, sigma two, where both sigma one and sigma two are odd? Let's see, folks, if you take two odd permutations and you multiply them, what do you get? An even. If I've got, this is the product of seven transpositions and this is the product of five transpositions and I nailed them together and I've got the product of twelve transpositions. An odd plus an odd is an even. So in this particular case, this is even because the product of two odd permutations is even. Well, let's see what happens over here. Well, let's see. If that's even, then by definition, tau of it's zero. But on the other hand, tau of sigma one plus tau of sigma two, sigma one is odd. So tau of it is one. Sigma two is odd. So tau of it is one. What happens if you add one plus one? Yeah, you zero, because the group that we're working in is z two. Perfect. So did that one work? Yeah, it did. And that was the interesting one. So in the end, tau's a homomorphism. So it's highly possible to have homomorphisms from non-abiding groups to abiding groups. It's possible to have homomorphisms from abiding groups to non-abiding groups. And we'll see a whole bunch of examples on Wednesday. All right, sorry to run over a little bit here. If you have specific questions on your...


First fears have now come true President Putin of Russia has unleashed war in our European continent. This crisis is about the right of a free, sovereign, independent European people to choose their own future. That is a right that the UK will always defend. The UK is announcing the largest and most severe package of economic sanctions that Russia has ever seen. We will continue on a remorseless mission to squeeze Russia from the global economy piece by piece day by day and week by week. Our Ukrainian friends, in this moment of agony, I say that we are with you and we are on your side. Now what we're doing is everything we can to support the Ukrainians with defensive military equipment. First European country to do so. We've trained 22,000 Ukrainian troops, we've set more than 2,000 anti-tank weapons and we will do more. 48 new hospitals, 50,000 more nurses, 50 million more GP appointments, 40 new diagnostic centres and fixing those backlogs with real change. Leveling up means fighting crime, putting more police out on the beat as we are, toughening sentences, giving the police the powers they need. The UK recovering faster than our major competitors. Stronger public finances with our debt under control. Stronger employment with fewer people out of work and more people in work. Growth up, jobs up and debt down. Our plan is working. Imagination, enthusiasm everywhere in this country, all of them evenly distributed, but opportunity is not. And it is our mission as Conservatives to promote opportunity with every tool we have. Please welcome the Prime Minister. Hi everybody. Good morning. Please sit down. Please sit down. Good morning everybody. It is absolutely fantastic to be black here in Blackpool. I first spoke here, I want you to know I first spoke here 25 years ago, 1997 I was the freshly defeated candidate for Cluid South. And I did the appeal, but I didn't think they could get Jeffrey Archer that day because as you will recall we've been more or less wiped out. And what a joy it is to come back here today quarter of a century on and find that we have more Conservative MPs than at any time since the 1980s. And that we not only hold Cluid South, we hold Blackpool South, my friends. And thank you. As we meet today, a tragedy continues to unfold in our European continent. A vicious and a barbarian attack on innocent civilians, the like of which we haven't seen since the 1940s. And Mr. Ambassador, so I know there you are. Thank you Mr. Ambassador, Ukrainian Ambassador. I want to repeat to you directly what I told your wonderful President Volodymyr Zelensky yesterday. We stand with the Ukrainian people and our hearts go out to them. And tens of thousands of people in this country are opening our homes to the people of Ukraine. We say thank you to them and we applaud them. And with every day that Ukraine's heroic resistance continues, it is clear that Putin has made a catastrophic mistake. And you have to ask yourself why he did it. Why did he decide to invade this totally innocent country? He didn't really believe that Ukraine was going to join NATO any time soon. He knew perfectly well there was no plan to put missiles on Ukrainian soil. He didn't really believe the semi-mystical guff. He wrote about the origins of the Russian people. You read it Nostradamus meets Russian Wikipedia. I think that wasn't what it was about. I think he was frightened of Ukraine for an entirely different reason. He was frightened of Ukraine because in Ukraine they have a free press. And in Ukraine they have free elections. And with every year that Ukraine progressed not always easily towards freedom and democracy and open markets. He feared the Ukrainian example and he feared the implicit reproach to himself. Because in Putin's Russia you get jailed for 15 years just for calling an invasion an invasion. And if you stand against Putin in an election you get poisoned or shot. And it's precisely, that's what happens. And it's precisely because Ukraine and Russia have been so historically close that he has been terrified of the effect of that Ukrainian model on him and on Russia. And he's been in a total panic about a so-called colour revolution in Moscow itself. And that's why he's trying so brutally to snuff out the flame of freedom in Ukraine. And that's why it is so vital that he fails. Because a victorious Putin will not stop in Ukraine and the end of freedom in Ukraine will mean the extinction of any hope of freedom in Georgia and in Moldova. It will mean the beginning of a new age of intimidation across the whole of Eastern Europe from the Baltic to the Black Sea. And if Putin succeeds in crushing Ukraine it will be the green light for auto-crats everywhere in the Middle East, in the Far East. This is a turning point for the world. And it's a moment of choice. It's a choice between freedom and oppression. And I know there are some around the world, even in some western governments who invoke what they call reale politique and who say that we're better off making accommodations with tyranny. I have to say I believe they are profoundly wrong. And that to try to re-normalise relations with Putin after this, as we did in 2014, would be to make exactly the same mistake again. And that is why Putin must fail. And I know that it's the instinct of the people of this country, like the people of Ukraine, to choose freedom every time. I can give you a couple of famous recent examples. When the British people voted for Brexit in such large numbers, I don't believe it was because they were remotely hostile to the foreigners. It's because they wanted to be free, to do things differently and for this country to be able to run itself. Give you another example, when the British population came forward to be vaccinated at such incredible speed, voluntarily, unlike many other countries. I'm sure it was, of course, partly because they wanted to avoid catching COVID. Very sensible thing to do, by the way. I hope you've all had your boosters. And you have. Well, we're getting ready for a fourth jab because we're going to need it. But I'll tell you why people did it. Why did the British people come forward? I mean, 90% we got entirely voluntarily. It was because they wanted to get on with their lives.  They were. We wanted to take back control of our lives. So I'm proud that this government has done the things that so many people said were impossible. We got Brexit done. I'm proud that we deliver the fastest vaccine rollout in Europe, not once, but twice, and the fastest booster rollout. And, of course, yes, I am proud that this government has been in the lead in sticking up for freedom in Ukraine. This was the first European country to send defensive weapons to help the Ukrainians. And now dozens are following our lead. And I'm very grateful to my friend Ben Wallace, also there in the front row, for his foresight many months ago and for getting me to read Putin's crazy essay, which I did. And we're talking continuously, Liz, Ben, and I were talking continuously to our colleagues in the Ukraine support group to discuss what more we can do. And those conversations go on literally every day, and there will be more. I'm proud of what we did on sanctions. We were in the lead in sanctioning Swift and the banks were banning Aeroflot. We've now sanctioned more banks and individuals than any other European country, and we will be restraining their yachts and their assets. And, of course, there is a cost to all these actions. Of course there is, but the cost of doing nothing would be far, far higher. Putin's war is intended to cause economic damage to the West and to benefit him. And he knows that with every dollar increase in the price of a barrel of oil, he gets billions more in revenues from the sales of oil and gas. And that's the tragedy of the situation. He's been preparing for this moment by pushing hydrocarbons on the West, like a backstreet pusher, feeding our addiction, creating dependency. And now he wants to weaken the collective will to resist by pushing up the cost of living, hitting us at the pumps and in our fuel bills. And so we must respond. And we've got to do everything we can to help people with their daily costs, help people with the cost of living. And of course that means doing all the things that we're doing, lifting the living wage, cutting council tax bills, helping with fuel costs, giving billions to councils, millions more to help people in particular hardship. The best possible answer, of course, is to make sure that this is the traditional conservative answer, to make sure that we have a strong economy and strong economic fundamentals with well-paid jobs. And thanks to the speed of that booster rollout, we have the fastest-growing economy in the G7. Youth unemployment now actually back to the level it was before the pandemic, virtually a record low, 3.9 percent youth unemployment at or near record lows. But if we're going to deal with the particular cost, the biggest cost that families now face and tackle these rising fuel bills, the energy spike, we must take the bold steps necessary to end our dependence on Putin's oil and gas. And that is what we are doing. In the immortal phrase, it is time to take back control of our energy supplies. After years of short-termism and hand-to-mouth solutions, we are setting out a British energy security strategy. And we will make better use of our own naturally occurring hydrocarbons rather than import them top dollar from abroad and put the money into Putin's bank account. That does not mean in any way that we will abandon our drive for a low-carbon future. We're going to make some bets on nuclear power, some big bets on nuclear power, not just the big projects but also the small modular reactors. And we're going to take that bullet-to-gate spirit of the booster rollout and use it to build more offshore wind at double-quick time and many other investments in clean green power. But if you ask me how we're going to pay for all this, I can tell you that I've been going around the world recently and I've been talking to international investors who are yearning to make colossal long-term investments in British green infrastructure, colossal. There's a reason why people want to come to this country, many, many reasons why they want to come, but you're seeing them invest massively in everything from tech to finance to green power. And that is that they know what this government is doing. They know about, I mean you may not believe it, but they do, they know about our levelling up agenda, my friends. They do. They've heard of it. They have. They're very well informed. They know that we have a plan to unleash the potential of this whole country and they can see how we are doing it. Making our streets safer with 20,000 more police, rounding up the county lines drugs gangs that cause such misery, stopping and searching the kids with the knives, giving the police the powers that they need, and by the way giving the criminals the serious sentences that they deserve for the crimes that they commit, tackling the middle class drugs use, by the way, that helps to drive so much of the consumption, but doing things, thank you for that small clap for that. I think it's about time that the government said that we don't tolerate this kind of concern, but it is driving misery across the whole country. It's driving the county lines gangs and we should be absolutely frank about it. But what we're doing goes far beyond that and people around the world can see what levelling up is doing. The massive extension of fibre optic broadband, and I'm just trying to look for Nadine where perhaps Nadine isn't yet, but Oliver, you were doing it, massive expansion when you were at DCMS of full fibre of gigabit broadband. It is a great thing for our country and we need to do it. We will go further and faster, investing massively in road and rail, not just the colossal schemes of the integrated rail plan, northern powerhouse rail, making sure that the midlands and the north of the country finally get the kind of commuter style rail networks that have been taken for granted for so long in the south-east. Huge schemes that we're doing, they'll be transformative for the UK economy, but look at the little things that we're doing as well, or relatively smaller schemes such as the new tram improvements in Blapple, which I was delighted to see the other day. I congratulate the Blackpool authorities on what they are doing because it's driving tourism, it's driving investment, even more investment here in Blapple, and that's what levelling up is. I went for a run, that's what you may not believe it, but I did. I went for a run this morning on the beach, absolutely beautiful. Better than anything in the Caribbean. That is not mere gallantry, it is true, and it's true. I mean, at that time in the morning, 6.45, sun coming up, unbelievable. It's tied right out, ribbed sand stretching for miles and miles, and absolutely beautiful. And as I ran along, I saw all the new hotels and the new attractions that will benefit from that new transport infrastructure going in, the new tram. Of course, government has a role, safer streets, better healthcare, better schools, better education, creating the conditions for that investment. But in the end, you need the private sector, the animal spirits of the private sector to come in and have the confidence to invest. That's what it's all about. That's the fundamental symmetry at the heart of our conservative vision. And as I was running along the sand, I saw a man who seemed to be prospecting with a kind of steel pole or tube, and I asked him what he was looking for. Hydrocarbons, perhaps. And what do you think he was looking for? Does anybody have any idea? I tell you, he wasn't an oil wellman. He was looking for lugworms. And he showed me some lugworms that he had caught. And I will tell you, my friends, that the lugworm is not perhaps the most beautiful of God's creatures. But bigger fish love lugworms. And I want you to know that we conservatives back everybody in this country who gets up early and invest their time and their skill and their energy and their effort in the hope of a bigger return. And that is... You need to use a lugworm to catch a bream, my friends. And I have a bream, as they say. I have a bream of a vision of a one-nation conservatism that takes that capitalist spirit and uses it, uses our wonderful free market system to make sure that we have the revenues for Rishi Sunak to pay for our fantastic NHS and the 50,000 more nurses and the thousands more doctors that are hiring. Pay for all the wonderful staff who are clearing the COVID backlogs. Pay for our defences. Pay for Ben Wallace's troops. Pay for our 20,000 more police officers. Pay for investment in skills. Like what we're doing with the Blackpool and Fylde College, the new multiversity that's opening, because that is what levelling up is all about. And it is those wonderful public services that create the climate of confidence, which means that private sector investment comes in. And it works... This formula levelling up, I believe, is right for the whole country. It's vital to understand this. It works everywhere, by unleashing talent everywhere across the UK. Still, under the old model, the most imbalanced European economy, by unleashing talent everywhere, you stop the overheating and the stress and the overdevelopment that is a part of the failed economic model. And we take the whole country forward together. That's what we're going to do. That's why Rishi, I think, is totally right to be driving a new age of post-Brexit entrepreneurialism. Tough word to say. You know what I mean. Low business taxes and other fiscal incentives, the eight new free ports and all the new freedoms that we're currently taking, driven by Jacob, who's our invigilator of these things, to do things differently and do things better now that we can. And I think it's because of the spirit that people can see in this country that we're seeing a surge of investment in the last few months. Another billion just this week from Al Fna, from Saudi Arabia in Teeside to make green aviation fuel, a billion from Nissan for a Gigafactory, a billion from Mubadala for life sciences, six billion for Ibadrola, a Spanish company in East Anglia and wind farms, 1.5 billion from Blackstone for labs and the creative sector. The list goes on and on. It's astonishing. A billion here, a billion there. You're talking about serious money. You're talking about tens of thousands of high-wage, high-skilled jobs. And there's another reason, of course, why investors come here. And when they think about the UK and what it's going to be like for themselves and their families, they think about the time that they're going to be spending in the UK. And I'll tell you something. It is the invincible strength of this country that we believe by and large and within the law that people should be able to do whatever they want provided they don't do any harm to anybody else. And that's called freedom. That's called freedom. And we don't need to be woke. We just want to be free. And that's why talented people are fleeing Russia, quite frankly, right now. And that's why they're flocking to the UK. And to get back to my theme, that's Putin's tragedy. That's his tragedy. Actually, there's a sense in which his disastrous error in Ukraine is itself an argument for democracy and freedom. I mean, seriously, if Putin had a free press, if he had the BBC on his case, well, let's be serious. He would know, whatever you may think, he would have known the truth. Or a version of it, he would have known the truth. OK, now come on. If he had free impartial responsible journalists, let me put it that way, then he would have known the truth that the Ukrainians are a proud, proud nation with a charismatic leader, and he would have known before he set out on this disastrous and inhuman venture that they would fight to defend their homeland. He would have known that. And he wouldn't have locked himself in this echo chamber of sicker fans. Which subject if Putin had to explain himself to a real parliament with real backbenchers, whom of course all leaders must have a very, very lively regard, who had to... Here, here. Backbenchers who had to justify themselves every day to their electors, and of course at elections. You know, I don't believe that he would have been capable of such a crescendo of disastrous and self-destructive mistakes. Isn't that the truth? Now look, I don't believe that democratic freedoms are going to sprout anytime soon in the Kremlin. Far from it. But with every day that passes, I think that Putin becomes a more glaring advertisement for the system that he hates and despises, and it becomes ever more obvious why we have to stick up for Ukraine. And we will. And that's why we will continue with absolute conviction to stick up for freedom under the law. Freedom under the law at home and abroad, even if it means making some tough decisions. We made a tough choice, for instance, over Christmas and New Year to keep going, to keep our economy open when some people said we should go back into lockdown. We made a tough choice to open up last July when I think that the Labour Party said we were being reckless. Never forget if we'd listened to Captain hindsight and the Labour Party, I never tire of telling you this, but it's true that we would still be in lockdown and we would certainly not be seeing the strong growth and employment that we're seeing today. When the Labour Party and the current leader were trying with might and main for a leader who wanted, I'm sorry, but I'm serious, to abolish NATO, we were already training Ukrainian troops to fight. And it's an absolutely incredible fact, but it's true. At a time when Russia is being led by a president who is capable of bullying and threats and who's plainly capable of making dangerous and irrational decisions, we have a Labour Party whose shadow cabinet is stuffed with people who only recently voted to abolish the UK's independent nuclear deterrent. That's right, eight of them, the shadow foreign secretary, unbelievable, the shadow levelling up secretary, shadow transport secretary, I haven't got the entire list, but you can find it out. That's them. Do we want them in charge, my friends, at this moment? Do we want them running up the white flag? Do you see them standing up to Putin's blackmail? By the way, in the next few weeks, do we want them running our councils where we know that up and down the country Labour councils cost you more and conservatives deliver better services? Do we want them in charge of the economy of this amazing country of ours when there has never been a Labour government that left with unemployment lower than when they arrived? Let me tell you, when you go out campaigning in the next few weeks, as we all will, joyfully, let me tell you that my message to everybody on the doorstep is that it is conservatives. It's conservatives who get things done even when they look difficult. And it's conservatives who take the tough decisions to help you, to be on your side, to help you with the cost of living. And it's conservatives who stand up for freedom against the blackmail of Vladimir Putin. And I'll tell you why we do it. We do it not out of ideology because we know through long experience that it's only by sticking up for freedom that we can deliver long-term prosperity and security. And that's what we will deliver together. Thank you all very much, and thank you for coming to the spring conference. Thank you. Thank you. Thanks, everybody.


Appalachian Wireless now has new forward pay pricing, where you can pay up front and get the data and features you desire. Five gigabytes of data, just $39.99. Unlimited, just $79.99. Better service, bigger savings. That's today's Appalachian Wireless. February 28th, 1958 is a date that many in Floyd County and throughout the region will never forget. It's on a Friday morning. And buses were pulling in to Prestonsburg High School, or this area, the grade school up here that used to be. But one bus didn't. At 8.10 a.m., bus number 27, carrying 48 students, plunged into the icy river near Prestonsburg. 22 students survived, but 26 students along with the bus driver died. At that time, there was no Floyd County Emergency and Rescue Squad in Prestonsburg or anywhere else in the state. But that crash changed everything. They saw the need to have an organization that helped people when they needed help the most. The paperwork was actually signed on the backs of people on that riverbank while they were searching. And there is when the charter members were formed and things began for the Rescue Squad. Less than two months after the deadliest school bus crash in the United States, the Floyd County Emergency and Rescue Squad was formed, which was the first of its kind in the state of Kentucky. Former Rescue Squad Captain Tim Cooley says he believes the Floyd County Rescue Squad's Swift Water Rescue Team remains the number one team in the state today because of those 27 lives that were lost. Because of 26 children and a driver and 22 that had to live with it the rest of their life that made it off that bus or survived, look at how many people that squad's probably saved. That's a testimony right there. Monuments stand in remembrance of the 27 who died on February 28, 1958, both at the Floyd County Emergency and Rescue Squad office and the actual location where the bus crashed into the river. In Prestonsburg, I'm Shannon Deskins, EKB News.


ELT – English Language Teaching In countries where British-based English is taught and spoken, ELT is used as an umbrella term that covers all teaching situations that are defined under the acronyms TEFL and TESOL. This usage corresponds directly to the way TESOL is used in American-based English. As English Language Teaching, ELT is rather a generic term, it is not unusual to find it used in many other situations. For example, it is sometimes used by large companies to refer to their language training departments or by publishing companies to refer to their language division.


Why the narcissist is jealous of you. The narcissist will flirt with other people. They will behave playfully as though they are attracted to other people. Which is very disrespectful to you. And very inappropriate. But if you even talk to someone else, they will be paranoid and they will get very angry. They can dish it out but they can't take it. Because they see you as their possession. They see you as something that belongs to them. So they can't stand to see you getting attention. They can't stand to see other people noticing you. They hate to see you making more money than them.  They will get very jealous and they will put you down. Because they're resentful of your achievements, possessions and advantages. They wish they had that. Which is why they may even feel entitled to steal from you. And they won't feel bad for doing that. Because in their minds, it's unfair. So they believe that you owe it to them. But it may seem strange to think that the narcissist is jealous of you.  They act like they have it all. So why would they be jealous of you? Why would they want anything that you have? But narcissists are very insecure. They lack confidence in themselves. They have no inner sense of value. So they have to get their value from other people. The narcissist is jealous of you. And they have no control of their jealousy. They will deny being jealous of you. They will treat you as unimportant. They will minimize anything that you have. But they're only doing that because they're insecure. And because they feel threatened. If you achieve something great in your work or you buy a new car. They won't be happy for you. They will be really jealous. They will get angry. Because they can't stand that something good has happened for you. Especially if they don't have anything going on in their lives. They may try to compete with you. Or they may compare you with other people. To try to make you think that you're not doing as well as you think. It's all about winning with a narcissist. They have to feel like they're better than you. So everything becomes a competition. But the things that they are jealous of are the things that they are insecure about. It activates their shame and it makes them feel inadequate. When the narcissist first targeted you. They saw everything that you had to offer. And they wanted it all for themselves. They wanted to take it from you. So they try to manipulate you into giving it to them. Because they're jealous of everything you have. Your job, your money, your resources. Even the empathy, love and affection that you give to them. Which is why they would often turn away and withhold sex from you. They're very bitter and resentful.  They have to manipulate people. They have to take everything for themselves. They have to use people. They have to steal. They have to destroy. But they've seen how you take care of yourself. They've seen how you treat other people. And they're jealous. But they also want to destroy it. Once they've gotten everything that they want. Because they don't want anyone else to have it after they're gone. They want it all for themselves. And that is why they isolated you from other people. That is why they turned other people against you. Or they turned you against them. That is why they targeted your self-esteem. And made you feel as though you weren't good enough.  Out of their own jealousy. Because they don't have what you have. And because they're very insecure. If they were confident in themselves they wouldn't have to manipulate you.  And they wouldn't have to discard you. Because they wouldn't be so jealous. But once you start getting stronger. And you begin to make more money. That is when they will really get jealous. And they may even try to get you back. Even if they're already with someone else. Because they're never satisfied with anything. They always want more. They always think the grass is greener on the other side.  Just like they did when they discarded you. They're never happy with what they have. Just think back throughout your relationship with a narcissist. Was there ever a time where they said they're in a good situation. And they're enjoying themselves. They could never do that. Because they were too busy looking at other people's lives. They were too busy looking at what everyone else is doing. They were jealous. They were comparing their situation with other people. Because they're never satisfied. Even if you gave them the picture perfect life. They still wouldn't be happy. You could buy them a car. But if someone else has a better car. They would feel like they need something better. Because they're jealous. They believe that they are inherently deserving of privileges and special treatment. They believe that they deserve the best. And if you achieve something great or you buy a new car. They're not going to compliment it. They can't do that. Because they're jealous. They believe that they deserve better than you. But they will never tell you that. They will deny it. But they are very uncomfortable with your achievements and success. They need to be better than you. Narcissists are very jealous people. They're always on social media looking at what other people have. And they're always trying to find ways to get it through manipulating people or stealing. Because they're always comparing themselves to other people. And they're like with this with everything. Whether it's cars, clothes, phones or houses. They're always comparing themselves. And they will do whatever it takes to get what they want. They will destroy people in the process. Because they habitually seek to harm, imitate and coerce people into doing what they want. Thank you for watching. I hope this video was interesting with you. Please like, comment, share and subscribe. If you would like to donate. My PayPal link is in the video description. Coaching inquiries. You can email me at coach.narcsurvivor.uk Thank you for watching and I'll talk to you soon.


Hello and welcome to this session in which we will discuss trading securities. Under the umbrella of investments we can break down our investments under debt investments and equity investments. Under debt investments we could have three categories health to maturity, why because bonds could be held to maturity until they mature available for sale and we already covered this category health to maturities and we already covered available for sale categories. In this session we will be looking at trading securities trading securities is only available for debt securities debt securities are US government municipal bond corporate bonds convertible that so on and so forth. We don't discuss any equity that when it comes to trading securities it was an old category that's no longer available. Before we proceed any further I have a public announcement about my company farhatlectures.com. Farhat accounting lectures is a supplemental educational tool that's going to help you with your CPA exam preparation as well as your accounting courses. My CPA material is aligned with your CPA review course such as Becker, Roger, Wiley, Gleam, Miles. My accounting courses are aligned with your accounting courses broken down by chapter and topics. My resources consist of lectures multiple choice questions true false questions as well as exercises. Go ahead start your free trial today no obligation no credit card required. So what are trading securities trading securities are securities investments that you plan to sell soon simply put you are flipping flipping means you want to buy them sell them in the near future to make a profit. They are reported through fair using fair value through net income. Fair value means what fair value means the investment might go up in value the investment might go down in value and as a result you have to report increases and decreases the question is where do you report those increases and decreases well if it's a trading securities and remember only that only that can be classified as trading securities any adjustment goes into net income in contrast now why am I emphasizing this point in contrast to securities that we consider available for sale which is a different category it doesn't go into net income so this is not available for sale we are dealing with trading securities so trading securities fair value through net income any discount any or premium is amortized because it's a bond any interest revenue is reported on the income statement because bonds would have interest revenue and we will report the interest revenue on the income statement the best way to illustrate the concept of trading securities is to take a look at an example this is a trading debt securities portfolio we have three investments the economic integration forum we paid 40 000 for it youtube corp we paid 180 bonds and Lenovo bonds we paid 36 the fair value of the economic integration is 41 so we have a plus a thousand youtube is a loss we bought it at 180 it's 173 loss of seven and the cost of the Lenovo we paid 36 the value is 46 we have a gain of 10 overall if we net them out we have a gain of 4 000 so the total fair value is 260 the total cost is 256 as a result we have a gain of 4 000 just to make it simple we have no prior adjustment simply put this is the first year now we need to do what mark our investment mark our portfolio to market mark our investment to market so how do we do that remember we learned this in the prior session it's the same concept we're going to have an account called fair value adjustment and we're going to have an account called unrealized holding gain or loss now specifically i'm going to emphasize this point i'm going to use it in different color this is an income account so it's unrealized holding gain slash loss income i'm going to emphasize income now what did we learn and available for sale how to adjust a portfolio it's the same concept if we have a gain if we have a gain it means we're supposed to have a debit balance and fair value adjustment for the amount of the net gain which is 4 000 now the prior gain is zero because we have no prior adjustment therefore the adjustment is debit fair value adjustment 4 000 and for any debit the corresponding any debit any adjustment the fair value the corresponding is unrealized holding gain or loss so if we debited this we are going to credit unrealized holding gain or loss and we have a 4 000 is a debit balance which is a gain and 4 000 unrealized holding gain or loss matches our 4 000 gain our 4 000 gain our 4 000 gain that we reported let's take a look at the journal entry debit fair value adjustment 4 000 credit unrealized holding gain loss income 4 000 remember the only reason the gain and the adjustments are the same because we had no prior adjustment remember if we have a prior adjustment the entry might be a little bit more involved and i'm not going to go over this because i'm keeping this example simple i looked at this in the prior session when we looked at available for sale and we would look at trading trading securities fair value adjustment from year to year in a different session now let's take a look at future scenarios adam sold the economic integration for a march 1st 20x1 so what we did we took this investment this bond and we sold it for 42500 at at at that time the amortized cost was 40 000 now we need to process the entries now i'm gonna show you two type of entries you can do to to make the sale your cpr view course or your intermediate accounting book might show you one or the other they're both the same but they break it differently the first thing the first method is i'm gonna adjust my bond to market what does that mean it means look i sold it for 42 500 it means this bond went from 41 by the time i sold it went to 42 500 i'm gonna write it up i'm gonna debit fair value adjustment 1500 credit unrealized holding gain loss income of 1500 so first thing i'm gonna write it up then i'm going to sell it and get the cash i'm gonna debit cash for what what i sold it for 42 500 credit fair value fair value investment 2500 and debit the credit the investment so i got rid of the investment i received the cash so now i need to explain what is this credit the fair value adjustment remember in the prior year we had 4000 and fair value adjustment this 4000 include a thousand from this bond then we added an additional 1500 so this bond has an fair value adjustment 2500 which since i'm selling the bond i have to remove the bond so that's one way of booking the transaction the other way to book the transaction is not to bring it up to date what does that mean means i'm gonna receive the cash for 2500 i'm going to remove the fair value investment remember this 4000 has a thousand for the economic integration bonds i'm gonna remove that i'm gonna credit my investment 40 000 i have to get rid of my cost i have to get rid of the thousand now i have to book also a gain of 1500 and this gain since the beginning of the year since the beginning of the year the value of the bond went from 41 to 42 500 went up by 1500 i have a realized gain on the sale of 1500 so once again the concept is the same we end up in the same place the total gain is 2500 for the whole life of the bond the same amount of cash and we remove the bond but you could see it differently make sure you understand how to treat this transaction from a two different perspective what should you do now go to farhat lectures and work mcqs and look at additional resources especially looking to adjust your portfolio from year to year that's important that's an important concept good luck study hard invest in yourself and invest in your career


Calculus began as a way to solve three problems, and one of those three problems is called the optimization problem. And this is the following. Given some function f defined over some interval, the optimization problem consists of finding the greatest and least values of f of x for x values in that interval. So it'll be useful to introduce some grammar and syntax because how you speak influences how you think. In this case, we say that the extreme value is at x equals a, which is a location, and is equal to f of a of value. It's convenient to distinguish between two types of extreme values. First, there are local extreme values. If you're running a race, you don't actually have to be faster than everyone.  Then there's also global extreme values. If you want to set a world record, you must be better than everyone who has ever run the race. So how do we find extreme values? A good way of searching is to figure out where something isn't located. Suppose the derivative at a point is greater than zero. Then we know, I mean, it's increasing until we get to x equals a, and that it's increasing after we pass x equals a. And so if a larger value of x exists, f of x will be greater than f of a, so f of a is not going to be a local maximum. On the other hand, if x less than a also exists, f of x will be less than f of a, so f of a is not a local minimum. And a similar thing happens if our derivative at a is less than zero. Then I know that f of x is decreasing, both until I get to a and after I pass through a. And so this leads to the following result. Suppose I have some value of x between a and b, and the derivative at that value is a non-zero real number. Then, because I have values that are larger than x zero and values that are smaller than x zero, f of x zero will not be a local maximum and will not be a local minimum. Well, this conclusion is based on assuming that there are values of x that are both higher and lower than a. Suppose there aren't. Suppose we're at the edge of the world, so to speak. So suppose the derivative is greater than zero. If there's no value of x that's greater than a, then our function is going to increase up until it hits x equal a, and then it will stop in some sense at f of a. And so f of a will be a local maximum value. And similarly, if there's no values of x that are smaller than a, then our function starts at f of a and increases after it, making our function value a local minimum. And a similar result is going to follow if the derivative is negative. And as a result, we get the following theorem. Suppose my function is defined and differentiable on the interval including a and b. Then f of a and f of b are local extreme values. And this gives us a lot of useful information. For example, suppose my function is restricted to the interval minus 10 to 10, and I know the values of the derivative. So for each value, we can try to determine whether we produce a local maximum, a local minimum, neither or both. So x equals minus 10 and 10 are boundaries. So they're guaranteed to correspond to local extreme values. The only question is what type? Well, we might begin with the observation that the derivative at negative 10 is 3. So we know that f of x is increasing after x equals minus 10, which means that f of minus 10 will be a local minimum. Similarly, at x equals 10, our derivative is 8. And so we know f of x is increasing until x equals 8. And so f of 10 will be a local maximum. How about at negative 5? Our derivative at negative 5 is 5. And so we know f of x is increasing until we hit negative 5, which means that f of negative 5 can't be a local minimum. We also know f of x is increasing from x equals negative 5. So that means f of minus 5 is not a local maximum. And what this means is that f of negative 5 is neither a local maximum nor a local minimum. And a similar argument can be made for f of 0 and f of 5. And so neither maximum nor minimum values can be found in these other places.


Hello, I'm Tom Sanderson, IT Infrastructure and Operations Manager and Fire Behavior Analyst at CFA. And here's the January update of the summer bushfire outlook. Despite being in the El Nino and positive Indian Ocean Dipole, which is typically very dry, rainfall during October to December 2023 was well above normal across much of Victoria, particularly in the North and in Gippsland. This is likely due to record warm global sea surface temperatures as well as the southward shift of weather patterns, bringing more tropical moisture down to Victoria. During January, tropical moisture drawn down from northern Australia has resulted in significant rainfall and flooding in central and northern Victoria. The first nine days of January had set the record for the wettest start of the year in Victoria since records began around 1900. Parts of Victoria have recorded three times January average rainfall, but there are pockets such as the outways that are below the average rainfall for January. The high rainfall has resulted in very much above average soil moisture in most areas. Catchments are now saturated and many water storages are close to capacity. This will mean forests right across Victoria would generally have lower potential for bushfires. Most grasslands and croplands in northwestern West regions are in the dry phase of curing. This extends into the northeast region and even parts of the southwest region. Due to recent rain and good soil moisture, we have seen grass start to green up and this has led to lower incidence of grass fires. In December, grass and scrub incidents have been below the long term average in all regions except for the northwest, which has been closer to the long term average. For the season so far, northwest region is running at between 500 and 5000 hectares, cumulative area burnt, while other regions remain less than 500 hectares, with the exception of the southeast, where large early season bushfires occurred in September. During December, approximately 800 hectares was burnt, most of this was in the grasslands in the northeast and northwest regions. The fire danger period is in place in all municipalities across Victoria. Looking forward for the period February to April 2024, it is likely that maximum temperatures will be warmer than the average. Following a wet January, it is expected that February to April will be average to dry than average, particularly in the north and west. In summary, heavy rainfall in early summer has reduced fire potential across Victoria, particularly in the forests. However, grasslands and dry woodlands and heaflands may rapidly dry out to a flammable condition with the onset of hot and dry weather. Victoria is expected to see a return to warmer and drier conditions February to April. Therefore, bushfire risk is likely to increase in this late summer period, particularly in the north and west of the state, while forests in the east will remain damper for longer. Wherever you live, work or travel. Now is the time to have a conversation with your family, friends and neighbours about what it means to live with fire. What can you do to be more prepared for this bushfire season?