 In this video, I actually want to start off a mini-series, the so-called Hunt for Red October. Wait, wait a second, that's not it. Oh, the Hunt for Non-Abelian Simple Groups. We learned in abstract algebra 1 that if you're an Abelian group, the only simple group possible is, in fact, the cyclic ones of prime order. That's the only Abelian Simple Group. And we revisited this issue, of course, in Algebra 2 here as well. But what about non-Abelian Simple Groups? It turns out there's a lot of them, but in some regard, they're sparse. They're scarce. There's not a lot of them. In fact, if you've been following this lecture series and you really have no other knowledge of abstract algebra, then the only other simple group we know besides the Abelian Cyclic ones would be basically A5. Well, we proved that one specifically. We also then argued that A sub n, the Alternate Group when n is greater than or equal to 5, is a non-Abelian Simple Group. There are, of course, many families of non-Abelian Simple Groups, and it will not be my, it's not our scope to discuss those in this lecture series, at least not at this time. What I want to talk about right now is that how do we know that A5 is the smallest non-Abelian Simple Group? Because A5, as a group, it's an Alternate Group, so its order is going to be, well, the order of the Symmetric Group is in factorial. If you're the Symmetric Group on Inletters, the Alternate Group is going to be half of that, and so as 5 factorials, 120, we see that the Alternate Group has an order, has order 60. Is there a non-Abelian Simple Group of order less than 60? And I claim that the answer is no. It's a very, very long argument, which is why we're going to break it up over several lectures. And in this very short video, I want to get rid of most of them, believe it or not. Because of what we, because of the equipment we already have, we've taken care of a lot of them. So what are some of the things we've already established? Why can't we have, why can't we have non-Abelian Simple Groups of smaller orders? Well, the first thing to mention here is that there's only one group of order one. It's the Trivial Group. It's not simple by construction, so that one's easy to rule out. The next one to talk about is groups of order of prime. Okay? Why can't we have a simple group of order of prime? Well, actually we can. The prime order groups are simple, but they're also sick like they're a billion. So if we're looking for non-Abelian Simple Groups, its order can't be prime. So we lose two, three, five, seven, 11, 13. I'll probably embarrass myself by forgetting a prime somewhere. 17, 19, let's see, 23, 29. Let's see, 31 is prime. I feel fairly confident about that. 37, 39. Let's see, 41, 43 are twin primes. 47, 49. I know you're seven squared. Don't trick me there. 51 is divisible by three. 53, 57. Nope, 57 is divisible by three. 59. There you go. So I think we've ruled out all of the prime numbers because there's no non-Abelian groups of order of prime. They're necessarily Abelian. Okay? But we can also elevate it one more. In fact, we can rule out, so we have no groups of order prime, right? We actually can also rule out there's no groups of order, excuse me, of any prime power order. Okay? Why is that? That follows from C law theory that when it comes to any group, the center of a group is always a normal subgroup, okay? Of G. And of course, if you're a P group, that is your orders of power of prime. In that situation, then we know that the center is non-trivial. That's what I meant to say. It could be the whole group. You could be Abelian in that situation. But in particular, the center of a P group is always non-trivial. It's not equal to the trivial subgroup. And so it's always a proper normal subgroup. It could be the whole thing. But in that situation, you're Abelian. And which case, we're looking for non-Abelian case, the groups in that situation. So if you're a non-Abelian P group, you have a non-trivial center. That's a normal subgroup. So you cannot be center in that situation. So we lose powers of primes as well. So no four, no eight, no nine. Let's see who else are we going to have here. We don't get 16. Let's just do powers of two first. 16, 30 to the next would be 64. So that's bigger than we care about. Powers of three, we get nine and 27. The next one's 81. That's too big. Powers of five, we get five and 25. The next one's 125. Seven, we're going to get seven squared is 49. And then there's no other powers of primes in that list there. Okay. So we've now ruled out powers of primes. But in the previous video for this lecture here, lecture number nine, it was a long video, but we discussed groups of order P times Q. So semi-prime order, we proved that a lot of the times these are cyclic. Sometimes you can be dihedral or a semi-direct product, but in particular, in each of those situations, you're not simple because you do have at least one unique Sealof Q subgroup, assuming Q is the larger of the two primes there. So your Sealof Q subgroup is going to be normal inside of G. It's also proper. And so you can't have any simple groups of order P times Q. So what's semi-primes do we got here? We got six, which is two times three, 10, which is two times five, 12. Can't say anything about that right now. 14 and 15, we did specifically in the examples we did before. And I'm actually going to do some highlights along the way here. So notice 12, right? 12 is two squared times three. That's not a semi-prime. 18 and 20. Let's look at those for a moment. 18, this is equal to two times three squared. 20 is equal to two squared times five, like so. So let's continue on. 21 is three times seven. That's a semi-prime. 22 is two times 11. That's a semi-prime. 24 will come back. 226 is two times 13. 28 and 30, those are not semi-primes. So we're going to highlight them here in our list. 24, 28, 30. So let's see. 24, that's equal to eight times three. So that's two cubed times three. Let's look at 28. That's four times seven. So two squared times seven. And then 30, it's an interesting one. It's actually a product of three primes, two times three times five, like so. Returning to semi-primes, 33 is three times 11. 34, what is that? That's two times 17. 35 is three times, excuse me, it's five times seven. That's gone. 36 is not a semi-prime. Neither is 40, but 38 is 38. That's going to be two times 19. So in that line, there were two numbers which were not canceled out so far. You got 36. That's four times nine. So two squared times three squared. And then 40, that's eight times five. So that's two cubed times five, like so. Going on to the next row in our list here, 42. Let's see. 42 is actually a product of three primes. That's two times three times seven. So that should be added to our list. 42 is two times three times seven. 44, that's going to be four times 11. So notice multiples of four are giving us a little bit of a problem right now. Two squared times 11. We'll deal with them later. Then 45, that is nine times five. So 45 is equal to three squared times five. Notice the bigger we're getting with our factorizations, the harder and harder it is to cancel them out. 46 is two times 23. So we remove it. The other two numbers on this row, we're going to have to keep around for a little bit. So you got 48. 48 is 16 times three. So that's two to the fourth times three. 50, of course, can be factored as two times 25, like so. Looking at the last row here, where are the semi primes? 51 is three times seven. So we remove it. 55 is five times 11. We have 57 is three times 19. And 58 is two times 29. And so the remaining numbers we have on our list, you're going to have 52. 52 factors as four times 13. We also have 54. 54 factors as two times 27. So three cubed. We have 56. 56 factors as eight actually times seven. So two cubed, two cubed times seven. And then the last one, of course, is 60 itself. We actually know there's a non-Abelian, non-Abelian simple group before 60. It's a five. So what we're going to do in our next lecture, and finishing up this discussion, this hunt for red October here, is we're going to then consider these other numbers 12 through 56 and explain why there cannot be a non-Abelian group, a non-Abelian simple group of those orders.