 Yeah, our next speaker is Andrew Nifoldein of Southern Utah University, and he's speaking on counting shawarings of the cycle groups. Alright, thank you everyone. Can everyone hear me okay? Yes. Alright, so like Anton, I'm here in Utah in the U.S., so it's like 4.30 in the morning so hopefully I don't say anything too ridiculous, and if I do, you'll assume I'm sleepy. Anyways, I wanted to begin just by thanking the organizers for the hard work they've put into creating this conference for everyone, you know, it's kind of an unusual time in the world so it's nice that there are still some opportunities for people like us to get together and talk about mathematics and such. And so in my talk, what I'm going to first want to do before I start talking about shawarings is give a little bit of motivation by comparing a similar problem that's quite well known in combinatorics. So if we were to talk a minute about the bell numbers named after Eric Bell, whom you can see on the slides right there, the bell numbers is a famous number in combinatorics, a famous sequence of numbers which we say in combinatorics that counts the number of ways we can partition a set, or we might call this a set partition, distinguish it from integer partitions which are related. And so a graphic I have here is if we were to take a set of five elements and we were to consider all the different types of partitions one could create for that set of five elements, there turns out to be 52 of these partitions and you can see in this graphic some different possibilities like on the left you see that you could just glue everything together in one cell, on the right you could have five single tens and then there's some different pictures in the middle here. And so a question that one could ask is, all right, we can partition sets but what if we want to do similar things in different categories, like what would it mean to partition a finite group and how many different partitions can we come up for said finite group. Now, if we were going to partition the group, the partition in some regard should respect the algebraic structure of the group, otherwise we're just partition a set. And so what are some types of partitions of a group that one would actually care to count if we want to list all of them. And so some sort of obvious examples that might come to mind if you think about that. The partition of a group via its conjugates is a fairly important partition that one could consider. We could also partition group elements via automorphisms of some kind, sort of a simple partition as you could glue together elements with its inverse. Co-sets also is another very important equivalence relation we put on groups of working with the non-normal subgroup, maybe a double coset or just another sort of very simple equivalence one could put on a group is you could just talk about membership to a subgroup or not. And I list these examples, this is certainly not an exhaustive list of group partitions one might be interested in. But all of these examples I've listed so far are all special types of what's referred to as a sure partition or what I'll refer to as a sure ring. These of course were named after sure. The phrase was coined by V-Land. Now what is a sure ring? So first of all, let's see a sure ring, it's going to be a partition of the group into, and this will be a finite group here, it's going to be a partition of the group into cells, say C1, C2 up to CR, and this partition, we're going to try to think of these as elements inside of the group algebra, we're going to take QG here. And so we're going to do this by identifying subsets of the group with elements of the group algebra. And this will just be by taking your set, you'll identify with the formal sum of all the elements inside that set as an element of the group algebra. So we're going to take some liberty of using the same symbol described, these subsets as we describe the elements of the group algebra itself. And so this partition can be identified with a submodule of the group ring, we're going to call that object S, and then to make it a sure ring, there's a few other conditions that we're going to have here. So the first condition, so with this partition here, the first condition we want, let me just get these all on the screen right here, the first condition is we want that the identity element of the group is going to be a singleton inside of this partition. The second condition we require is that if you take the set of inverses of any class in the provision, you want that also to be a class in the partition. And then finally, if we take the product of any two cells in our partition, we want the product to be a linear combination of the other cells in this partition here. And so when you look, and then so if a partition of a group satisfies these three conditions we refer to it as a sure ring. And the reason why we call these sure rings is that these objects, when you view it as a submodule of the group ring, is a genuine subring of the group algebra there. And so in addition to being a partition of the group, there's a very nice ring structure to this partition. And when you look at the three axioms on the screen right here, you can see that the axioms of a sure ring very much mimic the axioms of a group. We have to have an identity, we have an inverse-like object, we have closure under multiplication. And so these partitions of the group in many ways act like, they're group-like objects. And so in regard to considering partitions of a group, this is a very natural type of partition to study as these are very group-like themselves. And so I want to give a few examples of partitions that lead to sure rings. The simplest of all examples is just to take the partition where you put every group element by itself as a singleton, to sort of borrow a term from topology, we could refer to this as the discrete sure ring because every object is by itself. And so as an example, if you take the cyclic group of order six, then the group algebra itself is a sure ring. So sure rings do capture the notion of a group and generalizes it. On the other extreme, you could take a partition where the identity element is by itself because that's what it has to, that has to happen. And then you clump everything else together into one massive cell and we get what we'll call the industry sure ring or sometimes this is called the trivial sure ring. And again, you can see another example for the cyclic group of order six, the identity of by itself, but then the other five elements could glue together. Some other examples, you can build sure rings using automorphisms. If you have a subgroup of the automorphism group of your group, we'll call it H, then H will act on the group G via automorphisms. And the orbits associated to that group action give us a sure ring partition. And so you can see on the screen an example for the cyclic group of order seven. If we act on it by powers of two, you'll get this rank three sure ring. You can construct sure rings using a direct product factorization if our group G factors as H times K, you can take a sure ring over H, you can take a sure ring over K. And if you take the tensor product of those two sure rings, that'll give you a sure ring over the direct product H times K. And so we'll call these direct product sure rings. And then as a last example, this last one's a little bit more complicated to explain. So I'll just give a watered down version of it. We can build sure rings using normal subgroups and their quotients. If we have a normal subgroup H inside of G, then take a sure ring over the normal subgroup H, take a sure ring over the quotient group G mod H. And again, without going into the details, there is a way that we can glue together the sure ring over the normal subgroup and the quotient. And this is commonly referred to as a wedge product of the sure ring. And so this construction's going to involve cosets in great detail there. And so if I bring up that graphic again of the 52 partitions of a set of size five, if we take that five element set and identify it with the five complex roots of unity, then of those 52 partitions, only three of them will coincide with these sure ring partitions. You see on the left here, this would be the indiscreet sure ring. The one on the far right would be the discrete one. And then the one in the middle, this would be an automorphic sure ring that corresponds to identifying inverses or complex conjugates with each other. So one would actually anticipate that these sure rings are very sparse inside the set of set partitions. There's not a lot of them. Now, with that said, if one wanted to enumerate the sure rings for a group, it would be a very impractical approach to just search through all of the set partitions and check which ones are sure rings and which ones are not. Because the number of set partitions is going to grow rapid. It's going to grow very, very quickly. And it would be too exhausted to try to search for all of them one by one by one. So if we're trying to enumerate sure rings, we need a much more strategic approach. Now for cyclic groups, we have the advantage that we can actually classify all the possible sure rings for them. And so in the examples we listed, we had the indiscreet sure ring, the automorphic one, the direct product, and the wedge products. These four families of sure rings we're going to put together and recall these the traditional sure rings. And it was shown back in the 90s by Lungan Mon that all sure rings over a cyclic group will be traditional. They'll fall in one of these four families. And so using this classification theorem, we can use that to try to enumerate all of these possible partitions of the group. Now, when it comes to the direct products and the wedge products, these are built recursively off of smaller sure rings for proper subgroups. And so with this classification theorem in hand, one strategy we would use to enumerate all the sure rings is to try to grapple the recursive relationships from these constructions, which the direct product is fairly simple. The wedge product, I kind of mentioned earlier how I simplify the explanation. Well, the recursion there can get quite wild very quickly. And I don't want to say too much about that in this talk for the sake of time. But the other element we have to consider is how do we enumerate the so called indie composable sure rings? Those sure rings which cannot be broken into smaller sure rings using direct products or wedge products. And so I wanted to use the remainder of my time to get a little bit of understanding on how you try to enumerate the indie composable ones. For a cyclic group, you have the indiscrete one, which is going to be unique. There's one indiscrete sure ring per group. And so all the other ones are going to be automorphic. And so it really comes down to trying to identify automorphic sure rings for a group. And now for a cyclic group, the automorphic sure rings are going to be in one-to-one correspondence with subgroups of its automorphism group. So if you want to consider the indie composable ones, you have to consider what does the automorphism group for these groups look like. Now, if your group is a cyclic P group, the automorphism group is fairly tame. In the odd, if your prime is odd, your automorphism group is just going to be cyclic itself. And so if you take, for example, the cyclic group of order five, its automorphism group is going to be the cyclic group of order four. And the subgroups will correspond to the divisors of five minus one, and so you just see a simple chain lattice right there. If we increase the power from five to five squared, what you're going to see is you're just going to take the original lattice, copy it and stack it on top of it. And as you look at higher powers, five cubed, you're just going to use copy the original lattice and stack on top of it, five to the fourth. You see this pattern happening again. I like to think of it as building a stack of Legos, in which case, your tower just gets taller and taller and taller as you take higher powers of your prime. Similarly, if you do the cyclic group of order seven, the first layer is going to look like the divisor lattice of six. You copy that, put it on top of the next layer, then the next layer. And so from these examples, I want you to see that the automorphisms, the automorphism group of a cyclic group of prime power order when P is an odd prime, there's a very simple recursion to how these things develop. You need to have the divisor lattice for P minus one, and then there's this recurrence that occurs by building your towers bigger and bigger and bigger. So if we let omega of n denote the number of shear rings over the cyclic group Zn, and for an odd prime P, let the number x denote the number of divisors of P minus one, x there being the number of groups you see in the number of subgroups of the automorphism group, then we can get a formula, a recursive formula for omega P to the n, and it will recurse using omega Pn minus k. You'll multiply things by x. And also, interesting enough, the recursion from the wedge products actually produces the sequence of catalon numbers, which is a very famous number sequence in common torques there. Just very quickly, that formula can be expanded to polynomials that look like this. Again, this is nothing anyone needs to memorize here. Just sort of an interesting comment. Higman posed a problem about counting up to isomorphisms groups of P groups, finite P groups, and he conjectured that the number of P groups is a so-called pork function, polynomial over residue class function, which essentially means there's a number of polynomials that can be used to count the number of different groups up to isomorphism for various primes. And so I just want to make a comment here that, although Higman's conjecture has not been proven for some special subcases it has been, and it's interesting that the number of sure rings over cyclic groups, it turns out also that this is a special case of that conjecture that is a pork function after all. There's polynomials we need to use to describe this growth. And if you plug in specific primes, you'll get numbers like the following. The even prime P equals two is a little bit different here. As you look at, as the power of two increases, the lattice will start growing in the following manner. It's no longer a, the automorphism group is not a cyclic group, but we can still predict how this thing is going to grow. Each time you increase the power of two, you're going to get three new subgroups that grow onto the automorphism group there. And so using that recursion of the growth of the automorphism group, you can use that to come up with a formula for the number of sure rings of cyclic group of order two to the end. You can see that formula on the screen here. You can see that it's a whole lot more complicated than it was for PDN. It does recurse off of smaller omega values. The catalog numbers come into play again because of the wedge products. But also the shorter numbers come into play here. And that's a consequence of the fact that you're growing, the tower grows, adding three new subgroups each time. But the fact that we no longer have a cyclic automorphism group is essentially where these shorter numbers are coming from. Now, and then for some specific values, you can see those right there. So kind of concluding here, I wanted to mention that what would you do if you wanted to count these sure rings when you have multiple prime divisors? Because it turns out this problem can get much more complicated the more prime divisors you add. And so a natural example would be like, what if we take a semi-prime, like 15, which is three times five, it's just the product of two distinct primes with no repeated primes with sliver. In this case, the automorphism group is Z2 cross Z4, for which you get the lattice associated to Z4. You get the lattice associated to Z2. And when you take the direct product of their lattices, you get this parallelogram shape there. But these six groups don't account for all of the subgroups present. You have these two extra so-called diagonal subgroups that show up in the lattice there. And this actually is a pretty deep question in group theory, that if you take the direct product of cyclic groups or the direct product of abelian groups, can you find a closed formula to count all of the subgroups of said abelian group? And it turns out that if you take the product of two cyclic groups, we have a formula for that for three. And I think there is a closed formula for that. But for four, the rank four problem, I believe, is still very open. And so it's kind of limited on how well we can predict with a formula, the number of these subgroups of the automorphism group. But in the special case of a semi-prime, it doesn't get too out of control here. This, I should mention, this result you see on the screen is actually joint with two students of mine from Southern Utah University, Joseph Keller and Max Sullivan. We are able to come up with a closed formula for the number of sure rings for cyclic groups of order p times q. Again, it's a little complicated here and a lot of this comes down to how do you count the subgroups of the automorphism group. And then one last example that Joseph, Max and I were able to work on is if you take the factorization four times p, in that situation, your automorphism group would be z2 cross zp and minus one. You get as your lattice for the automorphism group, the lattice associated to z4. You'll also have the lattice associated to zp, its automorphism group. Taking the direct product, you'll get a lattice isomorphic to the original lattice. You'll glue it on top. But then you have also these diagonal subgroups that show up in the middle. These are the ones you have to worry about, but 4p is still fairly tame. So these diagonal subgroups do not get out of control in that context. And of all the formulas you've seen on the slides so far, this last one is the cutest one, and that's where we'll end. The number of sure rings of a cyclic group of order 4 to the p, you get this nice little formula right here where the x means what it did before. k is the number of times two divides p minus one. And so taking these techniques that I've mentioned here, one could extend this to potentially finishing the rank two problem. You could use these techniques to count the number of sure rings for any exponent as long as you have at most two prime divisors, p to the n times q to the m. We don't have a formula for it yet, but if you continue down this road, we could find something for that formula. And so it's natural to continue on and ask if you have more prime divisors, three primes, four primes. If you take a square free number, that's where you'd want to kind of push this thing forward. But trying to count sure rings, as I mentioned before, it's gonna be obstructed by this other group theory problem about cutting subgroups of an abelian group. And so this is a really fun problem that I've been able to work on over the last couple years. But like I said, it's kind of obstructed by this sort of bigger group theory problem that's sitting in the background about cutting subgroups. All right, and that's my talk today. Thanks everyone for listening. Are there any questions I can entertain? Let us just maybe send the speaker and then comment some questions, please. Any questions? No? Well, let us send the speaker again and