 Hello, my name is Joseph Keller, and today I'm going to be presenting my research on county insurings over the CICLA Group of Order 4P. This project was mentored by Dr. Andrew Missildine, and worked on together with my partner Max Sullivan. I would like to begin by giving a huge thanks to the Walter Maxwell Gibson Fellowship for Undergraduate Research. They have funded my research, and without them, this wouldn't be possible. We begin with a definition of a shoring given this particular group algebra with C1 through CR being what we call primitive sets or classes, and C1 bar through CR bar being simple quantities. This set S is a shoring. If one of the classes is the identity, there's some form of inverse operation that maps classes between each other, and we also have that this product of simple quantities is going to be the sum of scalar multiples of other simple quantities. Some helpful additional vocab would be that an S set is a union of those primitive sets or classes, an S subgroup is an S set that's a subgroup of the original group, and an S section is just a chain of those S subgroups. We will say that a S section is non-trivial if one K is not the trivial subgroup and H is not the whole group. There are a couple different types of shorings we'll be interested in, the first being a trivial shoring. This is the one wherein one class is the identity, and every other element belongs to the second class. We will denote this with a superscript zero. We also have automorphic shorings, given any automorphism or subgroup of the automorphism group, it partitions the group, and that partition is the automorphic shoring. If H is simply one, then each element is its own class, and we call that the discrete shoring, and if each element is together with its inverse, we call that the symmetric shoring. We also have direct product shorings. So given a group that is a direct product of other groups, given shorings over each of those groups, the direct product shoring will be the span of C and D, C and D being classes of each of those shorings. It's important to note that H and K are both S subgroups of this new shoring. We also have shear quotient rings, so given a normal S subgroup, since we're working with cyclic groups, any subgroup is a normal subgroup, and this natural quotient map from G to G mod K. If we take classes of the shoring S and map them according to fee, this will produce for us our shear quotient ring. A property of shorings that we're interested in is wedge decomposable. We say that it's wedge decomposable if there's a proper S section, such that every primitive set is a subset of H or a union of cosets of K. Otherwise we say it's wedge-indicomposable. This is going to be tied very heavily in with our wedge products, construction and counting for our particular problem. Speaking of, a wedge project shoring is going to be constructed in the following way. If you have a proper section of your group and you have shorings over H and G mod K respectively, as well as those three qualities being satisfied, then by taking the inverse map of the classes of T and adding them with S, we will construct our wedge product shoring. So those four previously mentioned, the wedge, direct, automorphic and trivial are what we call traditional shorings. That's very useful because according to Lungenmann, all of the shorings over cyclic groups are traditional. So now we have an exhaustive list, which if we can enumerate them and eliminate any overlap, we can count the number of shorings over the cyclic group of order 4p. It is important to note some of the overlap. So, for instance, direct products are automorphic when both products are automorphic. Direct products are wedge when one of them, one of the products is wedge. And we also have that automorphic or wedge decomposable only if they map into the bottom layer of the lattice of the automorphism of z4p for our particular case. So this would be the example of the lattice of z4p. This bottom half right here is going to be isomorphic to zp, that is the lattice of zp. And the top part of this lattice would certainly be zp, while the top part of this one would be z4p. And again, this top stretch right here will be also isomorphic to the lattice of zp. The thing I have thus far omitted is the central part. The central part is what we call the diagonal subgroups. The top layer referred to before contained both this top part of the lattice as well as the diagonal subgroups. The bottom layer referred to this portion of the lattice. And this guy right here is kind of up in the air. So we'll see why that is in a bit, but we'll come back to this picture. In order to enumerate the z4p case, we're going to enumerate the z4p case. We do need to understand the z2p case. So the z2p case is a special case of the semi-prime case, which was discussed by Max Sullivan. There are four possible forms, z2p wedge zp, zp wedge z2, and an isomorphic, or in this case, isomorphic and direct product are equivalent, or z2p not. So there are x of the sum of the sum of the sum of the first three types, and there's one trivial assuring. So that brings our total, which we will always denote with omega. So omega2p with the trivial subgroup will be 3x plus 1, while just for future reference, omega41 will be 3. That is, you have z4 you have z4 not, and you have z2 wedge z2. If the section is trivial, we will omit the subscript here for the wedge products. So in order to enumerate these wedge products, we want to enumerate all of them without double counting. So the way we do that is we consider indecomposable left factors. If we consider decomposable left factors, then by the associativity of the wedge product, we can potentially get double counts, even if they have different representations. So forcing the left product to be a indecomposable assuring will alleviate that issue. So when constructing them, we basically have a indecomposable left factor, which leads itself to potential sections, and potential sections will lead us to a assuring over t over this particular subgroup. So some quick examples of that. We have, if we pick a s to be over z2, there's only one option, just the indiscreet one. So we would have something like the following z2 wedge. And since z2 has no subgroups, the section is 2, 2. So we would have z2 wedge z2p, or rather some assuring over z2p. But again, since this section is 2, 2, it only has to contain the trivial subgroup, which as discussed previously will give us 3x plus 1. Similarly for s over zp, you have this pp section, which will again give you omega 4, 1, omega p, 1, and that'll give us the 3x as we were talking about earlier. Lastly we have two trivial cases. Trivial showings don't have any subgroups except the trivial and the whole group. So again this is the 4 to 4 section and the 2p to 2p section. And so this one will be z4 not wedge some assuring over zp, and this one will be z2 p not wedge some assuring over z2. So in this case we get x, and in this bottom case we get 1. Now some more interesting ones occur with z4 and z2p. So for z4, this has as a possible subgroup 4, 4, or rather possible section 4, 4, or 2, 4. Now if we recall back to our definition, the classes had to be subsets of z4 and also cosets of k. Well every coset of z4 is also going to be a coset of z2, so this 2, 4 section is preferable, and with this particular section in mind we'll get omega of 2p2. So for omega 2p of 2, if we go back to our 2p case, we want all the assurings over z2p that have z2 as a subgroup. So that is z2 wedge zp and all of the automorphic assurings, and that's why we get the 2x there. For the next case we have something like the following, z2 cross zp, that is assurings of this over each of those, taking the direct product, and we're going to, since that has z2 and zp and z2p as subgroups, we have all three of these possible sections. So in that case, like the 2, 4, 4 case, 2 to 2p is strictly smaller than 2p2p, and p to 2p is strictly smaller than, or rather 2p to 2p is strictly smaller than p to 2p. So using this inclusion exclusion principle we get that sigma of, or excuse me, omega of p1, and that selection of the prime shoring forces the selection of these. And again if we go back to our 2p case, there's two cases that have zp as the subgroup, and there's two cases that have z2 as the subgroup for a particular choice of shoring over zp. So that's why we get the 3x here. So that gives us all of the wedge product shorings. So for the automorphic, we're calling back to that picture above, we need to include at least as many as are in the top layer here, and maybe this guy over here. So this is more in depth in Max's talk, but this corresponds to the diagonal subgroups, and with our calculation of the diagonal subgroups, that portion of the lattice is 2k plus 1 over k plus 1x minus 1. However, the shoring that maps to the z4 there is zp naught cross z4. So for a prime, all of the shorings are automorphic. So this direct product here is an automorphic, which means that there are 2k plus 1k plus 1x plus 1, so there's 2k plus 1 over k plus 1x automorphic, shorings that are specifically not wedge products. For the direct products, we need it to not be automorphic, and we also need it to not be wedge, so that means both factors cannot be automorphic, and none of the factors can be wedge decomposable. So the only non-wedge decomposable, non-automorphic S-ring over z4 is the trivial, and that allows us to take any choice of shoring over zp, so there's going to be x direct products that are not automorphic and not wedge. That together with the last trivial shoring, which there's always one of, will give us this theorem that for any odd prime p equals 2k to the a plus 1, we get that omega of 4p is the following. So some questions that still remain in this particular research topic is obviously in this case, we knew how we constructed the shorings over z2p, so the omega rs recursion is pretty straightforward, but if we want to generalize this, having a better understanding of that recursion would be very helpful. We also want to generalize this to p squared q, zp squared q, or any rank to zpnqn, and obviously that last one is a bit more complicated than this one. But there's also more resources in the preprint of our paper. This was just the ones that had direct reference to stuff that was talked about in this presentation, and I just want to thank y'all for listening.