 Hello, my name is Max Sullivan and I'll be presenting these slides based on an upcoming paper I have with my research partner Joseph Keller and our research mentor, Dr. Andrew Misseldine, on counting sure-rings over cyclic groups of semi-prime order. So first off, what is a sure-ring? Well, for some finite group G and the group algebra over G with coefficients from the field F, we'll define elements from X by these sums of elements in that group algebra and we say that for some partition of primitive sets of the finite group G and a S, this funny symbol is an S, let S be the subspace of the group algebra spanned by these simple quantities and we'll say that S is a sure-ring if it fulfills these three conditions. And first, the identity is in a partition by itself. We have the second condition that basically forms quasi-group inverses over the ring and this third condition basically guarantees that it is closed under multiplication of these simple quantities. So we'll move on to a couple more definitions about sure-rings. So an S set for some sure-ring S and S set is a union of some of those primitive sets, some of those CIs. An S subgroup is an S set that just happens to be also a subgroup and an S section is for two S subgroups. We have this section KH is an S section of the sure-ring S, this particular one, where K is on the trivial subgroup is subgroup of K, which is a subgroup of H, which is a subgroup of G and K is normal in G. This section is proper if K cannot be equal to the trivial subgroup and H can't be all of G. So there are a couple of types of sure-rings. We'll identify what we call traditional sure-rings, the first of which is called the trivial sure-ring. In the trivial sure-ring it only has two classes, the first one has to be one since the identity is by itself, by definition, and then the rest is just everything else in G. For any finite group G, there's always a trivial sure-ring. We'll see later that it's unique and it's denoted with the G superscript zero. Next we have automorphic sure-rings. So for the automorphism group over G and H, which is some subgroup of G, will partition G into these CIs according to what H does when it acts on G. And this would be the automorphic sure-ring. And we'll denote that with G with the superscript H. In the case that H happens to be the trivial subgroup, this will form the discrete sure-ring where all the elements are in their own partitions. And for any abelian group, this G with the superscript plus minus is called the symmetric sure-ring. Basically we pair each element with its inverse. That will always be available in abelian groups, which is what we have to be dealing with as well. Next we'll move on to direct products of sure-rings. So if we have two sure-rings, S and I, if we can partition, or not partition, if we can divide up G into this direct product of H and K, where H and K happen to be S subgroups for S and I subgroups particularly, then S and I are sure-rings over H and K respectively, then we can just take their direct product and this itself will be a sure-ring constructed by the following formula. And so next we'll talk about sure-sub-rings. And so for an S, for a sure-ring S over G and H, which is some S subgroup, if we intersect S with the group algebra over H, we get a sure-sub-ring over that group algebra. And so of course if we choose the right H and I, we can just get all of the original sure-rings if we have the right sections. Moving on to our final, oh wait one more, so sure-quo-sion-rings will be, so if we have an S be a sure-ring over G and K normal S subgroup, we can define just the natural quotient map and just have a sure-ring over that quotient map. It's called the sure-quo-sion-ring. Of course with the right choice of sections we can just get the whole sure-ring. And then we'll have one more definition and that will be wedge decomposable. So for a sure-ring S, S will be wedge decomposable if we can find some proper S section for every primitive set. Either C is going to be a subset of H or just a union of cosets of K. Basically we can decompose it into these different sections. If we cannot find some proper S section of this, then the sure-ring S would be wedge in decomposable. So for some wedge decomposable sure-rings we have wedge product sure-rings. So for some sure-ring that would be wedge decomposable and that proper section that is attributed to it, an S and I over H and G mod K respectively, then we can form the wedge product by the following function between these two things and note that the wedge product always has to do with a specific S section. So we'll see later that this is a non-abelian operation as well and so counting this gets a little bit more complicated later on. So that brings us to the end of our types of sure-rings and so we'll basically classify those sure-rings as the traditional ones. So a sure-ring is traditional if it's either trivial, automorphic, a direct product sure-ring, or a wedge product sure-ring. And of course a theorem from Lang and Mann, all of the sure-rings over any cyclic groups are traditional and so since we'll be dealing with cyclic groups we only need to consider these traditional sure-rings when we're counting them. So moving on to an example, consider the group Z6, the trivial sure-ring, we just take one by itself and then everything else in the second partition. The symmetric sure-ring, we take Z1 and Z5, Z2 and Z4 and Z3 because they're all with their respective multiplicative inverses in the group. So we have the wedge product sure-ring Z2, wedge Z3 forms this one and the direct product between these two. So also Z2, Z3, and then this Z2 on the final line here. These are all the discrete sure-rings in this case. So not just those groups but just the discrete sure-rings of those. And so we can get this for the wedge product sure-ring and then the direct product sure-ring with the discrete over Z2 and the trivial over Z3 will form the direct product sure-ring listed here. Move on to our main problem which is actually counting them. So the goal is sort of to count the number of those traditional sure-rings and so using the principles of inclusion and exclusion to remove any double count as well. So we'll count those four types and then discount any overlap between them. So basically we just taking the following principle so we'll just count those four types and subtract any of the overlap between those types. So for the number of trivial sure-rings, so we'll move on to actual counting, the number of trivial sure-rings over a group of semi-prime order or pretty much any group there's only going to be one. And so it only adds one to our final formula and as an added bonus it's not going to overlap with any of the other types of sure-rings and so we can just count one and we don't have to worry about subtracting the intersection with automorphic direct product or wedge product sure-rings. And so we get the following added to our general formula that we'll have at the end plus one. So pretty nice there. Moving on to automorphic sure-rings. The automorphic sure-rings over a semi-prime group can be counted by counting the subgroups. It's not, so there's a nice bijection between those two sets of numbers and so we can use a nice counting argument there. And there's a source on sort of the methods of counting subgroups over groups of those orders. We use some of those methods to arrive at our conclusions, but a previous result proved by my research mentor Dr. Misseldine. The number of automorphic sure-rings over the automorphism group of Zp p or p is some prime to the k will just be nk and n is just the number of divisors of p-1 or just the totient of p. And so we'll use that along with some other results about automorphism groups. The automorphism group of Zpq can just be broken up into the direct product of the automorphism groups of Zp and Zq since they're relatively prime. And then those automorphism groups since p and q are prime will just be isomorphic to Zp-1 and Zq-1 respectively. So we'll go ahead and use these to develop our counting argument has to do with how many subgroups are going to be given over those. So it's going to deal a lot with the common factors of p-1 and q-1. So we started out by dealing with some specific cases like safe primes and formal primes, but now we've gotten to a point where we got a general formula for these and it looks something like this. And so where we can count sort of the number of, where we can actually count with the totient function and these finite sets, the actual number of automorphic sure-rings over that group. And so this will be added into our final formula and so this will be one piece of it, specifically the automorphism piece and then we'll have a piece for each of the different kinds. So moving on to the direct product ones, it's actually very nice because we have a result that any direct product s-ring will be automorphic if and only if it's a product of automorphic sure-rings. And we actually get that every direct product sure-ring is going to be an automorphic sure-ring since the product will have to combine each of the s-subgroups of each s-ring and since the right s-subgroups will be in the right places, each of them will be automorphic. So we actually don't have a separate formula. We've already counted the number of automorphic ones. The direct product sure-rings will be a proper subset of the automorphic sure-rings. And so we get that we don't actually have to add anything else to our formula. We've already counted three of the four types. So moving on to the final, possibly most complicated type, but you wouldn't know it looking at the formula. The wedge product sure-rings, all of them will be of some form where it's a sure-ring, this funny z, we're not denoting a sure-ring over those groups. So a sure-ring over a zp and a sure-ring over a zq or a zq wedge zp because this is a non-Abelian operation and depends on the sections. So the sections actually don't factor into these groups. And so we don't have to denote it on the bottom here. It's a little bit easier when they're both prime. We have just sort of a more specific wedge product. But anyway, so the number will actually just be the number of these and the number of these where we just count the number of sure-rings over zp and zq and that's actually a much easier operation. So we get a little recursive with it, but in the end we get this following formula. We get the number of wedge product sure-rings will just be two times basically the number of sure-rings over zp and times the number of sure-rings over zq. So arriving at our general results, we have to count the overlap first. So with the overlap with the auto-orific sure-rings we actually don't get any because any wedge product sure-ring of the form of zp wedge zq will contain all the s-subgroups of zp but not the right s-subgroups of zq. And so we'll not be able to be automorphic since an automorphic one will contain both the s-subgroups over zp and zq. And so none of the wedge product sure-rings over a semi-prime group could be automorphic. And so those three pieces of the puzzle are all that we need to complete our general formula. So we've counted all four and we get the number of sure-rings. So for p and q primes this omega pq's number of sure-rings over the number of sure-rings over the group zpq where this was the automorphic part plus the wedge product part and plus one for the trivial sure-ring. And then phi of course is the totient function, just counts the number of integers relatively prime to p. So that's it. So that's our main result. Some further work that we did. So this was just sort of counting all of them from that main theorem and verifying with a computer all of the sure-rings over semi-prime groups up to 100. And so these are all of the semi-prime groups up to 100 in this nice table. So that's pretty nice. So another result that we have, this is actually, there's a separate presentation over about the sure-rings over the 4p groups. And so that my research partner Joseph Keller should have presented. I'll probably be not too hard to find if you found this one. So for any prime of the form 2 to the k a plus 1. So some prime where it's 4 times some prime. We get the following formula for the number of sure-rings over that. So a little bit more neat right there. So it's kind of a nice symbol right there. There's a little bit more cumbersome notation in the paper, but I decided to omit it. But yeah, so this is a nice, nice type formula. There's, as I said, a separate presentation on that. So some further questions we had when we were working through this project. So what about groups over p squared q? So pq we figured out, then why don't we try another progression of that. So still working with primes, but p squared since the automorphism group of p to the k was already figured out. So for any k that would be easy. But what if we had another q on there? All about 2pq. So we have sort of the, you know, just 3 primes. But with the specific prime 2 since it'll make it a little bit easier to deal with it first. And sort of figuring out how to make a computer program work for higher numbers so that we can verify larger values of pq and 4p and those sort of things with a computer program just to further corroborate our results so far. Here's some of the things that I cited. Obviously there's quite a lot of secondary reading material that went into this presentation. I'll all be sort of cited in the upcoming paper. But these are the ones I directly referenced in this presentation. So yeah, that would be the end of our presentation. Thank you very much for listening. Let me know if you have any questions. I think we'll be putting this up on YouTube. So welcome to comment on it. And thank you very much.