 Hello, my name is Dr. Andrew Missildine and I'm a mathematics professor at Southern Utah University And this is my talk entitled counting show rings over cyclic groups To give some motivation on why we want to count show rings in the first place. Let me actually compare this to Comparable problem from common atorix the famous Bell numbers are actually a sequence of numbers which counts the number of ways one can partition a set of In many elements these so-called set partitions the first couple terms in the sequence are given right here These Bell name numbers are named after Eric Bell whose picture you can see on the screen right there I'd give you a visualization of what this means is consider consider a set of size of Five elements right there now it turns out there's gonna be 52 ways one can partition that well one option as you put everything together You could have everything isolated so when you see no color on the dot That means it's just a singleton and you get things like this where you put three things together leave two apart But you could decide on a different two elements which would be isolated from the rest and so this case There's a lot of possibilities here 52 for a five element set Now related to the Bell numbers are the so-called partition numbers which the partition numbers count the number of integer partitions of A non-negative integer and and so by an integer partition Well, what I mean is something like the following if we wanted to partition the number six We could partition as two plus two plus two or we could write as three plus three or we could write as three plus one Plus one plus one in other words We're looking for any way any any all the possible sums of the number express the sum of positive integers Now there's gonna be a lot fewer integer partitions and there are set partitions for the same size But the two things are related to each other as you can see We'll get to that in just a second here But when you look at say the part the integer partitions of five there are seven possibilities There's just the number five itself four plus one three plus two three plus one plus one You get the idea here and oftentimes these can be illustrated as young diagrams where the first row will be the first number The second row the number squares there will be the second number involved here and these are always written in descending order now the two things are connected to each other because the The set partitions have a very natural equivalence relationship on them We could actually say this is an isomorphism for set partitions because when one talks about a set isomorphism That is just a bijection the only thing that's left in variance by all possible Bijections is really the cardinality of the set and so if we look at set partitions and only focus on their cardinality's we can associate to each Partition one of these young diagrams so for example, we get Partitions over here where you put two together two together and one together and so you get this right here So all of these different partitions as set partitions are isomorphic and can be identified with a single integer partition I should also mention so like the integer partitions the integer numbers count the number of isomorphism classes There are of this set partitions which are counted by the Bell numbers The the number of classes is counted by the partition numbers And then if you are curious the number of elements in each class those are counted by the sterling numbers of the second kind And so there are some classic combinatorial number sequences used to analyze these questions about partitions So what I want to talk about today is what if we want to change this question to focus on finite groups? What can we do to start counting partitions of groups? Well, the first thing to kind of nail down here is that if we want to start enumerating partitions of groups What does it mean to have a partition of a group? We need a partition because the group itself is a set but with a with a binary operation Our partitions should have some respect of the binary operation and play here. Otherwise, it's just set partition Now there actually is a notion of a group Partition which in the category of groups. It's the partition object and that involves Covering a group using subgroups of that group and so that they intersect in trivial ways So there is a notion there and it seems like a strong candidate of what we want to be talking about right here But this idea of group partitions actually a very rare Phenomenon like you can do a fruit for previous groups But and others I think as well But it doesn't happen that often like for example a cyclic group could not be covered using proper subgroups. It's not possible At least not know. Yeah, you can't do with proper subgroups. And so that type of Partition where the cells themselves are subgroups. We got to do something else than that now some Partitions that do kind of respect the group structure that might be of interest counting here is take the the equivalency of Conjugates right two elements are in the same class if they're conscious of each other This is a very important partition of the group another equivalence relationship could just be automorphisms in general Sort of a stronger Relation than the conjugacy one. We could pair things together based upon their inverses Involutions being by themselves we could partition a group using cosets or double cosets if it's a non-normal group We could talk about membership of a subgroup like we could partition the group by you belong to the cell if you're in the subgroup And you part you belong to this other cell if you're not part of the subgroup And so these just give us a few examples of some partitions These are gonna set partitions that in some in some way or another the set partition is Measuring some part of the algebraic structure of the group Now all of these that you see on the screen right now are examples of a special type of partition called a sure partition or More likely called a sure ring These are named after you sigh sure Who first kind of came up with the idea and they're called sure rings as opposed to sure partitions because the written Because these objects are genuine rings sub rings inside of the group algebra Associate to the group and let me kind of explain how that is if we have some finite group g We can identify subsets of the group with elements of the of the group algebra qg and these are called simple quantities Where you basically just add together all the elements inside the subset via the formal sum of the group ring And that becomes at this simple quantity in the group Another bit of notation here if we take a subset or an element of the group ring It doesn't really matter what we do if we raise it to this negative one power inside the parentheses right here What this means is we're gonna be just taking the inverse of each element Inside that set there And so then with this bit of notation in this convention in hand We can define the notion of a sure ring so sure ring you start off with a partition of the group and This partition will make a sub module of the group ring We just take the subspace spanned by all of these simple quantities But then to make it a sure ring there's three extra conditions require First that one of the classes in the partition must be the group identity element all by itself the second Condition is that given any class in the partition It's inverse class must also belong to the partition and then finally if you take the product of any two classes in our partition the product will be a linear combination of Other of other classes in the partition so when you look at these things three things together Right, we're looking for a partition which has an identity which has inverses and which is closed under multiplication The the multiplication in play here will always be associative so we're looking for partitions that mimic the group axioms themselves and so these partitions are very group like objects and this is what we call a sure ring and So to give you an example a few examples of sure rings here One simple example is what we call the discrete sure ring and this is just you take the entire group and you partition into singletons So the group itself is a sure ring and so if we take the sickly group of order six We can partition it by just taking Everything by itself you see different colors for the different cells On the other extreme we can take what we call the indiscreet sure ring for which the Identities by itself because it has to be and then everything else is fused together Again looking at the sickly group of order six you have the identity by itself and then everyone else is in one very large class We can build partitions using automorphisms If we have an automorphism a subgroup of the automorphism group it acts on the group via That automorphic action and then the orbits associated to that group action form what we call an automorphic sure ring so for example if you take the sickly group of order seven and you take powers of two which is an automorphism you get the following three celled Automorphic sure ring as another example We can actually form sure rings using the direct product structure of the group if we have a sure ring Over we have a direct factor So she factors as h times k here if you have a sure ring over h and a sure ring over k Then you can take the tensor product of those two things and that'll form a sure ring over The direct product we call this the direct product sure ring So if you take z6 again take just take the subgroups z2 and z3 And so if we take a sure ring over z2 and as sure ring what you can see right here and a sure ring over Z3 which you can see right here take the all the possible products you get this direct product sure ring and Then as a final example here let's consider the Well, we can actually build sure rings using quotients of some kind normal subgroups So if we have a sub normal subgroup of some kind h inside of G Then take a sure ring over h and take a sure ring over g mod h and then there's a it's possible that We can kind of glue these two sure rings together to make a sure ring over The the group G and it's going to involve using cosets of the quotient and so If you were to identify the five or five points set before if we identify those points as the Five complex roots of unity then those of those 52 partitions only three of them will actually correspond to sure ring partitions We have the indiscreet one right here We have the discreet one right here and then lastly we have the This one right here. This is an automorphic one where you identify inverses together so these these four types of sure rings that mentioned so far Constituted what we call the traditional sure rings and it was shown in the 90s by London mod that all Sure rings over sickly groups are traditional. They're one of these four types And so if one wanted to try to count All the possible sure rings over a sickly group It would be very very difficult to try to go through all the partitions and decide this one's a sure ring This one's not because there's just think that that's the number of sure and the number of partitions just grows too rapidly So we need a different approach of deciding which finding which partitions form sure rings over the group And the two big things that one has to deal with here is that the direct the direct product and the wedge product constructions is a recursive Recursive construction based upon proper subgroups So you have to grapple that recursion and then also we have to be able to list the Indicomposable sure rings over a group that is those sure rings which cannot be written as a direct product or a wedge product of any kind And in this talk, I really want to focus on these Indicomposable problem because the recursion here can get a little bit crazy and we're limited on time And so I wanted to present just some examples here How does one decide how does one find the Indicomposable sure rings for a group? Well, the in the indiscreet sure ring always offers one and all the other ones are going to be Automorphic ones and there's gonna be a one-to-one correspondence between the automorphic sure rings and subgroups of the automorphism group for a cyclic group and so one has to study the automorphic subgroups of Set group. So if you take for example z5, its its automorphic group is Is a cyclic group order z4 and you're gonna see that the subgroups of z4 are gonna look like the following We'll just buy divisors you get one two and four when you go up to The automorphism group for five squared that's gonna give you z20 and you're gonna get all the subgroups You had before but you're also gonna gain Some new subgroups z5 z10 z20 and these are just gonna end up stacking on top of each other if you look for the subgroups of z10, which is the automorphism sorry z100, which is the automorphism group of z125 You're just gonna get these layers that's to start stacking on top of each other Isomorphic to each other as lattices and you see this happening over and over again If we looked at for example z7 its automorphism group is gonna look like z sick It's the cyclic group of this p-1 If you take this layer right here and you copy on top of it You get you get the automorphism lattice for z or for seven squared and for seven cubed same thing for 13 The base layer here is gonna look like the divisor lattice for p-1 And then we stack on top of it lattice isomorphic copies of that and so very recursively we can see very clearly what the automorphism groups for Ciclic groups are gonna be when it's a power of an odd prime And so if we take omega n to be the number of sureings over the cyclic groups the n We can then well for an odd prime Let x be the number of divisors of p-n that that's sort of this base case when it came to that The lattice at the bottom here We can then Using the recursion that comes from the wedge products and the recursion from that Automorphic lattice we saw on the screen a little bit ago one can actually pull apart our recursive formula for omega pn It'll depend on previous Omega p in minus ones this number x shows up a lot here again the number of divisors of p-1 And then also the the recursive relation brings out the catalog numbers There's sort of an interesting argument that I can't really go into right now The catalog number is a very famous number very famous number sequence in common torques If we were then to expound that recursive formula, we see the first 10 Iterations of that formula would be these 10 polynomials and some interesting things we can say about these polynomials They're always monic the leading coefficients one The last coefficient the constant terms always as follows the Fibonacci sequence kind of which is kind of fun and just some other things to mention that this actually shows that the number of sureings over a a Ciclic group of order pdn actually is a pork function And so this is kind of resemblance of Higman's pork conjecture that when one starts counting the Counting groups not isomorphic groups of order p to the n you have these polynomial over residue class functions turns out that sureings over cyclic P groups are also pork Which is kind of a nice connection there and if you plug in some specific numbers You can see here's a table of number of sureings for various powers of primes If we were to look like a power of two so two this has to be treated different differently as it's its even prime When you look at the growth of the automorphic lattice You see that when you get to eight it starts to stabilize that you get this little break this little v-shape That appears on each time it is because this group itself is not cyclic It's not a cyclic group like the previous ones were so it's a little bit more complicated But you can see that as the power of two increases you can you can predict how this Automorphism lattice is going to grow as well and as such you can use that to come up with the following formula This one's a whole lot more complicated as you can kind of see here as compared to the previous one Some things to notice is it is recursive right omega two to the n does depend on previous Omega values for powers of two we're looking at the proper subgroups of this group here It also relies on the catalan numbers But also it also the the Schroder numbers come out here another important number sequence That's actually closely related to the cat on long numbers the catalog numbers come back again This is mostly coming from the recursion from the wedge products involved for for p to the end There are no direct products possible But because of the non-cyclic nature of the this this lattice of automorphisms There is these Schroder numbers that come into play so these things can get complicated pretty quickly And again if you look at specific powers of two you get the following sequence of numbers So what if we want to look at counting the number of shurings when the order has multiple prime divisors Well a natural candidate, you know a sort of a simple example be look at z15 here If we try to look at the well for z15 you are gonna have direct products You're gonna have wedge products, but we want to figure out those indie composable ones How many automorphic ones can we have well the automorphism the automorphic group? The automorphism group for z15 is going to be z2 cross z4 So let's say the generator of z2 is a and the generator of z4 is b like this Then the lattice is gonna look something like the following you can look at the subgroup of z4 You'll have a the trivial one z2 z4 if you look at the group associated to z2 You just get the trivial subgroup and z2 itself and when you take the direct product of the lattice you get the following Parallelogram like shape this gives us six distinct subgroups Which will give us six distinct automorphic shurings, but the group has two extra subgroups These so-called diagonal subgroups they get their name Because of where they kind of appear inside of this lattice right here Now these diagonal subgroups come about from the following idea A is an element of order 2 and b squared is an element of order 2 if we Multiply these things together we'll get an element of order 2 and same thing when we put together a and b B has order 4 a has order 2 when you multiply them together you get a group of order 4 right there And so by both an abelian group if you multiply together elements of similar orders You can create new elements that are outside of this lattice And so this kind of becomes a tricky thing predicting when you take a direct product of abelian groups How many subgroups will you have this actually a very sophisticated problem and the the complete answer is actually not known yet? Turns out we only know a little bit about this formula of counting subgroups of general abelian groups Now if you use that you can apply that and get the following formula right here This this equate this part of the equation right here if we're counting if we're counting the number of shurings Whose order is a semi prime that is a product of two distinct primes? Then we get this right here. This is going to count the automorphisms Which comes down to counting all of the subgroups of the automorphism subgroup. We also have to count the wedge products right here and Then lastly we want to count the the indiscrete or sometimes called the trivial shirring There are direct products in this situation. It turns out every direct product is an automorphic one You don't have to count them twice now in order to get this form which is a little bit complicated here It does use Euler's tosian function and it has to do with how does the prime decompose? So if you write your prime with a it's sort of quote-unquote prime factorization if you subtract one from a prime It's no longer prime right with the prime factorization of P minus one that affects these this formula here And if you look at the this table this gives you just data for all these possible shurings right here There's we don't need to look at all of this here One last example. I want to I want to mention here is that? What have we looked at the lattice associated to for P four times a prime here? This is kind of a nice Predictable lattice here because if we take the lattice of Just Zp you're gonna get the trivial subgroup at the bottom plus something You know what is this some blob going on right there? I don't know exactly what it looked like But if we want to take the the automorphism group the lattice associated to the automorphism group of z4p Well, you're gonna get the lattice for z2 which comes from the z4 part You're gonna get the lattice associated to Zp, which is again this blob and when you take the direct product You're gonna get this this isomorphic lattice picture on top like there's this bottom layer And then there's this top layer that as lattice is going to look identical to each other But then you also have these diagonal subgroups that sit in the middle here And so if you want to count the shurings of the cyclic group of water for P There's gonna be some you have to watch out for these diagonals here. It turns out. It's not so it's not so Wicked in the situation of 4p and so as another as another kind of result here if we decompose our prime as 2k a plus one So we're trying to figure out how many multiples of two divide into two p minus one That's the factor of two is really gonna measure how many diagonal subgroups we're gonna have and so we put this together We got a cute little formula Compared to most of the formulas we've seen in this talk here. This one definitely is the cutest one here And I should should mention that These results you it was mentioned on the slides here, but these results the first couple about just the single pattern single primes Right p to the end 2 to the end. These were results proven by myself In the in a paper published earlier this year 2020 These last couple results about the the city prime p times q and this 4p case you can see on the screen here This was joint work of myself with also two undergraduate students at Southern Utah University Joseph Keller and Max Sullivan For which at the time of this recording these results will be published forthcoming probably 2021 assuming people will referee during the corona virus pandemic But we can make the projection that that preprint will be published sometime next year And so again, here's some numbers based upon that If you have if you apply that formula to the various four times p cases there And so finally just if we kind of just have some last last minute questions right here The techniques we've developed in this again, we were very very quick and we didn't talk about a lot of the recursion but In order to in order to enumerate these sure ring partitions It really could we have to be able to understand these We have to be able to predict how many in decomposals sure rings There's going to be and this has a lot to do with the number of subgroups of the automorphism subgroup of an abelian group And as I mentioned earlier, this is still a very open question We're using the so-called rank 2 problem associated to counting subgroups in abelian group Where it's just it's just a product of two Ciclit groups there P we can assume they're P groups and there there's work that's done for the rank 3 I think the rank 3 problem might have been solved I have to double-check myself before I could say that definitely the rank 4 Problem for counting subgroups in abelian group. I believe it's still quite open And so in terms of counting six assurance over Ciclit groups One's not going to be able to get any farther than the the the rank the subgroup counting problem Because that's going to be a subset of this larger problem we're in right now And so at the sort at the moment at the time of this recording The idea would be to be trying to carry this up to finish the rank 2 problem for these sure rings over Ciclit groups You know arbitrary order of P to the n times Q to the n Could we get a formula for all of those and I think there's really good hope of getting such a thing But it's it's still a little far off And then you can also ask well, what if you start getting more right just because the general Rank in problem hasn't been solved for counting subgroups and abelian groups There are special cases we could do like what if there's no repeated primes What if there's like PQR RP? You know that aren't just every of our many different primes Could we approach something like that as well? And of course, it's also natural Can we start looking at other types of groups other than cyclic groups? The problem there is that the cyclic groups are the only groups that have really a complete classification of all the sure rings over them even for small groups There are tons of partitions possible and therefore there's Potentially a lot of sure rings that could be done and we still have very kind of little knowledge on What type of sureings can appear for even even abelian groups? It's still a very very unknown thing which has some important applications One very important application would have to do is super character theory Theories because a super character theory for example, this is when you start gluing together characters of a group and you form character Theoretic like objects that we call these the super character theories and it was shown by Hendrickson that Super character theories correspond to central sure rings of the group and so be able to classify Sure rings is essentially the same problem is trying to classify super character theories And there are other applications of sure rings with algebraic commentatorics that I won't go into in this talk right now So that that finishes this talk. I appreciate everyone for watching it here I should mention that a lot of the images in this talk if they weren't made by myself They were borrowed from Wikipedia so like some of the some of the pictures of Bell and sure obviously I didn't draw those ones Though those are courtesy of Wikipedia. If you have any questions on these topics feel free to Comment in the comments below right if you like what you saw give it a give it a like click Or again if you want to have a more in-depth conversation about sure rings and topics related to that Feel free to send me an email Or again, just kind of reach out to me through the comments on this on this YouTube video. Alright, thanks everyone. Have a great day