
Aims and Scope
This seminar continues 2020 Ural Workshop on Group Theory and Combinatorics. The seminar aims to cover modern aspects of group theory (including questions of actions of groups on combinatorial objects), graph theory, some combinatorial aspects of topology and optimization theory, and related topics.
The seminar will be held on Tuesdays, usually one time in 2 weeks, with possible some exceptions. The list of talks can be found below.

Scientific Committee
Chair: Natalia Maslova (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University)
Alexander Makhnev (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University) Danila Revin (Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia and N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russia)

Organizers
Chair: Natalia Maslova (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University)
Ivan Belousov (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University)

Registration
To attend the seminar please register for free via this website. You do not need to register again if you was a participant of 2020 Ural Workshop on Group Theory and Combinatorics, to login to the seminar website you can use we the same login and password as for the workshop website.
In your registration form, you are welcome to give us some information on your mathematical interests.
We kindly ask invited speakers to register via this website to be available for mailings!

January 25, 2022
Time: 4 p.m. by Yekaterinburg
Topic: Diagonal structures and beyond
Abstract. Diagonal structures have been used in group theory since the midtwentieth century. Recent work uses them in various combinatorial contexts, including Latin squares, Hamming graphs, folded cubes, and other graphs. All of these depend on the theory of the partial order on partitions of the same set, so the first part of this talk describes this theory. The second part tells the story of some statisticians who developed part of this theory, not always using the words "partition" or "partial order", and not usually talking to pure mathematicians. The next two parts describe diagonal semilattices and diagonal graphs. The final section generalizes both of these in a way analogous to generalizing a Latin square to a set of mutually orthogonal Latin squares.

December 21, 2021
Date: December 21, 2021 Time: 4 p.m. by Yekaterinburg Speaker: Long Miao (Hohai University, Yangzhou University, Yangzhou, China) Topic: Some new ideas on the class of nonsolvable groups Abstract. Starting from the psolvable groups, some new class of nonsolvable groups are given through the chief factors of Sylow subgroups, commutator subgroups and Frattini subgroups of some nonsolvable groups. And some new information about nonsolvable groups is obtained by characterizing them with second maximal subgroups.

December 7, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Vladimir Trofimov (Krasovskii Institute of Mathematics and Mechanics UB RAS, and Ural Federal University, Yeketerinburg, Russia)
Topic: Symmetrical extensions of graphs
Abstract. A connected graph $\Sigma$ is a symmetrical extention of a graph $\Gamma$ by a graph $\Delta$ if there are a vertextransitive group $G$ of autumorphisms of $\Sigma$ and imprirmitivity system $\Sigma$ of $G$ on the vertex set of $\Sigma$ such that the quotient graph $\Sigma/\sigma$ is isomorphic to $\Gamma$ and blocks of $\sigma$ generate in $\Sigma$subgraphs isomorphic to $\Delta$. Symmetrical extensions of graphs are of interest for group theory, graph theory, topology, but also for crystallography and physics. In the talk the following question is discussed. Let $\Gamma$ be an infinite locally finite graph and $\Delta$ be a finite graph. Are there only finitely many (pairwise nonisomorphic) symmetrical extensions of $\Gamma$ by $\Delta$?Although in gene ral the question is answered in the negative, in some important cases of$\ Gamma$ and $\Delta$ the answer to the question is positive.

November 23, 2021
Date: November 23, 2021 Time: 4 p.m. by Yekaterinburg Speaker: Peter J. Cameron (University of St Andrews, UK) Topic: Generalizing EPPO groups by means of graphs Abstract. EPPO groups are finite groups in which all elements have prime power order. They were introduced by Higman in the 1950s, and the simple EPPO groups found by Suzuki in the 1960s, but the complete classification is more recent. There are two characterizations of EPPO groups in terms of graphs: they are the groups whose GruenbergKegel graph has no edges, and also groups whose power graph and enhanced power graph coincide. Based on these and similar ideas, I propose several problems involving classes of groups widere than EPPO groups, and give a few preliminary results.

November 9, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Anatoly Kondrat'ev (Krasovskii Institute of Mathematics and Mechanics UB RAS, Ural Federal University, and Ural Mathematical Center, Yeketerinburg, Russia)
Topic: On finite groups with given properties of GruenbergKegel graphs.
Abstract. The GruenbergKegel graph (or the prime graph) of a finite group G is a (labelled) graph in which the vertices are the prime divisors of the order of G, and two distinct vertices p and q are adjacent in this graph if and only if G contains an element of order pq. This graph is a fundamental arithmetical invariant of a finite group which have numerous applications. This talk is devoted to some problems and results on the study of finite groups with given properties of their GruenbergKegel graphs.

October 26, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Jinbao Li (Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing, China)
Topic: On weaker quantitative characterization of finite nonabelian simple groups.
Abstract. In the past forty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians, such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacyclasses, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups. In this talk, we will introduce some weaker quantitative characterizations of finite nonabeliansimple groups by their orders together with some special quantitative properties such as the largestelement orders, the largest conjugacy class sizes and the number of primeorder elements.

October 12, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Andrei Mamontov (Sobolev Institute of Mathematics SB RAS and Novosibirsk State University, Novosibirsk, Russia)
Topic: On periodic groups with a given spectrum.
Abstract. The spectrum of a periodic group is the set of its element order. A periodic group is called a group with a dense spectrum, or $OC_n$group, if its spectrum consists of all integers from 1 to some fixed number $n$. In the talk we discuss periodic $OC_n$groups ($n\leq 7$).

September 28, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Valeriy G. Bardakov (Sobolev Institute of Mathematics SB RAS and Novosibirsk State University, Novosibirsk, Russia)
Topic: Quandles: Algebraic theory and applications
Abstract. Quandle is a nonempty set with one binary algebraic operations which satisfies to three axioms. At first they arrive in Knot Theory, but now Quandle Theory is a part of Abstract algebra like Group Theory or Ring Theory. On my talk I give a definition and examples of quandles, explain their connection with groups, give a geometric interpretation of quandles, describe some interesting classes of quandles. We discuss connection of quandles with Knot Theory and with settheoretic solutions of the YangBaxter equation. Further I introduce some properties of quandles: residually finiteness, orderability, and formulate results on quandles which have these properties.

September 14, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Lev Kazarin (P.G. Demidov Yaroslavl State University, Yaroslavl, Russia)
Topic: Conjugacy class sizes and factorizations of finite groups
Abstract. The aim of the talk is to give a short survey concerning recent progress in the study of groups with factorizations and its relation with the structure of groups with an information on the sizes of a conjugacy classes of groups.

June 8, 2021
Time: 4 p.m. by Yekaterinburg
Topic: Recent results on the eigenvalue 1 problem for representations offinite groups of Lie type
Abstract. In this talk I shall discuss some aspects of the problem of determining unisingular iredicible representations of finite simple groups of Lie type over fields of describing characteristic. Some recent results will be exposed and commented. A representation $\phi$ of a group $G$ is called unisingular if 1 is an eigenvalue of $\phi(g)$ for every $g\in G$.

May 25, 2021
Time: 2 p.m. by Yekaterinburg
Speaker: Gareth Jones (University of Southampton, Southampton, UK)
Topic: Paley, Carlitz and the Paley graphs
Abstract. Anyone who seriously studies algebraic graph theory or finite permutation groups will, sooner or later, come across the Paley graphs and their automorphism groups. The most frequently cited sources for these are respectively Paley's 1933 paper for their discovery, and Carlitz's 1960 paper for their automorphism groups. It is remarkable that neither of those papers uses the concepts of graphs, groups or automorphisms. Indeed, one cannot find these three terms, or any synonyms for them, in those papers: Paley's paper is entirely about the construction of what are now called Hadamard matrices, while Carlitz's is entirely about permutations of finite fields.
The aim of this talk is to explain how this strange situation came about, by describing the background to these two papers and how they became associated with the Paley graphs. This involves links with other branches of mathematics, such as matrices, number theory, block designs, coding theory, finite geometry, polytopes and group theory, reaching back to 1509, with important contributions from Coxeter and Todd, Sachs, and Erd\H os and R\'enyi. I will also briefly cover some recent developments concerning surface embeddings of Paley graphs. A preprint is available at https://arxiv.org/abs/1702.00285.

May 11, 2021
Time: 4 p.m. by Yekaterinburg
Topic: Constructing tetravalent halfarctransitive graphs
Abstract. Halfarctransitive graphs are a fascinating topic, which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct halfarctransitive graphs with prescribed vertex stabilizers. In this talk, I'll focus on the tetravalent case, giving new constructions of halfarctransitive graphs with various vertex stabilizers. This sheds light on the larger problem of which groups can be the vertex stabilizer of a tetravalent halfarctransitive graph.

April 27, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Jack Koolen (The University Science and Technology of China, Hefei, China)
Topic: Improving Neumaier's Theorem on strongly regular graphs
Abstract. In this talk I will discuss a Theorem of Neumaier and some recent improvements.
This is based on joint work with Gary Greaves and Jongyook Park.

April 13, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Stephen Glasby (The University of Western Australia, Perth, Australia)
Topic: Recognizing classical groups
Abstract. Dr Who has been captured by the evil Celestial Toymaker. In order to be released, Dr Who must recognize a large (finite) classical group G (known only to the Toymaker) in under 5 minutes. The elements of G are encoded as strings of 0s and 1s, and so are not familiar dxd matrices over GF(q) preserving a certain nondegenerate form. Dr Who is allowed to 1. choose random elements, 2. multiply elements, 3. invert elements, and 4. test the order of elements of G, in order to (constructively and quickly) recognize G.
I shall first explain why the Toymaker's problem is central to computational group theory, and why a quick solution is highly desirable. In so doing, we will briefly review some key ideas for matrix group recognition before reducing the Toymaker's problem to the following geometric problem. Given two (smalldimensional) nondegenerate subspaces U, U' of a symplectic/unitary/orthogonal space V, what is the probability that the subspace U + U' is nondegenerate and of dimension dim(U) + dim(U')? (The sum U + U' is usually not perpendicular.)

March 30, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Alexander Buturlakin (Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia)
Topic: Structure of locally finite groups and some classes of subgroups
Abstract. We study the interplay between the structure of a locally finite group and some properties of its subgroups. Three classes of subgroups are considered: centralizers, cyclic subgroups, and Hall subgroups. We describe the structure of a locally finite group in which the lengths of chains of nested centralizers are finite and uniformly bounded. We finish a description of the spectra (the sets of orders of elements) of all finite simple groups and study the algorithmic aspect of the problem of recognition of finite simple groups by their spectra. Finally, we give a criteria for the existence of a solvable Hall subgroup in a finite group.

March 16, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Pablo Spiga (Department of Mathematics and Applications, University of MilanoBicocca, Italy)
Topic: A generalization of Sims conjecture for finite primitive groups and two point stabilizers
Abstract. In this talk we first discuss the classic Sims' conjecture on finite primitive groups. Then, we propose a refinement of Sims conjecture and we present some modest progress towards the proof of this refinement.
By analysing this refinement, when dealing with primitive groups of diagonal type, we construct a finite primitive group G on X and two distinct points x,y in X with G_x\cap G_y normal in G_x and G_x\cap G_y \ne 1, where G_x and G_y are the stabilizers of x and y in G. In particular, this example gives an answer to a question raised independently by Peter Cameron and by Anatily Fomin.

March 9, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Ilia Ponomarenko (St.Petersburg Department of V.A.Steklov Institute of Mathematics of RAS, St.Petersburg, Russia)
Topic: The 3closure of a solvable permutation group is solvable
Based on joint work with E.A. O'Brien, A.V. Vasil'ev, and E.P. Vdovin
Abstract. Let m be a positive integer and let V be a finite set. The mclosure of G<Sym(V)is the largest permutation group on V having the same orbits as G in itsinduced action on the Cartesian product V^m. The 1closure and 2closure of asolvable permutation group need not be solvable. We prove that the mclosureof a solvable permutation group is always solvable for m>2.

February 16, 2021
Time: 4 p.m. by Yekaterinburg
Topic: Maximality of Seidel matrices and switching roots of graphs
Abstract. In this talk, we discuss maximality of Seidel matrices with a fixed largest eigenvalue and fixed rank. We present a classification of maximal Seidel matrices of largest eigenvalue 3, which gives a classification of maximal equiangular lines in a Euclidean space with angle arccos(1/3). This may sound like a problem which has already been completed in 1970's by Seidel and others. However, maximality of equiangular lines with a fixed rank seems to be considered only recently. The use of a switching root, newly introduced in our work, facilitates the classification and puts the problem in the context of root systems in a canonical manner. Motivated by the maximality of the exceptional root system E_8, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal. Thus, the Seidel matrix of order 276 coming from the McLaughlin graph is strongly maximal. This is based on joint work with MengYue Cao, Jack H. Koolen and Kiyoto Yoshino.

February 2, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Gareth Jones (University of Southampton, Southampton, UK)
Topic: Primitive permutation groups of prime degree
Abstract. The study of transitive permutation groups of prime degree can be traced back two and a half centuries, through Burnside and Galois, to the work of Lagrange on polynomials of prime degree. It is sometimes asserted that the groups of prime degree are now completely known, as a consequence of the classification of finite simple groups: apart from a few interesting but easilyunderstood exceptions, there are infinite families of affine, alternating and symmetric groups, together with various projective groups related to $PSL_n(q)$, all acting naturally in those cases when their natural degree is prime. Although true, this assertion ignores an apparently difficult numbertheoretic problem, namely whether or not there exist infinitely many primes equal to the natural degree $(q^n1)/(q1)$ of $PSL_n(q)$. Such primes are also relevant to alternative versions of Waring's problem. In joint work with Sasha Zvonkin I shall present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one considers only prime values of $q$.

January 19, 2021
Time: 4 p.m. by Yekaterinburg
Topic: Finite edgetransitive Cayley graphs, quotient graphs and Frattini groups
Joint work with Behnam Khosravi, Institute of Advanced Studies in Basic Sciences, Zanjan, Iran
Abstract. The edgetransitivity of a Cayley graph is most easily recognisable if the subgroup of “affine maps” preserving the graph structure is itself edgetransitive. These are the socalled normal edgetransitive Cayley graphs. Each of them determines a set of quotients which are themselves normal edgetransitive Cayley graphs and are built from a very restricted family of groups (direct products of simple groups). We address the questions: how much information about the original Cayley graph can we retrieve from this set of quotients? And can we ever reconstruct the original Cayley graph from them: if so, then how? Our answers to these questions involve a type of “relative Frattini subgroup” determined by the Cayley graph, which has similar properties to the Frattini subgroup of a finite group – I’ll discuss this and give some examples. It raises many new questions about Cayley graphs.

December 22, 2020
Time: 2 p.m. by Yekaterinburg
Topic: Some applications of finite group theory in the design of experiments
Abstract. Group theory is used in (at least) two different ways in the design of experiments.
The first is in randomization, the process by which an initial design is turned into the actual layout for the experiment by applying a permutation of the experimental units, chosen at random from a certain group of permutations. Which group? What properties should it have?
The second is in design construction. The set of treatments is identified with a finite Abelian group, and the blocks are all translates of one or more initial blocks. The characters of this group form its dual group: they are the eigenvectors of the matrix that we need to consider to see how good the proposed design is.

December 8, 2020
Date: December 8, 2020 Time: 2 p.m. by Yekaterinburg Speaker: Peter J. Cameron (University of St Andrews, UK) Topic: Graphs on groups: old and new connections Abstract. Several graphs defined on the vertex set of a group have been studied. Theseinclude the commuting graph, introduced by Brauer and Fowler in 1955, the powergraph (Kelarev and Quinn 1999) and the enhanced power graph (Aalipour et al.2017). It turns out that there are connections with other topics in grouptheory, including the GruenbergKegel graph and Schur covers, as well asapplications in computational group theory. I will discuss some of thesethings, including the most recent, a graph which lies between the enhancedpower graph and the commuting graph.
