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With the aim of safiety of all the participants we kindly ask to avoid any political discussions during our meetings.

Aims and Scope
This seminar continues 2020 Ural Workshop on Group Theory and Combinatorics. The seminar covers 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 and aims to support communications between specialists on Group Theory and Combinatorics all over the World.
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.

Declarations
The situation in the World in the recent times is very complicate. However, we are going to keep the working of the Ural Seminar on Group Theory and Combinatorics as noncommercial and politicalfree meetings where people are able to share Math do not keeping in mind their nacionality, political positions, religion, gender, colour, and so on.
The official position of the Russian Academy of Sciences can be found in their official website. From our side, we declare that the participation in the Ural Seminar on Group Theory and Combinatorics cannot be considered directly as a declaration of any political or social position, except commitment to regular Humanitarian values and involving to Mathematical research. And with the aim of safiety of all the participants we kindly ask to avoid any political discussions during our meetings.
The only aim of the Ural Seminar on Group Theory and Combinatorics is to support communications between specialists on Group Theory and Combinatorics all over the world. We started in the tricky pandemic time and are going to continue in the same frame. Any specialist can register for free on the seminar website and attend the lectures online or see the records of the lectures. All the activities of speakers are for free and are aimed at develop of Mathematical research only.

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)
Mikhail Golubiatnikov (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!

Contributed talks
The option of 20minutes contributed talks is available for participants from all over the World. We can shedulle some such talks after the talk of the main speaker. If somebody wants to give a contributed talk, please contct Natalia Maslova via butterson[at]mail.ru.


May 24, 2022
Time: 4 p.m. by Yekaterinburg
A series of five contributed talks. The speakers are presented in the alphabet ordering.
4 p.m. Boris Durakov (Siberian Federal University, Krasnoyarsk, Russia) On periodic groups of 2rank one
4:25 p.m. Andrei Kukharev (Siberian Federal University, Krasnoyarsk, Russia) Simple groups with Brauer trees of principal blocks in the shape of a star
4:50 p.m. Nikolai Minigulov (Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russia) On finite groups which GruenbergKegel graphs are isomorphic to the paw
5:15 p.m. Dmitry Panasenko (Chelyabinsk State University, Chelyabinsk, Russia) Vertex connectivity of some classes of divisible design graphs
5:40 p.m. Ruslan Skuratovskii (National Aviational University of Ukraine and Igor Sikorsky Kiev Polytechnic Institution, Kiev, Ukraine) Verbal width by set of squares in alternating group A_n and Mathieu groups, squareness criterions in the group in PSL_2(F_p) and SL_2(F_p)

April 26, 2022
Time: 4 p.m. by Yekaterinburg
Speaker: Yaokun Wu (Shanghai Jiao Tong University, Shanghai, China)
Topic: Hurwitz primitivity and \v{C}ern\'{y} function
Abstract. For each positive integer $m$, we use $[m]$ for the set of first $m$ positive integers. Let $\mathcal{A} = (A_1, \ldots, A_m)$ be an $m$tuple of nonnegative $n\times n$ matrices. For each word $\alpha$ over $[m]$, say $\alpha=\alpha_1\cdots \alpha_s$, we write $\mathcal{A}_{\alpha}$ for the product $A_{\alpha_1}\cdots A_{\alpha_s} $. We call $\mathcal{A}$ primitive if $\mathcal{A}_{\alpha} > 0$ for a nonempty word $\alpha$ over $[m]$. We call $\mathcal{A}$ Hurwitz primitive provided there exists a nonnegative integer vector $\tau=(\tau(1),\ldots,\tau(m))$ such that for each $x,y\in [n]$ there exists a nonempty word $\alpha^{x,y}$ over $[m]$ such that $\mathcal{A}_{\alpha^{x,y}}(x,y)>0$ and the number of occurrence of $i$ in $\alpha^{x,y}$ is $\tau(i)$ for each $i \in [m]$. The $m$tuple $\tau$ satisfying the above property is named a Hurwitz primitive vector of $\mathcal{A}$.
Let $\mathsf{NZ}_1$ denote the set of nonnegative matrices without zero rows and let $\mathsf{NZ}_2$ denote the set of nonnegative matrices without zero rows/columns. We give a unified combinatorial proof for the ProtasovVonyov characterization (2012) of primitive $\mathsf{NZ}_2$matrix tuples and the Protasov characterization (2013) of Hurwitz primitive $\mathsf{NZ}_1$matrix tuples. By establishing a connection with synchronizing automata, for any Hurwitz primitive $m$tuple $\mathcal{A}$ of $n\times n$ $\mathsf{NZ}_1$matrices we give an $O(n^3m^2)$time algorithm to find a Hurwitz primitive vector $\tau$ of $\mathcal{A}$ such that $\sum_{i\in [m]}\tau (i) = O(n^3)$. For any given $m$tuple of $n\times n$ $\mathsf{NZ}_2$matrices, we present an $O(n^2m)$time algorithm to test whether or not it is primitive. We also report results on ergodic and Hurwitz ergodic matrix tuples.
This talk is based on a joint paper with Yinfeng Zhu, the current version of which can be found here.
Contributed talks:
5 p.m. Vikotor Panshin (Sobolev Institute of Mathematics SB RAS and Novosibirsk State University, Novosibirsk, Russia) On recognition of $A_6 \times A_6$ by the set of conjugacy class sizes
5:25 p.m. Mikhail Golubiatnikov (Krasovskii Institute of Mathenatics and Mechanics UB RAS ans Ural Federal University, Yekaterinburg, Russia) Deza graphs related to intersections of conjugate subgroups in groups SL2(q)s

April 12, 2021
Time: 4 p.m. by Yekaterinburg
Speaker: Fedor Dudkin (Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia)
Topic: Recent results on generalized BaumslagSolitar groups
Abstract. A finitely generated group G acting on a tree with all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (GBSgroup). These groups turned out to be of interest because of their geometric, algorithmic, and grouptheoretic properties. They have been actively studied during the last twenty years. Our goal is to tell about some recent results on GBS groups: outer automorphism group, description of the centralizer dimension, the problem of universal equivalence, Kresiduality, connection with knot groups. Some open problems will be discussed at the end of the talk.
Contributed talks:
5 p.m. Dmitry Churikov (Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia) The minimal size of a generating set for primitive 3/2transitive groups
5:25 p.m. Boris Durakov (Siberian Federal University, Krasnoyarsk, Russia) On infinite groups saturated with finite Frobenius groups of even orders

March 29, 2022
Time: 4 p.m. by Yekaterinburg
Speaker: Grigory Ryabov (Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia)
Topic: On Cayley representations of central Cayley graphs over almost simple groups
Joint work with J. Guo, W. Guo, and A. Vasil'ev Abstract. A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$ associated with the subgroups of $Sym(G)$ induced by left and right multiplications of $G$. We also provide an algorithm which, given a central Cayley graph $\Gamma$ over an almost simple group $G$ whose socle is of a bounded index, finds the full set of pairwise nonequivalent Cayley representations of $\Gamma$ over $G$ in time polynomial in size of $G$

March 15, 2022
Time: 4 p.m. by Yekaterinburg
Topic: Graphs with a symmetrical Euler cycle
Joint work with: Jiyong Chen, Cai Heng Li, and ShuJiao Song
Abstract. This work was motivated by our wish to understand arctransitive embeddings of graphs in surfaces, where the graphs might have more than one edge between adjacent vertices. So the graphs should be undirected, of finite valency with no loops, and admitting a subgroup of automorphisms which is transitive on arcs (incident vertexedge pairs). Apart from some degenerate cases, the boundary of a face in such an embedding is a sequence of pairwise distinct edges which form a cycle in the graph, and the stabiliser of this cycle admits a cyclic group of automorphisms which is either regular or biregular on edges. Such a cycle is called a symmetrical Euler cycle of its edgeinduced subgraph. We were curious: what kinds of graphs (with multiple edges allowed) admit a sequencing of all their edges into a symmetrical Euler cycle? And secondly (but this is a bit beyond the lecture), what kinds of symmetrical Euler cycles arise in arctransitive maps? Answering the first question involved developing a group theoretic model for edgetransitive graphs. It was the basic tool (though not sufficient) we used to classify all graphs with a symmetrical Euler cycle.
Contributed talks:
5 p.m. Alexey S. Vasil'ev (Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russia and Sobolev Institute of Mathematics and Mechanics SB RAS, Novosibirsk, Russia) Relatively maximal oddindex subgroups of symmetric groups
5:25 p.m. Artem Kravchuk (Novosibirsk State University, Novosibirsk, Russia) Spectrum of the Transposition graph
5:50 p.m. Ruslan Skuratovskii (Igor Sikorsky Kiev Polytechnic Institution, Kiev, Ukraine) Normal subgroups of iterated wreath products of symmetric groups and alternating with symmetric groups

February 15, 2022
Time: 4 p.m. by Yekaterinburg
Speaker: Alexander Makhnev (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University, Yekaterinburg, Russia)
Topic: On distanceregular graphs $\Gamma$ of diameter 3 such that $\Gamma_3$ is strongly regular
Abstract. If $\Gamma$ is a simple graph, then define $\Gamma_3$ to be a graph with the same vertex set as $\Gamma$ in which two different vertices are adjacent if and only if they are at distance $3$ in $\Gamma$. If $\Gamma$ be a distanceregular graph with the second large eigenvalue $\theta_2=1$, then the complement to $\Gamma_3$ is a pseudogeometric graph for $pG_{c_3}(k,b_1/c_2)$ (MakhnevNirova). We investigate distanceregular graphs $\Gamma$ for which the complement to $\Gamma_3$ are pseudogeometric for: 1) a net $pG_{t}(s,t)$ (MakhnevGuoGolubyatnikov); 2) a dual $2$disign $pG_{t+1}(s,t)$ (BelousovMakhnev); 3) a generalized quadrangle (MakhnevNirova). Moreover, we investigate distanceregular graphs $\Gamma$ with $\Gamma_3$ strongly regular without triangles (BelousovMakhnevPaduchikh).

February 1, 2022
Time: 4 p.m. by Yekaterinburg
Speaker: Vladislav Kabanov (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University, Yekaterinburg, Russia)
Topic: New construction of strongly regular graphs with parameters of symplectic graphs
Abstract. A. Abiad and W.H. Haemers in [1] used GodsilMcKay switching to obtain strongly regular graphs with the same parameters as Sp(2d,2) for all d at least 3. F. Ihringer in [2] provided a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. Recently, A.E. Brouwer, F. Ihringer and W.M. Kantor in [3] described a switching operation on collinearity graphs of polar spaces to obtain graphs that satisfy the 4vertex condition if the original graph belongs to a symplectic polar space.
In this talk we present new construction of strongly regular graphs with parameters of the complement of symplectic graphs. For our construction, we use new construction of divisible design graphs and do not use any switching.
[1] A. Abiad and W.H. Haemers, Switched symplectic graphs and their 2ranks, Des. Codes Cryptogr., 81 (2016) 3541. [2] F. Ihringer, A switching for all strongly regular collinearity graphs from polar spaces, J. Algebr. Comb., 46 (2017) 263274. [3] A. E. Brouwer, F. Ihringer, W. M. Kantor, Strongly regular graphs satisfying the 4vertex condition, arXiv:2107.00076v1 [math.CO] [4] W. H. Haemers, H. Kharaghani, M. Meulenberg, Divisible design graphs, J. Combinatorial Theory A, 118 (2011) 978992.

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
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
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.
