# Ural Seminar on Group Theory and Combinatorics

### Yekaterinburg-Online, Russia

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## Reminding!

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 non-commercial and political-free 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)

Vladislav Kabanov (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS)

Anatoly Kondrat'ev (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS)

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)

Mikhail (Misha) Volkov (Ural Federal University, 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)

Alla Dobroserdova (Ural Federal University)

Nikolai Minigulov (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS)

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 20-minutes 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.

## Records of the talks

Records of all the talks are available both on this website after log in and on the website of the 2020 Ural Workshop on Group Theory and Combinatorics for registered participants after log in.

## November 22, 2022

Time: 4 p.m. by Yekaterinburg

Speaker: Ilya Gorshkov (Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia)

Topic: The structure of finite groups with restrictions on the set of conjugacy classes

Abstract can be found here.

The slides can be found  here.

## November 8, 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: Classification strongly regular graphs with \lambda=0 and \mu=2

This is joint work with Mikhail Golubyatnikov

Abtsract. There is a potentially infinite series of intersection arrays of distance-regular graphs \Gamma=\Gamma(a,\mu) of diameter 5: (b + \mu, b + \mu -1, b, \mu,1; 1, \mu, b, b+\mu-1, b+\mu), where b=b_2(\Gamma) = (a^2-1)\mu/4. Antipodal quotient of \Gamma is a strongly regular graph with parameters ((a^2\mu + 2a - \mu + 6)(a^2\mu - 2a - \mu + 6)\mu)/16, (a^2 - 1)\mu^2/4 + \mu, 0, \mu). Thus, only in the case \mu \in \{2, 4, 6\} we have an infinite series of intersection arrays of \Gamma. In this talk, we prove that a distance-regular graph with \mu=2 and intersection array (a^2+1, a^2, a^2-1, 2, 1; 1, 2, a^2-1, a^2, a^2+1) exists only if a \in \{2, 3\}. In particular, a strongly regular graph with parameters ((a^4+3a^2+4)/2, a^2+1, 0, 2) does not exist for a>3.

## October 25, 2022

Time: 4 p.m. by Yekaterinburg

Speaker: Mercede Maj (Dipartimento di Matematica Universit`a di Salerno, Fisciano (Salerno), Italy)

Topic: Detecting properties of a finite group through the study of somefunctions on element orders

Abstract can be found here.

The slides can be found  here.

## October 11, 2022

Time: 4 p.m. by Yekaterinburg

Speaker: Michael Khachay (N.N. Krasovskii Institute of Mathematics and Mechanics UB RAS and Ural Federal University, Yekaterinburg, Russia)

Topic: Towards a polynomial time approximation of a symmetric routing problems within fixed ratios

Abstract. In this talk, we will discuss the first fixed-ratio polynomial time approximation algorithms are proposed for a series of asymmetric routing combinatorial optimization problems including the Steiner Cycle Problem (SCP), Rural Postman Problem (RPP), Generalized Traveling Salesman Problem (GTSP), Capacitated Vehicle Routing Problem with Unsplittable Customer Demands (CVRP-UCD), and Prize Collecting Traveling Salesman Problem (PCTSP).
The presented results are shared the common property, all of them rely on polynomial-timel cost-preserving reduction to appropriate settings of the Asymmetric Traveling Salesman Problem (ATSP) and the seminal (22+ε)-approximation algorithm for this classical problem proposed by O. Svensson and V. Traub in 2019.

## June 7, 2022

Time: 4 p.m. by Yekaterinburg

Speaker: Gang Chen (School of Mathematics and Statistics, Central China Normal University, Wuhan, China)

Topic: Schur rings over some classes of groups

Abstract. An Schur ring A over a group G is called a traditional Schur ring if it is either a discrete Schur ring, a trivial Schur ring, a Schur ring of tensor product type, an orbit Schur ring, or a Schur ring of wedge product type. A group is called traditional if any Schur ring over it is traditional. Let $Z$ denote the infinite cyclic group of the set of integers with respect to addition of numbers. In this talk, it is proved that $Z \times Z_p$ is traditional, where $p$ is a prime with $p \le 13$; neither $Z \times Z$ nor $Z \times Z_2 \times Z_2$ is traditional. Finallly, Schur rings over alternating group on five letters which contain the set of 3-cycles as an S-set are classfied.

Contributed talks:

5 p.m. Ivan Timofeenko (Siberian Federal University, Krasnoyarsk, Russia), Generation of the Chevalley groups $E_l$ over the ring of integers by three involutions two of which commute

5:25 p.m. Yinfeng Zhu (Shanghai Jiao Tong University, Shanghai, China; Imperial College London, London, UK), Automata with an almost constant rank property on words

## 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 2-rank 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 Gruenberg-Kegel 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 Protasov-Vonyov 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 Baumslag-Solitar 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 (GBS-group). These groups turned out to be of interest because of their geometric, algorithmic, and group-theoretic 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, K-residuality, 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 3/2-transitive 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

Speaker: Cheryl E. Praeger (The University of Western Australia, Perth, Australia)

Topic: Graphs with a symmetrical Euler cycle

Joint work with: Jiyong Chen, Cai Heng Li, and Shu-Jiao Song

Abstract. This work was motivated by our wish to understand arc-transitive 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 vertex-edge 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 bi-regular on edges. Such a cycle is called a symmetrical Euler cycle of its edge-induced 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 arc-transitive maps? Answering the first question involved developing a group theoretic model for edge-transitive 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 odd-index 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 distance-regular 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 distance-regular graph with the second large eigenvalue $\theta_2=-1$, then the complement to $\Gamma_3$ is a pseudo-geometric graph for $pG_{c_3}(k,b_1/c_2)$ (Makhnev-Nirova).

We investigate distance-regular graphs $\Gamma$ for which the complement to $\Gamma_3$ are pseudo-geometric for:

1) a net $pG_{t}(s,t)$ (Makhnev-Guo-Golubyatnikov);
2) a dual $2$-disign $pG_{t+1}(s,t)$ (Belousov-Makhnev);
3) a generalized quadrangle (Makhnev-Nirova).

Moreover, we investigate distance-regular graphs $\Gamma$ with $\Gamma_3$ strongly regular without triangles (Belousov-Makhnev-Paduchikh).

## 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 Godsil-McKay 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 4-vertex 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 2-ranks, Des. Codes Cryptogr., 81 (2016) 35-41.
[2] F. Ihringer, A switching for all strongly regular collinearity graphs from polar spaces, J. Algebr. Comb., 46 (2017) 263-274.
[3] A. E. Brouwer, F. Ihringer, W. M. Kantor, Strongly regular graphs satisfying the 4-vertex condition,  arXiv:2107.00076v1 [math.CO]
[4] W. H. Haemers, H. Kharaghani, M. Meulenberg, Divisible design graphs, J. Combinatorial Theory A, 118 (2011) 978-992.

## January 25, 2022

Time: 4 p.m. by Yekaterinburg

Speaker: Rosemary A. Bailey (University of St Andrews, UK)

Topic: Diagonal structures and beyond

Abstract.  Diagonal structures have been used in group theory since the mid-twentieth 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 semi-lattices 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.

The slides:  rabDiag.pdf

## 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 p-solvable 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 vertex-transitive 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 non-isomorphic) 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 Gruenberg-Kegel 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 Gruenberg-Kegel graphs.

Abstract. The Gruenberg-Kegel 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 Gruenberg-Kegel 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 prime-order 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 non-empty 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 set-theoretic solutions of the Yang-Baxter 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

Speaker: Alexandre Zalesskii (University of East Anglia, Norwich, UK)

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

Speaker: Binzhou Xia (The University of Melbourne)

Topic: Constructing tetravalent half-arc-transitive graphs

Abstract. Half-arc-transitive 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 half-arc-transitive graphs with prescribed vertex stabilizers. In this talk, I'll focus on the tetravalent case, giving new constructions of half-arc-transitive graphs with various vertex stabilizers. This sheds light on the larger problem of which groups can be the vertex stabilizer of a tetravalent half-arc-transitive 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 non-degenerate 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 (small-dimensional) non-degenerate subspaces U, U' of a
symplectic/unitary/orthogonal space V, what is the probability that
the subspace U + U' is non-degenerate 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 Milano-Bicocca, 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 3-closure 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 m-closure 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 1-closure and 2-closure of asolvable permutation group need not be solvable. We prove that the m-closureof a solvable permutation group is always solvable for m>2.

## February 16, 2021

Time: 4 p.m. by Yekaterinburg

Speaker: Akihiro Munemasa (Tohoku University, Sendai, Japan)

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 Meng-Yue 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 easily-understood 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 number-theoretic problem, namely whether or not there exist infinitely many primes equal to the natural degree $(q^n-1)/(q-1)$ 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

Speaker: Cheryl E. Praeger (The University of Western Australia, Perth, Australia)

Topic: Finite edge-transitive Cayley graphs, quotient graphs and Frattini groups

Joint work with Behnam Khosravi, Institute of Advanced Studies in Basic Sciences, Zanjan, Iran

Abstract. The edge-transitivity of a Cayley graph is most easily recognisable if the subgroup of “affine maps” preserving the graph structure is itself edge-transitive. These are the so-called normal edge-transitive Cayley graphs.  Each of them determines a set of quotients which are themselves normal edge-transitive 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

Speaker: Rosemary A. Bailey (University of St Andrews, UK)

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 Gruenberg--Kegel 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.