# Ural Seminar on Group Theory and Combinatorics

### Yekaterinburg-Online, Russia

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from 7 Dec 2020 till 30 Dec 2021
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In Yekaterinburg 19:40
Sunday, 13 June 2021
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## !!!

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

## 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)

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)

## 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!

## 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: R. 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 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.