### Donald S. Passman (University of Wisconsin-Madison, USA)

**Title: **Polynomial identities, permutational groups, and rewritable groups

**Abstract:** We first study groups whose group algebras satisfy a polynomial identity. We then consider permutational groups and rewritable groups. We discuss the known characterizations of such groups and the relationships between these three group-theoretic properties and also between the proofs of their corresponding main theorems. Finally, we discuss certain parameters associated with these conditions and we mention a number of examples of interest.

Downloads: pdf

### Dan Segal (University of Oxford, UK)

**Title:** Groups, rings, logic

**Abstract:** In group theory, interesting statements about a group usually cant be expressed in the language of
rst-order logic. It turns out, however, that some groups can actually be determined by their
rst-order properties, or, even more strongly, by a single
rst-order sentence. In the latter case the group is said to be
nitely axiomatizable. I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of p-adic analytic pro-p groups, within the class of all pro
nite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain R; the group G(R) is bi-interpretable with the ring R: This means in particular that
rst-order properties of the group G(R) correspond to
rst-order properties of the ring R. As many rings are known to be
nitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every
nitely generated group of the form G(R).

Download: pdf

### Andrea Lucchini (Universita di Padova, Italy)

**Title:** The non-F graph of a finite group (and the generating and independent graphs)

**Abstract:** Recent results and open questions will be presented, related with some graphs encoding generating properties of finite and profinite groups. In particular, given a family F of groups, we will consider the graph whose vertices are the elements of a group G and where two vertices x, y in G are adjacent if and only if the subgroup generated by them does not belong to F.

Download: pdf

### Andrey V. Vasilev (Sobolev Institute of Mathematics and Novosibirsk State University, Russia)

**Title:** Finite groups and orders of their elements

**Abstract: **The elementary assertion that groups of exponent 2 are abelian and one of the most complicated theorems in the finite group theory on solvability of groups of odd order are similar in the following sense. In both cases, arithmetic, i.e., expressed by numeric parameters, properties of a group allow to conclude on its structure. In this talk I will concentrate on the connection between properties of a finite group G and the set of orders of its elements, called the spectrum of G. We consider a general structure of groups isospectral to (i.e., having the same spectrum as) a given finite group with trivial solvable radical (especially, to a nonabelian simple group). Some related questions will be discussed, in particular, we give a new criterion of nonsolvability of a finite group. Some open problems will be posed.

Download: pdf

### Frank Lübeck (RWTH Aachen, Germany)

**Title**: Generic results for groups of Lie type

**Abstract**: Series of finite groups of Lie type are sets of finite groups like {GL(n,q)| q prime power} (fixed n) or {E_8(q)| q prime power}. In this talk we will sketch how such infinite sets of groups can be parameterized by simple combinatorial data, called a root datum. Then we will discuss some examples of data from these groups which can be computed and described "generically", that is with the prime power q as parameter: conjugacy classes, centralizers, character degrees, character tables, representations in defining characteristic.

Download: pdf

**Tim Burness **(University of Bristol, UK)

**Title:** The soluble graph of a finite group

**Abstract:** Let G be a finite insoluble group with soluble radical R(G). The vertices of the soluble graph of G, denoted Gamma_S(G), are labelled by the elements in G\R(G), with x and y adjacent if they generate a soluble subgroup of G. This is a natural generalisation of the widely studied commuting graph of G. In this talk I will report on recent joint work with Andrea Lucchini and Daniele Nemmi, which establishes several new results on Gamma_S(G). Our main result states that this graph is always connected, with diameter at most 5. We can construct examples with diameter 4, so our upper bound on the diameter is close to best possible. I will explain some of the main ideas and I will present a number of open problems.

Download: pdf

**Benjamin Sambale **(Leibniz Universitat Hannover, Germany)

**Title:** Character counting conjectures for π-separable groups

**Abstract: **One of the most important open problems in the representation theory of finite groups are the local-global conjectures by McKay, Alperin and Dade. These conjectures count certain irreducible characters by means of local subgroups with respect to a prime p. In the first part of my talk I will give precise definitions and indicate relations between the three conjecture. In the second part, I show how the conjectures can be generalized (and proven) for π-separable groups where π is a set of primes. This is joint work with Gabriel Navarro.

Download: pdf

**Rachel Camina **(University of Cambridge, UK)

**Title:** Conjugacy classes of maximal cyclic subgroups of finite p-groups.

**Abstract:** Let G be a finite p-group. We consider the following questionwhich I was asked back in 2017. Does the number of conjugacy classes ofmaximal cyclic subgroups of G increase with the order of G? If p=2 this is clearly not true, consider, for example, the family of dihedralgroups of 2-power order all of which have just 3 conjugacy classes ofmaximal cyclic groups. In this talk we discuss the answer for p odd andhow we reached the answer by considering pro-p groups. The talk is basedon joint work with Y. Barnea, M. Bianchi, M. Ershov, M.L. Lewis and E.Pacifici.

Download: pdf

### Nguyen Ngoc Hung (University of Akron, USA)

**Title:** Some lower bounds for the number of p'-degree characters of finite groups and their p-blocks

**Abstract:** Bounding the number of conjugacy classes and irreducible characters of a finite group is a classical problem in group representation theory, with several applications in other problems in the area. In 2016, Maroti proved a remarkable result that the number of conjugacy classes of a finite group of order divisible by p is always at least 2\sqrt{p-1}. This bound and the well-known McKay conjecture imply the same bound but for the number of p'-degree irreducible characters of the group, a result that was confirmed later by Malle and Maroti. I will discuss similar bounds for the number of almost p-rational irreducible characters of p'-degree and the number of height zero characters in the principal p-block of the group. These bounds arise naturally from Maroti's result and the McKay-Navarro and Alperin-McKay conjectures. Along the way I also discuss a new result on principal blocks with few height zero characters. The talk is based on joint works with Malle and Maroti, and with Schaeffer Fry and Vallejo.

Download: pdf

### Lucia Sanus Vitoria (Universität de Valencia, Spain)

**Title:** On character degrees

**Abstract:** Under some separability conditions, many results on character degrees of finite groups, such as the Ito-Michler or Thompson theorems, can be unified in a single statement. We present a characterization, under some separability conditions, of when Irrπ(G) = Irrρ(G) where π and ρ are sets of primes and Irrπ(G) is the set of irreducible characters χ of G such that all the primes dividing χ(1) lie in π.

Download: pdf

### Mahdi Ebrahimi (IPM, Isfahan Branch, Iran)

**Title: **Some properties of character graphs

**Abstract:** Let G be a finite group. Also let cd(G) be the set of all character degrees of G, that is, cd(G) = {χ(1)| χ ∈ Irr(G)}, where Irr(G) is the set of all complex irreducible characters of G. The set of prime divisors of character degrees of G is denoted by ρ(G). A useful way to study the character degree set of a finite group G is to associate a graph to cd(G). One of these graphs is the character graph Δ(G). Its vertex set is ρ(G) and two vertices p and q are joined by an edge if the product pq divides some character degree of G. The main questions in this research area concern the relationships between the group structure of G and certain graph-theoretical features of Δ(G). In this talk, we present some properties of G can be recognized by Δ(G) and focus on the structure of Δ(G).

Download: pdf

**Majid Arezoomand (**University of Larestan, Iran)

**Title:** On problems concerning 2-closure of permutation groups

**Abstract: **The concept of 2-closure of a permutation group is introduced by Wielandt in 1969. Let G be a permutation group on a set Ω. The 2-closure of G^(2) is the set of permutations of Ω leaving invariant each G-orbit in the induced G-action on ordered pairs from Ω. It is important to note that G^(2) is not intrinsic to the group G. It can depend on the nature of the action on Ω. We will discuss some group theoretic properties of G^(2) inherited by G. Let G ≤ Sym(G). One can see that G ≤ G^(2). The group G is called 2-closed if G^(2)=G. An abstract group G is called totally 2-closed if it is 2-closed in all of its faithful permutation representations. We classify all totally 2-closed solvable and also non-solvable groups with trivial fitting subgroup. Furthermore, we talk about some problems concerning 2-closure of permutation groups such as the polycirculant conjecture and the isomorphism problem of Cayley graphs.

Download: pdf

### S. Mohsen Ghoraishi (Shahid Chamran University, Iran)

**Title:** The noninner conjecture: a review

**Abstract:** The Noninner Conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. This talk aims to review existing results on the conjecture. The speaker has made a little contribution to some of the results. As an instance, recently, he has reduced the the verification of the conjecture to the case of finite nonabelian p-groups G in which the coclass of G is greater than the minimum number of generators of G.

Download: pdf