Winter Semester 2020

• September 24th, 18:00 (CEST time): I. Martin Isaacs (University of Wisconsin-Madison)

Title: Carter subgroups and characters

Carter subgroup is a self-normalizing nilpotent subgroup, and a theorem of Carter asserts that every solvable group contains a unique conjugacy class of Carter subgroups. Given a solvable group G with Carter subgroup C, we construct a family of injective maps from the set of linear characters of C into the set Irr(G). There are usually many maps in this family, but all of them have exactly the same image. This common image is thus a canonically determined set of irreducible characters of G, and we refer to the members of this set as the "head characters" of G. They are characterized by their behavior with respect to the C-composition series of G.

Each head character of G has degree dividing|G:C|, and each of them restricts irreducibly to the nilpotent residual of G. Experiment suggests that the head characters of G are exactly those irreducible characters that have nonzero values on all elements of C, but this remains unproved.

• October 1st, 17:00 (CEST time): Andrea Lucchini (Università degli Studi di Padova)

Title: Intersections of maximal subgroups in finite groups

We will investigate a series of questions in group theory which, although with different motivations, are all related with the study of lattice M(G) consisting of the subgroups of a finite group that can be obtained as intersection of maximal subgroups.

• October 8th, 17:00 (CEST time): Colva Roney-Dougal (The University of St. Andrews)

Title: Finite simple groups and complexity class NP

This talk will describe connections between structural results about the finite simple groups and the complexity of computational algorithms for permutation groups.

The first part of the talk will discuss the base size of a permutation group, an invariant which determines the complexity of many permutation group algorithms. We will present a new, optimal, bound on the base size of the primitive groups that are not large base. After this, we will discuss some group-theoretic questions for which there is no known polynomial time solution. In particular, we shall present a new approach to computing the normaliser of a primitive group G in an arbitrary subgroup H of Sn. Our method runs in quasipolynomial time O(2^{log^3 n}), whereas the previous best known algorithm required time O(2^n).

This is partly joint work with Mariapia Moscatiello (Padova), and partly with Sergio Siccha (Siegen).

• October 15th, 17:00 (CEST time): Andrei Jaikin-Zapirain (Autonomous University of Madrid)

Title: Intersection of subgroups in a surface group

Let G be a surface group, i.e the fundamental group of a compact surface. Denote by d(G) the number of generators of G and by χ(G) the Euler characteristic of G. We put χ(G) = max{0, −χ(G)}.

In this talk I will explain the following two results. In the first result we prove that for any two finitely generated subgroups U and W of G,

From this we obtain the Strengthened Hanna Neumann conjecture for non-solvable surface groups. In the second result we show that if R is a retract of G, then for any finitely generated subgroup H of G,

d(R ∩ H) ≤ d(H).

This implies the Dicks-Ventura inertia conjecture for free groups. The talk is based on a joint work with Yago Antolín.

• October 22nd, 17:00 (CEST time): Anitha Thillaisundaram (University of Lincoln)

Title: The congruence subgroup property for multi-EGS groups

It was proved by G. A. Fernández-Alcober, A. Garrido and J. Uria-Albizuri that the branch Grigorchuk-Gupta-Sidki (GGS) groups possess the congruence subgroup property. This result was extended to all branch multi-GGS groups by A. Garrido and J. Uria-Albizuri. The extended Gupta-Sidki (EGS) groups, which were the first examples of finitely generated branch groups without the congruence subgroup property, were constructed by Pervova. In this talk, we consider a natural generalisation of multi-GGS and EGS groups, and demonstrate their unexpected behaviour concerning the congruence subgroup property. This is joint work with J. Uria-Albizuri.

• October 29th, 17:00 (CET time): Bettina Eick (Technical University of Braunschweig)

Title: Groups and their integral group rings

The integral group ring ZG of a group G plays an imporant role in the theory of integral representations. This talk gives a brief introduction to this topic and then shows how such group rings can be investigated using computational tools. In particular, the quotients I^n(G)/I^{n+1}(G), where I^n(G) is the n-th power ideal of the augmentation ideal I(G), are an interesting invariant of the group ring ZG and we show how to determine them for given n and given finitely presented G. We then exhibit a variety of example applications for finite and infinite groups G.

• November 5th, 17:00 (CET time): Derek J. S. Robinson (University of Illinois at Urbana-Champaign)

Title: The seriality problem for Sylow-permutable subgroups in locally finite groups

A subgroup H of a group G is said to be weakly Sylow permutable in G if HP=PH for all Sylow subgroups P of G and all primes p dividing orders of elements of H. Otto Kegel proved that if G is finite, then H is subnormal in G. This does not hold for infinite groups. The Seriality Problem is whether Kegel’s theorem can be extended to locally finite groups if “subnormal” is replaced by “serial”. I will discuss the background to the problem and recent progress towards its solution.

• November 12th, 17:00 (CET time): Alexander Bors (Johann Radon Institute for Computational and Applied Mathematics)

Title: Groups with few automorphism orbits

Let G be a group, and consider the natural action of the automorphism group of G on G. The orbits of this action are called the automorphism orbits of G. In this talk, we will give an overview of known results concerning groups with finitely many automorphism orbits, including results where G is assumed to have a concrete, small number of automorphism orbits, such as 3. We will then speak in more detail about a result, achieved in collaboration with Stephen Glasby from UWA (Perth), which provides a full classification of the finite 2-groups with exactly three automorphism orbits.

• November 19th, 17:00 (CET time): Ramón Esteban-Romero (University of Valencia)

Title: Triply factorised groups and skew left braces

The Yang-Baxter equation is a consistency equation of the statistical mechanics proposed by Yang [6] and Baxter [1] that describes the interaction of many particles in some scattering situations. This equation lays the foundation for the theory of quantum groups and Hopf algebras. During the last years, the study suggested by Drinfeld [2] of the so-called set-theoretic solutions of the Yang-Baxter equation has motivated the appearance of many algebraic structures. Among these structures we find the skew left braces, in troduced by Guarnieri and Vendramin [3] as a generalisation of the structure of left brace defined by Rump [4]. It consists of a set B with two operations + and ·, not necessarily commutative, that give B two structures of group linked by a modified distributive law.

The multiplicative group C = (B, ·) of a skew left brace (B, +, ·) acts on the multiplicative group K = (B, +) by means of an action λ: C −→ Aut(K) given by λ(a)(b) = −a + a · b, for a, b ∈ B. With respect to this action, the identity map δ : C −→ K becomes a derivation or 1-cocycle with respect to λ. In the semidirect product G = [K]C = {(k, c) | k ∈ K, c ∈ C}, there is a diagonal-type subgroup D = {(δ(c), c) | c ∈ C} such that G = KD = CD, K ∩ D = C ∩ D = 1. This approach was presented by Sysak in [5] and motivates the use of techniques of group theory to study skew left braces.

We present in this talk some applications of this approach to obtain some results about skew left braces. These results have been obtained in collaboration with Adolfo Ballester-Bolinches.

This work has been supported by the research grants PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovación y Universidades (Spanish Government), the Agencia Estatal de Investigación (Spain), and FEDER (European Union), and PROMETEO/2017/057 from the Generalitat (Valencian Community, Spain).

References here.

• November 26th, 17:00 (CET time): Urban Jezernik (Alfréd Rényi Institute of Mathematics)

Title: Diameters of groups

The diameter of a finite group G equipped with a generating set S is the smallest number k so that every element of G can be written as a product of at most k elements from S. We will take a look at how large or small these diameters can (conjecturally) be, and what the generic situation is like.

• December 3rd, 17:00 (CET time): Gareth Tracey (University of Oxford)

Title: On the Fitting height and insoluble length of a finite group

A classical result of Baer states that an element x of a nite group G is contained in the Fitting subgroup F(G) of G if and only if x is a left Engel element of G. That is, x is in F(G) if and only if there exists a positive integer k such that [g, x, ..., x] (with x appearing k times, and using the convention [x1, x2, x3, ..., xk] := [[... [[x1, x2], x3], ...], xk]) is trivial for all g in G. The result was generalised by E. Khukhro and P. Shumyatsky in a 2013 paper via an analysis of the sets E(G(k))= {[g, x, ..., x]: g in G}.

In this talk, we will continue to study the properties of these sets, concluding with the proof of two conjectures made in said paper. As a by-product of our methods, we also prove a generalisation of a result of Flavell, which itself generalises Wielandt's Zipper Lemma and provides a characterisation of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a nite group inverted by an involutory automorphism. Joint work with R.M. Guralnick.

• December 10th, 17:00 (CET time): Donald S. Passman (University of Wisconsin-Madison)

Title: Polynomial Identities, Permutational Groups and Rewritable Groups

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.

• December 17th, 17:00 (CET time): Francesco Matucci (Università di Milano-Bicocca)

Title: On Finitely Presented Groups that Contain Q

It is a consequence of Higman's embedding theorem that the additive group Q of rational numbers can be embedded into a finitely presented group. Though Higman's proof is constructive, the resulting group presentation would be very large and unpleasant. In 1999, Martin Bridson and Pierre de la Harpe asked for an explicit and "natural" example of a finitely presented group that contains an embedded copy of Q. In this talk, we describe some solutions to the Bridson - de la Harpe problem related to Richard Thompson's groups F, T, and V. Moreover, we prove that there exists a group containing Q which is simple and has type F infinity. This is joint work with Jim Belk and James Hyde.

• January 14th, 16:00 (CET time): Alex Lubotzky (Hebrew University)

Title: Stability, non-approximated groups and high-dimensional expanders

Several well-known open questions, such as: "are all groups sofic or hyperlinear?" , have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, some of these versions, showing that there exist finitely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm and many other norms. The strategy is via the notion of "stability": Some higher dimensional cohomology vanishing phenomena is proven to imply stability. Using Garland method (a.k.a. high dimensional expanders as quotients of Bruhat-Tits buildings), it is shown that some non-residually-finite groups are stable and hence cannot be approximated. These groups are central extensions of some lattices in p-adic Lie groups (constructed via a p-adic version of a result of Deligne). All notions will be explained. Based on joint works with M. De Chiffre, L. Glebsky and A. Thom and with I. Oppenheim.

• January 21st, 17:00 (CET time): Mima Stanojkovski (Max Planck Institute for Mathematics in Sciences)

Title: On the modular isomorphism problem for groups of class 3

Let G be a finite group and let R be a commutative ring. In 1940, G. Higman asked whether the isomorphism type of G is determined by its group ring RG. Although the Isomorphism Problem has generally a negative answer, the Modular Isomorphism Problem (MIP), for G a p-group and R a field of positive characteristic p, is still open. Examples of p-groups for which the (MIP) has a positive solution are abelian groups, groups of order dividing 2^9 or 3^7 or p^5, certain groups of maximal class, etc.

I will give an overview of the history of the (MIP) and will present recent joint work with Leo Margolis for groups of nilpotency class 3. In particular, our results yield new families of groups of order p^6 and p^7 for which the (MIP) has a positive solution and a new invariant for certain 2-generated groups of class 3.

• January 28th, 17:00 (CET time): Iker de las Heras (Heinrich Heine University Düsseldorf)

Title: Hausdorff dimension and Hausdorff spectra in profinite groups

The Hausdorff dimension is a generalisation of the usual concept of dimension which allows to define the dimension of fractal sets in metric spaces. In the last decades, this notion has led to fruitful applications in the context of countably based profinite groups, as these groups can be naturally seen as metric spaces with respect to a given filtration series.

In this talk we will give a brief introduction to this topic and we will overview some of the main related properties. Finally, we will present some results concerning the so-called (normal) Hausdorff spectra of a given profinite group, which reflect the range of Hausdorff dimensions of closed (normal) subgroups.

Joint work with Benjamin Klopsch and Anitha Thillaisundaram.

• February 4th, 17:00 (CET time): Eugenio Giannelli (Università di Firenze)

Title: On a Conjecture of Malle and Navarro

Let G be a finite group and let P be a Sylow subgroup of G. In 2012 Malle and Navarro conjectured that P is normal in G if and only if the permutation character associated to the natural action of G on the cosets of P has some specific structural properties. In recent joint work with Law, Long and Vallejo we prove this conjecture. We will start this talk by describing the problem and its relevance in the context of representation theory of finite groups. Then we will introduce and review some recent results on Sylow Branching Coefficients for symmetric groups. Finally we will talk about the crucial role played by these objects in our proof of the conjecture.

• February 11th, 17:00 (CET time): Kıvanç Ersoy (Free University of Berlin)

Title: On the centralizer depth in simple locally finite groups

Abstrat here.

• February 18th, 19:00 (CET time): Norberto Gavioli (Università degli Studi dell'Aquila)

Title: Thin subalgebras of Lie algebras of maximal class

Joint work with M. Avitabile, A. Caranti, V. Monti, M. F. Newman and E. O'Brien.

Let L be an infinite dimensional Lie algebra which is graded over the positive integers and is generated by its first homogeneous component L_1. The algebra L is of maximal class if dim(L_1)=2 and dim(L_i)=1 for i larger than 1. The algebra L is thin if it is not of maximal class, dim(L_1)=2 and L_{i+1}=[x,L_1] for any nontrivial element x in L_i.

Suppose that E is a quadratic extension of a field F and that M is a Lie algebra of maximal class over E. We consider the Lie algebra L generated over the field F by an F-subspace L_1 of M_1 having dimension 2 over F. We give necessary and sufficient conditions for the Lie algebra L to be a thin graded F-subalgebra of the F-algebra M. We show also that there are uncountably many such thin algebras that can be constructed by way of this “recipe”, attaining the maximum possible cardinality.

The authors started this project almost independently since 1999 and their partial results have been luckily and duly recorded by A. Caranti. Only recently we have been able to jointly develop thorough and concise results for this research.

• February 25th, 17:00 (CET time): Agnieszka Bier (Silesian University of Technology)

Title: On weak Sierpinski subsets in groups

A subset E in a group G is called a weak Sierpinski subset if for some g, h in G and a different from b in E, we have gE = E \ {a} and hE = E \ {b}. In the talk we discuss the subgroup generated by g and h, and show that either it is free over (g,h) or it has presentation G(k) = <g, h | (h^{-1}g)^k>. We also characterize all weak Sierpinski subsets in the groups G(k). This is joint work with Y. Cornulier and P. Slanina.

• March 4th, 17:00 (CET time): Cristina Acciarri (University of Brasilia)

Title: A stronger version of Neumann’s BFC-theorem

A celebrated theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G’ is finite.

In this talk we will discuss a stronger version of Neumann’s result and some corollaries for finite and profinite groups. Based on a joint work with Pavel Shumyatsky.

• March 11th, 17:00 (CET time): Gunter Malle (Technical University of Kaiserslautern)

Title: Conjugacy class numbers and π-subgroups

We will discuss relations between the number of conjugacy classes of a finite group and that of proper subgroups. On the way, we'll encounter the so-called almost abelian groups (a term coined by J. Thompson). We then connect this to obtaining estimates for the number of Brauer characters in a Brauer block of a finite group. This is joint work with Gabriel Navarro and Geoffrey Robinson.

• March 18th, 17:00 (CET time): Britta Spaeth (University of Wuppertal)

Title: Representation Theory above Spin Groups - Another Step towards the McKay Conjecture

In the representation theory of finite groups it is suspected that the representation theory of a group is already determined by its local subgroups. This lead to numerous conjectures like the McKay conjecture. During the last decade substantial progress in a final proof of the McKay conjecture has been made. After an overview of the development I sketch the open questions, that are mainly regarding the representation theory of spin groups and some progress made on one of those questions.

• March 25th, 17:00 (CET time): Ischia Group Theory 2020/2021

Go to Schedule Spring Semester for the Schedule of the Spring Semester of GOThIC!