Fall 2021

  • October 14th, 17:00 (CEST time): Robert Guralnick (University of Southern California)

Title: Topological Generation of Algebraic Groups

We consider the problem of generation of (mostly simple) algebraic groups G in the topological setting using the Zariski topology. In particular, we will discuss the problem of how many conjugates of a given element are needed. We will give applications to some generation problems for inite groups of Lie type and to generic stabilizers.

This is joint work with Tim Burness and Spencer Gerhardt.

  • October 21st, 18:00 (CEST time): Ángel del Río (Universidad de Murcia)

Title: A negative solution to the Modular Isomorphism Problem

Let R be a ring. The Isomorphism Problem for group rings over R asks whether the isomorphism type of a group G is determined by the isomorphism type of the group ring RG. The special case where R is a field with p elements and G is a finite p-group, for p prime, is known as the Modular Isomorphism Problem.

The history of the Isomorphism Problem goes back to a seminal paper of G. Higman in the 1940s. The Modular Isomorphism Problem appeared in a survey paper by R. Brauer in 1963. While many relevant instances of the general Isomorphism Problem have been already resolved, the Modular Isomorphism Problem resisted until now.

In cooperation with Diego García and Leo Margolis we discovered recently two non-isomorphic groups of order 2^9 whose group algebras over any field of characteristic 2 are isomorphic. We will present this example and give an overview of the state of the art on the Isomorphism Problem.

  • October 28th, 18:00 (CEST time): Daniele D'Angeli (Univ. Niccolò Cusano - Roma)

Title: Graph automaton groups

In this talk I will review some basic and interesting properties of automaton groups, i.e. groups generated by the action of a transducer on a finite alphabet. Then I will explain a new construction (introduced in collaboration with M. Cavaleri, A. Donno and E. Rodaro) to obtain automaton groups starting from finite graphs. This class of "Graph Automaton groups" contains classic examples of automaton groups and other groups exhibiting interesting combinatorial and spectral properties.

  • November 4th, 18:00 (CET time): Michael Vaughan-Lee (Christ Church Oxford)

Title: Schur's exponent conjecture

If G is a finite group and we write G=F/R, where F is a free group, then the Schur multiplier M(G) is (R ∩ F')/[F,R]. There is a long-standing conjecture attributed to I. Schur that the exponent of M(G) divides the exponent of G. It is easy to show that this is true for groups G of exponent 2 or exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. However the truth or otherwise of this conjecture has remained open up till now for groups of odd exponent.

In my talk I describe counterexamples to the conjecture of exponent 5 and exponent 9. I also give some suggestions for further counterexamples, and explore the possibilities for alternative conjectures.

  • November 11th, 18:00 (CET time): Clara Franchi (Università Cattolica del Sacro Cuore)

Title: Majorana representations of finite groups

The concept of Majorana representations of finite groups have been introduced by A.A. Ivanov in 2009 as a tool to better understand the Monster and its representation on the Conway-Norton-Griess algebra.

In my talk I will review the principal results of the theory of Majorana representations of finite groups. In particular, I will focus on the representations of the symmetric groups, presenting some joint work with A.A. Ivanov and M. Mainardis.

  • November 18th, 18:00 (CET time): Thomas Z. Viji (Indian Institute of Science Education and Research Thiruvananthapuram)

Title: Schurs Exponent Conjecture and Related Problems

Assume G is a finite p-group, and let S be a Sylow p subgroup of Aut(G) with exp(S) = q. We prove that if G is of class at most p^2 − 1, then exp(G)|p^2q^3, and if G is a metabelian p-group of class at most 2p − 1, then exp(G)|pq^3. To obtain this result, we will first speak about Schur’s exponent conjecture and related problems. This is joint work with my PhD student P. Komma.

  • November 25th, 18:00 (CET time): Sandro Mattarei (University of Lincoln)

Title: Graded Lie algebras of maximal class

Interest in Lie algebras of finite coclass begun in the 1990's, when Shalev and Zelmanov discovered both similarities and striking differences with the recently proved coclass conjectures of Leedham-Green and Newman for pro-p groups. Later work by Caranti, Newman, Jurman, Vaughan-Lee led to classifications of infinite-dimensional Lie algebras of maximal class (meaning of coclass 1, also called filiform in some areas) having gradings of certain types over the positive integers. After surveying those developments I will present generalizations of some of those results, recently completed in collaboration with my former PhD students Simone Ugolini, Claudio Scarbolo and Valentina Iusa.

  • December 2nd, 18:00 (CET time): Alejandra Garrido (Universidad Autónoma de Madrid - ICMAT)

Title: On various profinite completions of groups acting on rooted trees

Groups that act faithfully on rooted trees can be studied via their finite quotients. There are several natural collections of finite quotients that can be chosen for this. The mathematical object that encodes all these finite quotients and the maps between them is the profinite completion of the group (with respect to the chosen collection). Taking all possible finite quotients of the group gives *the* profinite completion of the group, and this maps onto each of the other completions. Determining the kernels of these maps is known as the congruence subgroup problem. This has been studied by various authors over the last few years, most notably for self-similar groups and (weakly) branch groups. In the case of self-similar regular branch groups, much insight can be gained into this problem using a symbolic-dynamical point of view. After reviewing the problem and previous work on it, I will report on work in progress with Zoran Sunic on determining the dynamical complexity of these completions and calculating some of these kernels with relative ease. Examples will be given. No previous knowledge of profinite, self-similar or branch groups is required.

  • December 9th, 18:00 (CET time): M. D. Pérez-Ramos (University of Valencia)

Title: Carter and Gaschütz theories revisited

Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. We present an extension of these facts to π-separable groups, for sets of primes π, by proving the existence of a conjugacy class of subgroups in π-separable groups, which specialize to Carter subgroups within the universe of soluble groups.

The approach runs parallel to the extension of Hall theory from soluble to π-separable groups by Cunihin, regarding existence and properties of Hall subgroups. Our Carter-like subgroups enable an extension of the existence and conjugacy of injectors associated to Fitting classes to π-separable groups, in the spirit of the role of Carter subgroups in the theory of soluble groups. This is joint work with M. Arroyo-Jordá, P. Arroyo-Jordá, R. Dark and A.D. Feldman.