Spring Semester 2020

  • April 15th, 11:00 (CEST): Eamonn O’Brien (University of Auckland)

Title: Constructing composition factors for linear groups

A recent result of Holt, Leedham-Green and O'Brien shows that we are finally in a position where, subject to certain assumptions, we can construct in polynomial time the composition factors of a subgroup of GL(d, q). The principal components are "constructive recognition" and presentations on "standard generators" for the finite simple groups. We survey this work.

  • April 22nd, 17:00 (CEST): Thomas Keller (Texas State University)

Title: Character degrees, conjugacy class sizes, and element orders: three primes

There are many results that give information on the structure of a finite group in terms of properties that refer to its character degrees/class sizes/element orders and at most two primes. In this talk we present a first attempt to extend some of these results considering three primes. We concentrate on bounds for the Fitting height of solvable groups. (This is joint work with Alex Moreto).

  • April 29th, 17:00 (CEST): Evgeny Plotkin (Bar-Ilan University)

Title: Logical equivalences of linear groups

We will survey a series of recent developments in the area of first order descriptions of groups. The goal is to illuminate the known results and to pose new problems relevant to logical characterizations of linear groups. We also dwell on the principal problem of isotipicity of finitely generated groups.

  • May 6th, 17:00 (CEST): Natalia Maslova (Russian Academy of Sciences)

Title: On pronormality of subgroups of odd index in finite groups

In this talk we discuss a recent progress in research of pronormality of subgroups of odd index in finite groups.

A subgroup H of a group G is pronormal in G if for any element g from G, subgroups H and H^g are conjugate in the subgroup <H, H^g> generated by H and H^g. Some problems in Finite Group Theory, Combinatorics, and Permutation Group Theory were solved in terms of pronormality (see, for example, remarkable results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description of families of pronormal subgroups in finite groups is of interest. Well-known examples of pronormal subgroups in finite groups are normal subgroups, maximal subgroups, Sylow subgroups, Carter subgroups, Hall subgroups of solvable groups, and so on.

In 2012, E.P. Vdovin and D.O. Revin proved that the Hall subgroups are pronormal in finite simple groups and conjectured that the subgroups of odd index are pronormal in finite simple groups. This conjecture was disproved by A.S. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. Moreover, the question of pronormality of a subgroup of odd index in an arbitrary finite group can be partially reduced to questions of pronormality of some subgroups of odd indices in its chief factors.


This talk is partially based on joint results with S. Glasby, A.S. Kondrat’ev, C.E. Praeger, and D.O. Revin.

  • May 13th, 17:00 (CEST): Laurent Bartholdi (University of Göttingen)

Title: Dimension series and homotopy groups of spheres

The lower central series of a group G is defined by γ_1=G and γ_n = [G, γ_(n-1)]. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers:

δ_n = {g: g-1 belongs to the n-th power of the augmentation ideal}.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δ_n > γ_n, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ_4/γ_4 cyclic of order 2. On the positive side, Sjogren showed that δ_n/γ_n is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proven by Gupta) that only 2-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δ_n/γ_n; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups π_n^(S^m) of spheres; more precisely, the quotient δ_n/γ_n is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π_(2p)^(S^2) due to Serre.

  • May 20th, 17:00 (CEST) Thomas Weigel (University of Milano - Bicocca)

Title: Maximal pro p-quotients of absolute Galois groups

Abstract here.

  • May 27th, 17:00 (CEST) Peter P. Palfy (Alfréd Rényi Institute of Mathematics)

Title: Galois and PSL

In his "testamentary letter" Galois claims (without proof) that PSL(2,p) does not have a subgroup of index p whenever p>11, and gives examples that for p = 5, 7, 11 such subgroups exist. The attempt by Betti in 1853 to give a proof does not seem to be complete. Jordan's proof in his 1870 book uses methods certainly not known to Galois. Nowadays we deduce Galois's result from the complete list of subgroups of PSL(2,p) obtained by Gierster in 1881.

In the talk I will give a proof that might be close to Galois's own thoughts. Last October I exchanged a few e-mails on this topic with Peter M. Neumann. So the talk is in some way a commemoration of him.

  • June 3rd, 17:00 (CEST) John S. Wilson (University of Oxford)

Title: A first-order perspective on finite groups

The finite axiomatizability of classes of finite groups, and the definability of naturally occurring subgroups, have attracted considerable attention. In this talk, some of the results, positive and definite, will be discussed, and it will be shown that the strikingly different behaviour of certain properties seems to be reflected in (non-first-order) studies of these properties.

  • June 10th, 17:00 (CEST) Rachel Skipper (Ohio State University)


Title: The Cantor-Bendixson rank of the Grigorchuk group


The space of subgroups of a group has a natural Polish topology and understanding this space can help to understand the group. In this talk, we will consider the Cantor-Bendixson derivative and rank for the space of subgroups of the Grigorchuk group, using it to stratify the subgroups of this group. This is a joint work with Phillip Wesolek.

  • June 17th, 17:00 (CEST) Dan Segal (University of Oxford)

Title: Groups, Rings, Logic

In group theory, interesting statements about a group usually can’t be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely 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 profinite 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 first-order properties of the group G(R) correspond to first-order properties of the ring R. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form G(R).

  • June 24th, 17:00 (CEST) Nikolay Nikolov (University of Oxford)

Title: On profinite groups with positive rank gradient

In this talk I will introduce rank gradient of groups and discuss open questions about groups with positive rank gradient. In the second part I will focus on the profinite situation and sketch a proof that a profinite group G with positive rank gradient does not satisfy a group law.

  • July 1st, 17:00 (CEST) Rachel Camina (University of Cambridge)

Title: Word problems for finite nilpotent groups

We consider word maps on finite nilpotent groups and count the sizes of the fibres for elements in the image. We consider Amit’s conjecture and its generalisation, which say that these fibres should have size at least |G^(k−1)| where the word is on k variables. This is joint work with Ainhoa Iñiguez and Anitha Thillaisundaram.

  • July 8th, 17:00 (CEST) Giles Gardam (University of Münster)

Title: Kaplansky's conjectures

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

  • July 15th, 15:00 (CEST) Cindy Tsang (Ochanomizu University, Tokyo)

Title: The multiple Holomorph of centerless groups

The holomorph Hol(G) of a group G may be defined as the normalizer of the subgroup of left translations in the group of all permutations of G. The multiple holomorph NHol(G) of G may in turn be defined as the normalizer of the holomorph. Their quotient T(G) = NHol(G)/Hol(G) has been computed for various families of groups G, and interestingly T(G) turns out to be elementary 2-abelian in many of the known cases. In this talk, we consider the case when G is centerless, and we will present our new result that T(G) has to be elementary 2-abelian unless G satisfies some fairly strong conditions. For example, our result implies that T(G) is elementary 2-abelian when G is any (not necessarily finite) centerless perfect/almost simple/complete group.

  • July 22nd, 17:00 (CEST) Pavel Zalesski (University of Brasilia)

Title: Finitely generated pro-p groups acting on pro-p trees

I shall discuss various results on splitting of a pro-p group as a free amalgamated pro-p product or HNN-extension in the spirit of the Bass-Serre theory of groups acting on trees.