Summer Semester 2020

May 7th, 15:00 (CEST time): Cheryl Praeger (The University of Western Australia)

Title: Diagonal Structures and Permutation Groups

Permutation groups are often best understood by the structures they preserve on the underlying point set. For example, several families of maximal subgroups of finite symmetric groups arise as stabilisers of subsets, or partitions, or Cartesian decompositions of the point set. Among the finite primitive groups, one family (apart from the almost simple groups) resisted such a description until now, namely the maximal simple diagonal groups. We introduce the concept of a diagonal structure associated with any group T, finite or infinite, and any integer m at least 2. We prove that the stabilizer of the diagonal structure is precisely the corresponding diagonal group, and that this group is primitive essentially when T is characteristically simple.

This work began as a collaboration with Csaba Schneider, and over the past year the collaboration includes also R. A. Bailey and Peter Cameron. We have introduced also a diagonal graph, built from a collection of Hamming graphs: its automorphism group is the diagonal group. And there are close connections with Latin squares, and with certain kinds of higher dimensional Latin hypercubes. We continue to learn more about the combinatorial links.

May 14th, 15:00 (CEST time): Francesco de Giovanni (Università degli Studi di Napoli Federico II)

Title: Accessible group classes

If X is a class of groups, a group G is said to be minimal non-X (or an opponent of X) if it is not an X-group but all its proper subgroups belong to X. A group class X is called accessible if it admits no infinite locally graded opponents, or equivalently if every infinite locally graded group whose proper subgroups belong to X is likewise an X-group.

The aim of this talk is to discuss the accessibility of certain relevant group classes, and to illustrate a method to embed an arbitrary class of groups in a suitable accessible class.

May 21st, 17:00 (CEST time): Martyn Dixon (The University of Alabama)

Title: Groups with many permutable subgroups

Recent research of mine and others has been concerned with groups many of whose subgroups have certain properties. In this talk I shall give an outline of some of the results obtained when at least one of these properties is permutability. I shall also give details of joint work with Maria Ferrara, Yalcin Karatas and Marco Trombetti which has recently appeared.

May 28th, 17:00 (CEST time): Rostislav Grigorchuk (Texas A&M University)

Title: Asymptotic properties of Burnside groups

My talk will be dedicated to the memory of Sergei Adian who passed away on May 5th 2020 and will partially have a historic character.

I will begin with interesting developments initiated in two 1959 articles by Harry Kesten on random walks on groups and in particular with his idea how to give a counterexample to what he called "The Burnside Conjecture". Then I will briefly recall whatis The Burnside Problem and what is Adian’s contribution to it, based on his outstanding book The Burnside problem and identities in groups. I will recall his result on non-amenability of the free Burnside groups B(m,n) for odd n>665 presented in the 1982 article Random Walks on free periodic groups. It uses the cogrowth criterion of amenability invented by the speaker in 1977, whereas consequences of Adian’s result include results on the values of the Tarski numbers for groups B(m,n) obtained by T. Ceccherini-Silberstein, P. de la Harpe and the speaker. After a short discussion of growth, I will finally formulate a few new results on groups of Burnside type and on groups associated with automatically generated sequences, which, surprisingly, appear in the proof of the Novikov-Adian theorem.

June 4th, 17:00 (CEST time): Gustavo A. Fernández-Alcober (University of the Basque Country UPV/EHU)

Title: Elementary equivalence for partially commutative nilpotent groups and algebras

For a fixed algebraic structure (groups, rings, algebras, Lie algebras...), two instances of that structure are said to be elementary equivalent if they satisfy the same first-order sentences in the language corresponding to the structure. This way of identifying algebraic objects is weaker than isomorphism, in the sense that isomorphic objects are elementary equivalent, but not necessarily vice versa. On the other hand, Philip Hall introduced the concept of nilpotent R-group, where R is binomial domain, i.e. an integral domain containing the binomial coefficients of its elements.

In this talk we will give an exposition of a joint work with Montserrat Casals, Ilya Kazachkov, and Vladimir Remeslennikov, where we determine all groups/algebras that are elementary equivalent to a partially commutative nilpotent R-group/R-algebra. This is done by building a more general class of well-structured groups/algebras, for which we solve the problem of elementary equivalence.

June 11th, 17:00 (CEST time): Tatiana Smirnova-Nagnibeda (University of Geneva)

Title: On maximal and weakly maximal subgroups in infinite finitely generated groups

Margulis and Soifer proved that a finitely generated linear group has all maximal subgroups of finite index (aka has the MF property) if and only if it is virtually solvable. Otherwise it has uncountably many maximal subgroups of infinite index. Pervova later proved that outside the world of linear groups Grigorchuk's group also has the MF property. However it is known to have a huge variety of weakly maximal subgroups. I will discuss some questions stemming from the aforementioned results about the richness versus rigidity of maximal and weakly maximal subgroups in various classes of groups, in particular in branch groups and in Thompson groups.

June 18th, 17:00 (CEST time): Martin Evans (The University of Alabama)

Title: Nilpotent groups in which all proper subgroups have "much" smaller class

Let G be a nilpotent group and suppose that G has distinct maximal subgroups M and N of class at most n. Then Fitting’s theorem implies that G has class at most 2n. It is known that if n ≥ 3 is a positive integer and p is a prime with p ≥ 2n, then there exists a finite p-group Hn of class 2n in which all maximal subgroups have class at most n. Such groups Hn are necessarily 2-generator.

In this talk we'll consider some questions like the following: Let G be a nilpotent group minimally generated by d elements and suppose that all maximal subgroups of G have class at most n. What is the largest possible class of G? (The results above give information about the case d = 2) Our methods involve answering similar questions for Lie algebras.

June 25th, 17:00 (CEST time): Manoj K. Yadav (Harish-Chandra Research Institute)

Title: The Schur Multiplier of Central Product of Groups

Let G be the central product of two of its normal groups H and K amalgamating a given central subgroup A. That the Schur multiplier of G admits the abelian tensor product of H/A and K/A as a subgroup was shown by J. Wiegold in 1971, in case G is finite. The same conclusion for such arbitrary groups was derived by B. Eckman, P. J. Hilton and U. Stammbach in 1973.

In this talk, on the one hand, I’ll discuss on the refinements of these results and, on the other hand, establish an embedding from the Schur multiplier of G into an explicit group constituted by the second cohomology groups of certain quotients of H and K, and abelian tensor product of H and K. When the said embedding is an isomorphism, then a precise formula for the Schur multiplier is obtained. This talk is based on a joint work with L. R. Vermani and Sumana Hatui.

July 2nd, 17:00 (CEST time): Aner Shalev (Hebrew University of Jerusalem)

Title: Subsets products and derangements

In the past two decades there has been considerable interest in products of subsets in finite groups. Two important examples are Gowers' theory of Quasi Random Groups and its applications by Nikolov, Pyber, Babai and others, and the theory of Approximate Subgroups and the Product Theorem of Breuillard-Green-Tao and Pyber-Szabo on growth in finite simple groups of Lie type of bounded rank, extending Helfgott's work. These deep theories yield strong results on products of three subsets (covering, growth). What can be said about products of two subsets?

I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem, focusing on products of two normal subsets of finite simple groups. As a main application we show that any element of a sufficiently large simple transitive permutation group is a product of two derangements. The proofs apply Fulman-Guralnick's work on the proportion of derangements, and involve algebraic geometry, representation theory and other tools.