Title: On Gruenberg-Kegel graphs of finite groups

Time and place: 4pm Friday 18 Oct 2019, Weatherburn LT

Abstract: Let G be a finite group. The spectrum of G is the set of all its element orders. The Gruenberg-Kegel graph of G is a graph whose vertex set is the set of all prime divisors of the order of G, and two distinct vertices are adjacent in this graph if and only if their product is an element order of G. The spectrum is a very important invariant of a finite group. For example, many finite simple groups are defined by their spectra up to isomorphism. The concept of the Gruenberg-Kegel graph of a finite group widely generalizes the concept of the spectrum. In this talk we discuss some characterizations of finite groups by properties of their spectra and Gruenberg-Kegel graphs.

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

Time and place: 4pm Friday 25 Oct 2019, Weatherburn LT

Abstract: 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 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, results by L. Babai, P. Palfy, Ch. Praeger, and others). Thus, the question of description 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. An important problem is to describe families of pronormal subgroups in finite simple groups.

In 2012, E. Vdovin and D. 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. Kondrat'ev, the speaker, and D. Revin in 2016. However, in many finite simple groups the subgroups of odd index are pronormal. In this talk we discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal and some connected questions.

Title: Subdegrees of primitive permutation groups

Time and place: 4pm Friday 01 Nov 2019, Robert Street LT

Abstract: A subdegree of a permutation group is the length of an orbit of a point stabiliser. The study of subdegrees of primitive permutation groups has a long history and has attracted the attention of many researchers. In this talk I will survey some results in this area and discuss some recent work on small subdegrees, constant subdgrees and coprime subdegrees. I will also discuss some applications to graph theory.

Cheese and wine to follow in the Maths common room.

Abstract: We all have our doubts off and on if life is really so wonderful. But that is not what I want to address here. Watching the Jimmy Stewart movie with this title, there was one scene which captured my imagination: the Guardian Angel shows George Bailey how the world would have been without him. Personally, I never had much need to know how the world would have looked without me. However, all other things equal, how would life have been if I had lived in a different time and place, would be something of interest to me! This is the stuff of movies and fairy tales. But at least it is possible to play this as an intellectual game. I was born and raised in Germany before WW II. After getting my Ph.D. in 1962, I married a fellow mathematician and we immigrated to the US one year later, where we taught at a university until our retirements, first at Ohio State and then at Binghamton University. What would life have been if I stayed in Germany, did not get married, were born fifty or one hundred years earlier, or were born in another country? Looking at actual and potential role models over the centuries helped me answer some of these questions. In essence, it got me back to the roots of what shaped my life.