Arithmetic properties of a finite group and connection with the structure of the group

Andrey V. Vasil'ev,
IM SB RAS, Novosibirsk, Russia.

The elementary assertion that groups of exponent 2 are abelian and one of the most complicated theorems in the finite group theory on solvability of groups of odd order are similar in the following sense. In both cases arithmetical, i.e., expressed by numbers, properties of a group allow to conclude on its structure. The strongest conclusion that can be made in this direction is that a group is uniquely, up to isomorphism, determined by a set of number parameters. In such case the group is said to be recognizable by this set. 

The set of element orders of a group is called the spectrum of the group. In this course of lectures we consider a problem of recognition of a finite group by its spectrum in the class of finite groups. Though for every finite group with a nontrivial normal solvable subgroup there exist infinitely many finite groups with the same spectrum, many finite simple groups are recognizable by spectrum. For example, the only finite group with spectrum {1,2,3,5} is the alternating group of degree 5. In spite of the apparent specificity of the problem, methods of its solution, which will be discussed, touch more or less on most fields of modern finite group theory. So, in some way, the recognition problem is just an occasion to talk about related interesting results. Nevertheless, the course has a specific ultimate goal. Namely, the last lecture is intended to outline a proof of the following recent result: if a finite simple group and a finite group have the same spectra and orders then they are isomorphic. 

It is planned to accompany the lectures by a number of seminars in order to support theory by concrete examples. In particular, we will find the spectra of some simple groups and investigate a number of recognizable and non-recognizable groups.

This course will consist of six lectures and two practical traninigs. The participans should know basic intro to classical groups andbasic facts about perm groups. It suffices to know

1. D. Gorenstein, Finite Groups, 1968 (Part I and  Chapters 10-12 of Par II).
2. R. Carter, Simple Groups of Lie Type, 1972.
3. R. Carter, Finite Groups of Lie Type, 1985 (Chapters 1-3).