Nikolai Gordeev

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveCrd=G, wherer=rankG.

Products of conjugacy classes in Chevalley groups I. Extended covering numbers

This paper is concerned with products of conjugacy classes in Chevalley groups. We prove that in any quasisimple Chevalley groupG proper or twisted, over any field, the extended covering number is bounded above linearly in terms of the rank ofG, that is, for some constante, for any Chevalley groupG, the product of anye · rank(G) non-central classes covers all ofG. We give estimates for the constante in different cases.

A commutator description of the solvable radical of a finite group

We are looking for the smallest integer k> 1providing the following characteri- zation of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a1 ;a 2 ;:::;a k 2 G the subgroup generated by the elements g; ai ga � 1 i , i D 1; :::;k , is solvable. We consider a similar problem of finding the smallest integer `>1 with the property that R.G/ coincides with the collection of all g 2 G such that for anyelements b1 ;b 2 ;:::;b ` 2 G the subgroup generated by the commutators Œg; bi � , i D 1; :::;` , is solvable. Conjecturally, k DD 3. We prove that both k andare at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

Gauss decomposition with prescribed semisimple part in chevalley groups iii: Finite twisted groups

Continuing the investigations of [EG] and [EGII], we shall show that Theorem 1 below is also valid for twisted CheMLley groups over finite fields. Let G be such a group. Here we consider only groups G 2 ZF/2, where z is a universal Chevalley group over s finite field K, F is an automorphism of z, and Z is a subgroup of G contained in the center z(@). Suppose B = HU is a Bore1 subgroup of G. Let I? be a group generated by G and some element o normalizing G in I? and acting on G as diagonal automorphism.

Products of conjugacy classes in Chevalley groups II. Covering and generation

In this paper we establish a relationship between generating numbers and covering numbers of conjugacy classes in Chevalley groups over algebraically closed fields.