A commutator description of the solvable radical of a finite group

Abstract

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.