A description of Baer–Suzuki type of the solvable radical of a finite group

Abstract

We obtain the following characterization of the solvable radical R ( G ) of any finite group G : R ( G ) coincides with the collection of all g ∈ G such that for any 3 elements a 1 , a 2 , a 3 ∈ G the subgroup generated by the elements g , a i g a i − 1 , i = 1 , 2 , 3 , is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on { 2 , 3 } ′ -elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).