Baer–Suzuki theorem for the solvable radical of a finite group

Abstract

We prove that an element g of prime order q > 3 belongs to the solvable radical R ( G ) of a finite group if and only if for every x ∈ G the subgroup generated by g and x g x − 1 is solvable. This theorem implies that a finite group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup. To cite this article: N. Gordeev et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).