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).
Bibliographical noteFunding Information:
Gordeev was supported in part by the INTAS grant N-05-1000008-8118 and RFBR grant N-08-01-00756-A. A substantial part of this work was done during the visits of Kunyavski˘ı and Plotkin to MPIM (Bonn) in 2007 and of Grunewald to Israel in 2008. It was discussed by all the coauthors during the international conference hosted by the Heinrich-Heine-Universität (Düsseldorf) and by the last three of them during the Oberwolfach meeting “Profinite and Asymptotic Group Theory” in 2008. The visits of Grunewald, Kunyavski˘ı, and Plotkin were supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. The support of all above institutions is highly appreciated.