## 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).

Original language | English |
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Pages (from-to) | 250-258 |

Number of pages | 9 |

Journal | Journal of Pure and Applied Algebra |

Volume | 213 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2009 |

### Bibliographical note

Funding Information:Gordeev was supported in part by the INTAS grant N-05-1000008-8118 and RFBR grant N-08-01-00756-A. Kunyavskiĭ and Plotkin were supported in part by the Ministry of Absorption (Israel), the Israel Academy of Sciences grant 1178/06, and the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. A substantial part of this work was done during the visit of Kunyavskiĭ and Plotkin to MPIM (Bonn) in 2007 and discussed by all the coauthors during the international conference hosted by the Heinrich-Heine-Universität (Düsseldorf). The support of these institutions is highly appreciated.