## Abstract

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R (G) of any finite group G: R (G) consists of the elements g such that for any k elements a_{1}, a_{2}, ..., a_{k} ∈ G the subgroup generated by the elements g, a_{i} g a_{i}^{-1}, i = 1, ..., k, is solvable. Our method is based on considering a similar problem for commutators: find the smallest integer ℓ > 1 with the property that R (G) consists of the elements g such that for any ℓ elements b_{1}, b_{2}, ..., b_{ℓ} ∈ G the subgroup generated by the commutators [g, b_{i}], i = 1, ..., ℓ, is solvable. To cite this article: N. Gordeev et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Original language | English |
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Pages (from-to) | 387-392 |

Number of pages | 6 |

Journal | Comptes Rendus Mathematique |

Volume | 343 |

Issue number | 6 |

DOIs | |

State | Published - 15 Sep 2006 |

### Bibliographical note

Funding Information:Kunyavski˘ı and Plotkin were partially supported by the Ministry of Absorption (Israel), the Israeli Science Foundation founded by the Israeli Academy of Sciences – Center of Excellence Program, the Minerva Foundation through the Emmy Noether Research Institute of Mathematics, and by the RTN network HPRN-CT-2002-00287. A substantial part of this work was done during Gordeev’s visits to Bar-Ilan University in May 2005 and May 2006 (partially supported by the same RTN network) and the visit of Grunewald, Kunyavski˘i and Plotkin to MPIM (Bonn) during the activity on “Geometry and Group Theory” in July 2006. The support of these institutions is highly appreciated. We are very grateful to J.N. Bray, B.I. Plotkin, and N.A. Vavilov for useful discussions and correspondence.