On the number of conjugates defining the solvable radical of a finite group

Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiǐ; Eugene Plotkin

doi:10.1016/j.crma.2006.08.005

On the number of conjugates defining the solvable radical of a finite group

Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskiǐ, Eugene Plotkin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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 a₁, a₂, ..., 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 b₁, b₂, ..., 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
Pages (from-to)	387-392
Number of pages	6
Journal	Comptes Rendus Mathematique
Volume	343
Issue number	6
DOIs	https://doi.org/10.1016/j.crma.2006.08.005
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.

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title = "On the number of conjugates defining the solvable radical of a finite group",

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 a1, a2, ..., ak ∈ G the subgroup generated by the elements g, ai g ai-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 b1, b2, ..., bℓ ∈ G the subgroup generated by the commutators [g, bi], i = 1, ..., ℓ, is solvable. To cite this article: N. Gordeev et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).",

author = "Nikolai Gordeev and Fritz Grunewald and Boris Kunyavskiǐ and Eugene Plotkin",

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{\textquoteright}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.",

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AU - Plotkin, Eugene

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N2 - 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 a1, a2, ..., ak ∈ G the subgroup generated by the elements g, ai g ai-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 b1, b2, ..., bℓ ∈ G the subgroup generated by the commutators [g, bi], i = 1, ..., ℓ, is solvable. To cite this article: N. Gordeev et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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On the number of conjugates defining the solvable radical of a finite group

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