A description of Baer-Suzuki type of the solvable radical of a finite group

Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskii; Eugene Plotkin

doi:10.1016/j.jpaa.2008.06.006

A description of Baer-Suzuki type of the solvable radical of a finite group

Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskii, Eugene Plotkin

Department of Mathematics

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

17 Scopus citations

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 a₁, a₂, a₃ ∈ 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
Pages (from-to)	250-258
Number of pages	9
Journal	Journal of Pure and Applied Algebra
Volume	213
Issue number	2
DOIs	https://doi.org/10.1016/j.jpaa.2008.06.006
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.

Funding

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.

Funders	Funder number
INTAS	N-05-1000008-8118
Minerva Foundation
Russian Foundation for Basic Research
Ministry of Aliyah and Immigrant Absorption
Israel Academy of Sciences and Humanities	1178/06

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title = "A description of Baer-Suzuki type of the solvable radical of a finite group",

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 a1, a2, a3 ∈ G the subgroup generated by the elements g, ai g ai- 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).",

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

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-Universit{\"a}t (D{\"u}sseldorf). The support of these institutions is highly appreciated. ",

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doi = "10.1016/j.jpaa.2008.06.006",

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N1 - 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.

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N2 - 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 a1, a2, a3 ∈ G the subgroup generated by the elements g, ai g ai- 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).

AB - 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 a1, a2, a3 ∈ G the subgroup generated by the elements g, ai g ai- 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).

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A description of Baer-Suzuki type of the solvable radical of a finite group

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