Baer-Suzuki theorem for the solvable radical of a finite group

Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiǐ; Eugene Plotkin

doi:10.1016/j.crma.2009.01.004

Baer-Suzuki theorem for 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

20 Scopus citations

Abstract

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).

Original language	English
Pages (from-to)	217-222
Number of pages	6
Journal	Comptes Rendus Mathematique
Volume	347
Issue number	5-6
DOIs	https://doi.org/10.1016/j.crma.2009.01.004
State	Published - Mar 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. 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.

Funding

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.

Funders	Funder number
Russian Foundation for Basic Research
Minerva Foundation
INTAS	N-05-1000008-8118

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10.1016/j.crma.2009.01.004

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title = "Baer-Suzuki theorem for the solvable radical of a finite group",

abstract = "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).",

author = "Nikolai Gordeev and Fritz Grunewald and Boris Kunyavskiǐ 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. 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-Universit{\"a}t (D{\"u}sseldorf) 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.",

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AU - Plotkin, Eugene

N1 - Funding 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.

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N2 - 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).

AB - 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).

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ER -

Baer-Suzuki theorem for the solvable radical of a finite group

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