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

18 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.

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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.",

year = "2009",

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doi = "10.1016/j.crma.2009.01.004",

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AU - Gordeev, Nikolai

AU - Grunewald, Fritz

AU - Kunyavskiǐ, Boris

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.

PY - 2009/3

Y1 - 2009/3

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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SN - 1631-073X

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JO - Comptes Rendus Mathematique

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IS - 5-6

ER -

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

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