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

Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiĭ; Eugene Plotkin

doi:10.1016/j.crma.2009.01.004

Group Theory

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

Presented by: Jean-Pierre Serre

Nikolai Gordeev ¹; Fritz Grunewald ²; Boris Kunyavskiĭ ³; Eugene Plotkin ³

¹ Department of Mathematics, Herzen State Pedagogical University, 48 Moika Embankment, 191186 St. Petersburg, Russia
² Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
³ Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel

Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 217-222.

Abstract
Résumé

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 $x \in G$ 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.

Nous démontrons qu'un élément g d'ordre premier $q > 3$ appartient au radical résoluble $R (G)$ d'un groupe fini G si et seulement si pour tout $x \in G$ le sous-groupe engendré par x et $x g x^{- 1}$ est résoluble. Ce théorème implique qu'un groupe fini G est résoluble si et seulement si dans chaque classe de conjugaison de G tout couple d'éléments engendre un sous-groupe résoluble.

Received: 2008-09-05
Accepted: 2009-01-09
Published online: 2009-02-20

DOI: 10.1016/j.crma.2009.01.004

Author's affiliations:

Nikolai Gordeev ¹; Fritz Grunewald ²; Boris Kunyavskiĭ ³; Eugene Plotkin ³

¹ Department of Mathematics, Herzen State Pedagogical University, 48 Moika Embankment, 191186 St. Petersburg, Russia
² Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
³ Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel

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Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiĭ; Eugene Plotkin. Baer–Suzuki theorem for the solvable radical of a finite group. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 217-222. doi : 10.1016/j.crma.2009.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.004/

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