Group Theory
Baer–Suzuki theorem for the solvable radical of a finite group
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 217-222.

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 $xgx−1$ 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 $x∈G$ le sous-groupe engendré par x et $xgx−1$ 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2009.01.004

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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