On the number of conjugates defining the solvable radical of a finite group

Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiĭ; Eugene Plotkin

doi:10.1016/j.crma.2006.08.005

Group Theory

On the number of conjugates defining 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 343 (2006) no. 6, pp. 387-392.

Abstract
Résumé

We are looking for the smallest integer $k > 1$ providing the following characterization of the solvable radical $R (G)$ of any finite group G: $R (G)$ consists of the elements g such that for any k elements $a_{1}, a_{2}, \dots, a_{k} \in G$ the subgroup generated by the elements $g, a_{i} g a_{i}^{−1}$ , $i = 1, \dots, k$ , is solvable. Our method is based on considering a similar problem for commutators: find the smallest integer $ℓ > 1$ with the property that $R (G)$ consists of the elements g such that for any ℓ elements $b_{1}, b_{2}, \dots, b_{ℓ} \in G$ the subgroup generated by the commutators $[g, b_{i}]$ , $i = 1, \dots, ℓ$ , is solvable.

Nous cherchons le plus petit entier $k > 1$ caractérisant le radical résoluble $R (G)$ d'un groupe fini G comme suit : $R (G)$ est l'ensemble des éléments g tels que pour toute partie à k éléments ${a_{1}, a_{2}, \dots, a_{k}} \subset G$ le sous-groupe engendré par les élements $g, a_{i} g a_{i}^{−1}$ , $i = 1, \dots, k$ , est résoluble. Notre méthode s'appuie sur la considération d'un problème similaire pour les commutateurs. Nous cherchons le plus petit entier $ℓ > 1$ ayant la propriété suivante : $R (G)$ est l'ensemble des éléments g tels que pour toute partie à ℓ éléments ${b_{1}, b_{2}, \dots, b_{ℓ}} \subset G$ le sous-groupe engendré par les commutateurs $[g, b_{i}]$ , $i = 1, \dots, ℓ$ , est résoluble.

Received: 2006-07-12
Accepted: 2006-08-23
Published online: 2006-09-15

DOI: 10.1016/j.crma.2006.08.005

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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     title = {On the number of conjugates defining the solvable radical of a finite group},
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Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskiĭ; Eugene Plotkin. On the number of conjugates defining the solvable radical of a finite group. Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 387-392. doi : 10.1016/j.crma.2006.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.08.005/

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