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

Gordeev, Nikolai; Grunewald, Fritz; Kunyavskiĭ, Boris; Plotkin, Eugene

doi:10.1016/j.crma.2009.01.004

Group Theory

Baer–Suzuki theorem for the solvable radical of a finite group
[Le théorème de Baer–Suzuki pour le radical résoluble d'un groupe fini]

Présenté par : Jean-Pierre Serre

Gordeev, Nikolai ¹ ; Grunewald, Fritz ² ; Kunyavskiĭ, Boris ³ ; Plotkin, Eugene 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, Tome 347 (2009) no. 5-6, pp. 217-222.

Résumé
Abstract

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.

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.

Reçu le : 2008-09-05
Accepté le : 2009-01-09
Publié le : 2009-02-20

DOI : 10.1016/j.crma.2009.01.004

Affiliations des auteurs :

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

@article{CRMATH_2009__347_5-6_217_0,
     author = {Gordeev, Nikolai and Grunewald, Fritz and Kunyavski\u{i}, Boris and Plotkin, Eugene},
     title = {Baer{\textendash}Suzuki theorem for the solvable radical of a finite group},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {217--222},
     publisher = {Elsevier},
     volume = {347},
     number = {5-6},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.01.004/}
}

TY  - JOUR
AU  - Gordeev, Nikolai
AU  - Grunewald, Fritz
AU  - Kunyavskiĭ, Boris
AU  - Plotkin, Eugene
TI  - Baer–Suzuki theorem for the solvable radical of a finite group
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 217
EP  - 222
VL  - 347
IS  - 5-6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.01.004/
DO  - 10.1016/j.crma.2009.01.004
LA  - en
ID  - CRMATH_2009__347_5-6_217_0
ER  -

%0 Journal Article
%A Gordeev, Nikolai
%A Grunewald, Fritz
%A Kunyavskiĭ, Boris
%A Plotkin, Eugene
%T Baer–Suzuki theorem for the solvable radical of a finite group
%J Comptes Rendus. Mathématique
%D 2009
%P 217-222
%V 347
%N 5-6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.01.004/
%R 10.1016/j.crma.2009.01.004
%G en
%F CRMATH_2009__347_5-6_217_0

Gordeev, Nikolai; Grunewald, Fritz; Kunyavskiĭ, Boris; Plotkin, Eugene. Baer–Suzuki theorem for the solvable radical of a finite group. Comptes Rendus. Mathématique, Tome 347 (2009) no. 5-6, pp. 217-222. doi : 10.1016/j.crma.2009.01.004. http://www.numdam.org/articles/10.1016/j.crma.2009.01.004/

Bibliographie
Cité par

[1] Alperin, J.; Lyons, R. On conjugacy classes of p-elements, J. Algebra, Volume 19 (1971), pp. 536-537

[2] Baer, R. Engelsche Elemente Noetherscher Gruppen, Math. Ann., Volume 133 (1957), pp. 256-270

[3] Carter, R.W. Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, John Wiley & Sons, Chichester, 1985

[4] Ellers, E.; Gordeev, N. Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math., Volume 214 (2004), pp. 245-261

[5] Flavell, P. Finite groups in which every two elements generate a soluble group, Invent. Math., Volume 121 (1995), pp. 279-285

[6] Flavell, P. A weak soluble analogue of the Baer–Suzuki theorem http://web.mat.bham.ac.uk/P.J.Flavell/research/preprints (preprint, available at)

[7] Flavell, P. On the fitting height of a soluble group that is generated by a conjugacy class, J. London Math. Soc. (2), Volume 66 (2002), pp. 101-113

[8] P. Flavell, S. Guest, R. Guralnick, Characterizations of the solvable radical, submitted for publication

[9] Gordeev, N.; Grunewald, F.; Kunyavskiĭ, B.; Plotkin, E. On the number of conjugates defining the solvable radical of a finite group, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 387-392

[10] Gordeev, N.; Grunewald, F.; Kunyavskiĭ, B.; Plotkin, E. A commutator description of the solvable radical of a finite group, Groups, Geometry, and Dynamics, Volume 2 (2008), pp. 85-120

[11] Gordeev, N.; Grunewald, F.; Kunyavskiĭ, B.; Plotkin, E. A description of Baer–Suzuki type of the solvable radical of a finite group, J. Pure Appl. Algebra, Volume 213 (2009), pp. 250-258

[12] Gordeev, N.; Saxl, J. Products of conjugacy classes in Chevalley groups, I: Extended covering numbers, Israel J. Math., Volume 130 (2002), pp. 207-248

[13] Gorenstein, D.; Lyons, R. The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc., Volume 42 (1983) no. 276

[14] Gorenstein, D.; Lyons, R.; Solomon, R. The Classification of the Finite Simple Groups, Number 3, Math. Surveys and Monographs, vol. 40, Amer. Math. Soc., Providence, RI, 1998

[15] Gow, R. Commutators in finite simple groups of Lie type, Bull. London Math. Soc., Volume 32 (2000), pp. 311-315

[16] S. Guest, A solvable version of the Baer–Suzuki theorem and generalizations, Ph.D. Thesis, USC, May 2008

[17] S. Guest, A solvable version of the Baer–Suzuki theorem, Trans. Amer. Math. Soc., in press

[18] Guralnick, R.M.; Saxl, J. Generation of finite almost simple groups by conjugates, J. Algebra, Volume 268 (2003), pp. 519-571

[19] Kemper, G.; Lübeck, F.; Magaard, K. Matrix generators for the Ree groups ${}^{2}G_{2} (q)$ , Comm. Algebra, Volume 29 (2001), pp. 407-413

[20] Kleidman, P. The maximal subgroups of the Chevalley groups $G_{2} (q)$ with q odd, the Ree groups ${}^{2}G_{2} (q)$ , and their automorphism groups, J. Algebra, Volume 117 (1988), pp. 30-71

[21] Levchuk, V.M.; Nuzhin, Ya.N. Structure of Ree groups, Algebra i Logika, Volume 24 (1985) no. 1, pp. 26-41 (English transl. in Algebra and Logic, 24, 1985, pp. 16-26)

[22] Liebeck, M.W.; Saxl, J.; Seitz, G.M. Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc., Volume 65 (1992), pp. 297-325

[23] Malle, G. The maximal subgroups of ${}^{2}F_{4} (q^{2})$ , J. Algebra, Volume 139 (1991), pp. 52-69

[24] Springer, T.; Steinberg, R. Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin–New York, 1970, pp. 167-266

[25] R. Steinberg, Lectures on Chevalley Groups, Yale Univ., 1967

[26] Suzuki, M. On a class of doubly transitive groups, Ann. Math. (2), Volume 75 (1962), pp. 105-145

[27] Suzuki, M. Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. of Math. (2), Volume 82 (1965), pp. 191-212

[28] Thompson, J. Non-solvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 383-437

[29] Weigel, T.S. Generation of exceptional groups of Lie-type, Geom. Dedicata, Volume 41 (1992), pp. 63-87

Cité par Sources :