Abstract
Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O 2(G), andTεSyl2(G), thenG=C(G; T)K(G), whereC(G; T)=〈N G (T 0)|1≠T 0 charT〉 andK(G) is the product of all near components ofG of typeL 2(2n) orA 2 n +1. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Aschbacher,On the failure of the Thompson factorization in 2-constrained groups, Proc. London Math. Soc.(3)43 (1981), 425–449.
M. Aschbacher,A factorization theorem for 2-constrained groups, Proc. London Math. Soc.(3)43 (1981), 450–477.
M. Aschbacher,GF(2)-representations of finite groups, Amer. J. Math.104 (1982), 683–771.
M. Aschbacher and G. Seitz,Involutions in Chevalley groups over fields of even order, Nagoya J. Math.63 (1976), 1–91.
B. Baumann,Über endliche Gruppen mit einer zu L 2(2n)Isomorphen Faktorgruppe, Proc. Amer. Math. Soc.74 (1979), 215–222.
J. H. Conway et al.,An Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
G. Glauberman,Failure of factorization in p-solvable groups, Quart. J. Math.24 (1973), 71–77.
G. Glauberman,Central elements in core-free groups, J. Algebra4 (1966), 403–420.
G. Glauberman and R. Niles,A pair of characteristic subgroups for pushing-up finite groups, Proc. London Math. Soc.(3)46 (1983), 411–453.
D. Gorenstein,Finite Groups (2nd edition), Chelsea Publishing Co., New York, 1980.
D. Gorenstein and R. Lyons,The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc.276 (1983), 1–731.
L. Kaloujnine,La structure des p-groupes de Sylow des groupes symétriques finis, Ann. Sci. Ecole Norm. Sup.65 (1948), 239–276.
J. McLaughlin,Some subgroups of SL n (F 2), Ill. J. Math.13 (1969), 108–115.
U. Meierfrankenfeld and G. Stroth,On quadratic GF(2)-modules for Chevalley groups over fields of odd order, Arch. Math.55 (1990), 105–110.
R. Niles,Pushing-up in finite groups, J. Algebra57 (1979), 26–63.
R. Steinberg,Lecture notes on Chevalley Groups, Mimeographed notes, Yale University, 1967.
B. Stellmacher,Einfache Gruppen die von einer Konjugiertenklasse von Elementen der Ordnung 3 erzeugt werden, J. Algebra30 (1974), 320–356.
B. Stellmacher,Pushing up, Arch. Math.46 (1986), 8–17.
J. G. Thompson,Nonsolvable finite groups all of whose local subgroups are solvable, I. Bull. Amer. Math. Soc.74 (1968), 383–437.
F. G. Timmesfeld,Groups with weakly closed TI-subgroups, Math. Zeit.143 (1975), 243–278.
Author information
Authors and Affiliations
Additional information
For John Thompson
Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.
Rights and permissions
About this article
Cite this article
Gorenstein, D., Lyons, R. On aschbacher’s localC(G; T) theorem. Israel J. Math. 82, 227–279 (1993). https://doi.org/10.1007/BF02808113
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02808113