Abstract
The normalizer of each Sylow subgroup of a finite group G has a nilpotent Hall supplement in G if and only if G is soluble and every tri-primary Hall subgroup H (if exists) of G satisfies either of the following two statements: (i) H has a nilpotent bi-primary Hall subgroup; (ii) Let π(H) = {p, q, r}. Then there exist Sylow p-, q-, r-subgroups H p , H q , and H r of H such that H q ⊆ N H (H p ), H r ⊆ N H (H q ), and H p ⊆ N H (H r ).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Huppert B. and Blackburn N., Finite Groups. III, Springer-Verlag, Berlin; Heidelberg; New York (1982).
Guo W., “On normalizers of Sylow subgroups,” Dokl. Akad. Nauk BSSR, 37, No. 4, 22–24 (1993).
D’Aniello A., Vivi De, and Giordano G., “Saturated formations and Sylow normalizers,” Bull. Austral. Math. Soc., 69, 522–548 (2004).
Kondrat’ev A. S., “A criterion for 2-nilpotency of finite groups,” in: Subgroup Structure of Groups [in Russian], Sverdlovsk, 1988, pp. 82–84.
Chigira N., “Number of Sylow subgroups and p-nilpotence of finite groups,” J. Algebra, 201, 71–85 (1998).
Zhang J., “Sylow numbers of finite groups,” J. Algebra, 176, 111–123 (1995).
Guo W., “Finite groups with given indices of normalizers of Sylow subgroups,” Siberian Math. J., 37, No. 253–257, (1996).
Guo W. and Shum K. P., “A note on finite groups whose normalizers of Sylow 2-, 3-subgroups are prime power indices,” J. Appl. Algebra Discrete Struct., 3, 1–9 (2005).
Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin; New York (1992).
Guo W., The Theory of Classes of Groups, Sci. Press-Kluwer Acad. Publ., Beijing; New York; Dordrecht; Boston; London (2000).
Wielandt H., “Zum Satz von Sylow,” Math. Z., Bd 60, 407–408 (1954).
Wielandt H., “Über das Produkt paarweise vertauschbarer nilpotenter Gruppen,” Math. Z., Bd 55, 1–7 (1951).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2009 Li B., Guo W., and Huang J.
The authors were supported by the NNSF of P. R. China (Grant 10771180), the Scientific Research Fund of the Sichuan Provincial Education Department (Grant 08zb059), and the Research Program of the Chengdu University of Information Technology (Grant KYTZ200909).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 4, pp. 841–850, July–August, 2009.
Rights and permissions
About this article
Cite this article
Li, B., Guo, W. & Huang, J. Finite groups in which Sylow normalizers have nilpotent Hall supplements. Sib Math J 50, 667–673 (2009). https://doi.org/10.1007/s11202-009-0075-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-009-0075-7