We call H an SΦ-supplemented subgroup of a finite group G if there exists a subnormal subgroup T of G such that G = HT and H ∩ T ≤ Φ(H), where Φ(H) is the Frattini subgroup of H. In this paper, we characterize the p-nilpotency and supersolubility of a finite group G under the assumption that every subgroup of a Sylow p-subgroup of G with given order is SΦ-supplemented in G: Some results about formations are also obtained.
Similar content being viewed by others
References
A. N. Skiba, “On weakly s-permutable subgroups of finite groups,” J. Algebra, 315, No. 1, 192–209 (2007).
A. N. Skiba, “On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups,” J. Group Theory, 13, 841–850 (2010).
A. N. Skiba, “A characterization of the hypercyclically embedded subgroups of finite groups,” J. Pure Appl. Algebra, 215, No. 3, 257–261 (2011).
B. Huppert, Endliche Gruppen I, Springer, New York–Berlin (1967).
B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin–New York (1982).
D. Gorenstein, Finite Groups, Chelsea, New York (1968).
D. J. S. Robinson, A Course in the Theory of Groups, Springer, New York–Berlin (1996).
Deyu Li and Xiuyun Guo, “The influence of c-normality of subgroups on the structure of finite groups,” J. Pure Appl. Algebra, 150, 53–60 (2000).
Huaquan Wei and Yanming Wang, “On c*-normality and its properties,” J. Group Theory, 10, No. 2, 211–223 (2007).
K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter, Berlin–New York (1992).
M. Asaad, “Finite groups with certain subgroups of Sylow subgroups complemented,” J. Algebra, 323, No. 7, 1958–1965 (2010).
M. Weinstein (editor), Between Nilpotent and Solvable, Polygonal, Passaic, NJ (1982).
W. Guo, The Theory of Classes of Groups, Kluwer, Dordrecht (2000).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64 No. 1 pp. 92–99 January 2012.
Rights and permissions
About this article
Cite this article
Li, X., Zhao, T. SΦ-supplemented subgroups of finite groups. Ukr Math J 64, 102–109 (2012). https://doi.org/10.1007/s11253-012-0632-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0632-2