Solvability of finite groups

Abstract

H is called an {{\displaystyle \mathcal{M}}}_p-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that H_p ∈ Syl_p (B) and B is {{\displaystyle \mathcal{M}}}_p-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use {{\displaystyle \mathcal{M}}}_p-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that \mathrm{1}\langle d\le \left|P\right| and d divides |P|. If every subgroup H of P with \left|H\right|=d is {{\displaystyle \mathcal{M}}}_5-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5'-group, (3) I/C\cong A_5

Keywords

Composition factor / {{\displaystyle \mathcal{M}}}_p-embedded subgroup / primary subgroup / solvable

References

[1] AradZ, WardM B. New criteria for the solvability of finite groups. J Algebra, 1982, 77(1): 234–246

[2] Ballester-BolinchesA, WangY, GuoX.C-supplemented subgroups of finite groups. Glasg Math J, 2000, 42(3): 383–389

[3] ChenZhongmu. Inner Outer Σ-Group and Minimal Non Σ-Group. Chongqing: Southwest Normal University Press, 1988 (in Chinese)

[4] DoerkK, HawkesT. Finite Solvable Groups. Berlin: Walter de Gruyter, 1992

[5] GuoW. The Theory of Classes of Groups. Beijing/New York: Science Press/Kluwer Academic Publishers, 2000

[6] HallP. A characteristic property of solvable groups. J Lond Math Soc, 1937, 12: 198–200

[7] HelielA A. A note on c-supplemented subgroups of finite groups. Comm Algebra, 2014, 42(4): 1650–1656

[8] HuppertB, BlackburnN. Finite Groups III. Berlin: Springer-Verlag, 1982

[9] MiaoL, LempkenW. On M-supplemented subgroups of finite groups. J Group Theory, 2009, 12(2): 271–287

[10] MiaoL, QianG. A condition for the solvability of finite groups. Sib Math J, 2009, 50(4): 687–691

[11] MonakhovV S, ShnyparkovA V. On the p-supersolvability of a finite group with an M-supplemented Sylow p-subgroup. Sib Math J, 2009, 50(4): 681–686

[12] TangJ, MiaoL. On the Mp-supplemented subgroups of finite groups. Comm Algebra, 2013, 41(5): 1913–1922

[13] TateJ. Nilpotent quotient groups. Topology, 1964, 3: 109–111

[14] ZhangJ, MiaoL. On Mp-embedded primary subgroups of finite groups. Studia Sci Math Hungar, 2016, 53(4): 429–439

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag GmbH Germany