On p-CAP-subgroups of finite groups

Abstract

Suppose that G is a finite group and H is a subgroup of G. H is said to be a p-CAP-subgroup of G if H either covers or avoids each pd-chief factor of G. We give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups of G.

Keywords

Maximal subgroups / 2-maximal subgroups / p-CAP-subgroups

References

[1] Ballester-Bolinches A, Ezquerro L M. Classes of Finite Groups. Berlin: Springer, 2006

[2] Gillam J D. Cover-avoid subgroups in finite solvable groups. J Algebra, 1974, 29: 324–329

[3] Doerk K, Hawkes T. Finite Solvable Groups. Berlin: Walter de Gruyter, 1992

[4] Ezquerro L M. A contribution to the theory of finite supersolvable groups. Rend Semin Mat Univ Padova, 1993, 89: 161–170

[5] Fan Y, Guo X, Shum K P. Remarks on two generalizations of normality of subgroups. Chinese Ann Math Ser A, 2006, 27: 169–176 (in Chinese)

[6] Gaschütz W. Praefrattinigruppen. Arch Math, 1963, 13: 418–426

[7] Guo X, Shum K P. Cover-avoidance properties and the structure of finite groups. J Pure Appl Algebra, 2003, 181: 297–308

[8] Huppert B, Blackburn N. Finite Groups III. Berlin-New York: Springer-Verlag, 1982

[9] He X, Wang Y. On p-cover-avoid and S-quasinormally embedded subgroups in finite groups. J Math Res Exp, 2010, 30: 743–750

[10] Liu X, Ding N. On chief factors of finite groups. J Pure Appl Algebra, 2007, 210: 789–796

[11] Tomkinson M J. Cover-avoidance properties in finite solvable groups. Canad Math Bull, 1976, 19(2): 213–216

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Higher Education Press 2020