On \(\mathcal F\)-supplemented subgroups of finite groups.

*(English)*Zbl 1172.20019All groups involved in this review are finite. Recall that a formation is a class of groups which is closed under taking epimorphic images and subdirect products. If \(\mathcal F\) is a formation and \(G\) is a group, a subgroup \(H\) of \(G\) is said to be \(\mathcal F\)-supplemented in \(G\) if there exists a subgroup \(T\) of \(G\) such that \(G=TH\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathcal F\)-hypercenter \(Z^{\mathcal F}_\infty(G/H_G)\) of \(G/H_G\), that is, the largest normal subgroup of \(G/H_G\) for which every chief factor of \(G/H_G\) below it is \(\mathcal F\)-central in \(G/H_G\). Of course normal or complemented subgroups (more generally, c-supplemented subgroups introduced by X. Guo, Y. Wang and the reviewer [in Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]) are examples of \(\mathcal F\)-supplemented subgroups for every non-empty formation \(\mathcal F\).

In this paper, the influence of \(\mathcal F\)-supplemented subgroups on the structure of a group is analyzed. For instance, a group \(G\) is soluble if and only if every subgroup of prime order (respectively, every Sylow subgroup) is supplemented in \(G\) with respect to the formation of all soluble groups (Theorems 4.1 and 4.2). The author also proves that if \(\mathcal F\) is a subgroup-closed saturated formation containing all supersoluble groups and \(G\) is a group with a subgroup \(E\) containing the \(\mathcal F\)-residual of \(G\), then \(G\) belongs to \(\mathcal F\) provided that every subgroup of prime order or order \(4\) of \(E\) is supplemented in \(G\) with respect to the formation of all supersoluble groups. The same conclusion follows if the subgroups of prime order or order \(4\) are replaced by the family of all maximal subgroups of the non-cyclic Sylow subgroups of \(E\) not having a supersoluble supplement in \(G\). A series of known results about normal, c-normal and c-supplemented subgroups are unified and generalized.

Remarks: It should be noted that the concept of c-supplemented subgroup was introduced by Guo, Wang and the reviewer [in loc. cit.] and not by Y. Wang [in J. Algebra 224, No. 2, 467-478 (2000; Zbl 0953.20010)], as the author claims in the paper.

In this paper, the influence of \(\mathcal F\)-supplemented subgroups on the structure of a group is analyzed. For instance, a group \(G\) is soluble if and only if every subgroup of prime order (respectively, every Sylow subgroup) is supplemented in \(G\) with respect to the formation of all soluble groups (Theorems 4.1 and 4.2). The author also proves that if \(\mathcal F\) is a subgroup-closed saturated formation containing all supersoluble groups and \(G\) is a group with a subgroup \(E\) containing the \(\mathcal F\)-residual of \(G\), then \(G\) belongs to \(\mathcal F\) provided that every subgroup of prime order or order \(4\) of \(E\) is supplemented in \(G\) with respect to the formation of all supersoluble groups. The same conclusion follows if the subgroups of prime order or order \(4\) are replaced by the family of all maximal subgroups of the non-cyclic Sylow subgroups of \(E\) not having a supersoluble supplement in \(G\). A series of known results about normal, c-normal and c-supplemented subgroups are unified and generalized.

Remarks: It should be noted that the concept of c-supplemented subgroup was introduced by Guo, Wang and the reviewer [in loc. cit.] and not by Y. Wang [in J. Algebra 224, No. 2, 467-478 (2000; Zbl 0953.20010)], as the author claims in the paper.

Reviewer: Adolfo Ballester-Bolinches (Burjasot)

##### MSC:

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D40 | Products of subgroups of abstract finite groups |

##### Keywords:

finite soluble groups; supersoluble groups; saturated formations; supplemented subgroups; complemented subgroups; Sylow subgroups
Full Text:
DOI

##### References:

[1] | AlSheik Ahmad A.: Finite groups with given c-permutable subgroups. Algebra Discrete Math 3, 74–85 (2004) · Zbl 1067.20018 |

[2] | Arad Z., Ward M.B.: New criteria for the solvability of finite groups. J. Algebra 77, 234–246 (1982) · Zbl 0486.20018 |

[3] | Ballester-Bolinches A., Xiuyun G.: On complemented subgroups of finite groups. Arch. Math 72, 161–166 (1999) · Zbl 0929.20015 |

[4] | Ballester-Bolinches A., Wang Y.: Finite groups with C-normal minimal subgroups. J. Pure Appl. Algebra 153, 121–127 (2000) · Zbl 0967.20009 |

[5] | Buckley J.: Finite groups whose minimal subgroups are normal. Math. Z 15, 15–17 (1970) · Zbl 0202.02303 |

[6] | Doerk K., Hawkes T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992) · Zbl 0753.20001 |

[7] | Guo W.: The Theory of Classes of Groups. Science Press/Kluwer, Beijing/New York (2000) · Zbl 1005.20016 |

[8] | Guo W., Shum K.P., Skiba A.N.: G-covering subgroup systems for the classes of supersoluble and nilpotent groups. Israel J. Math. 138, 125–138 (2003) · Zbl 1050.20009 |

[9] | Hall P.: A characteristic property of soluble groups. J. London Math. Soc 12, 188–200 (1937) · JFM 63.0069.02 |

[10] | Huppert B.: Endliche Gruppen I. Springer, Berlin (1967) · Zbl 0217.07201 |

[11] | Kegel, O.H.: On Huppert’s characterization of finite supersoluble groups. In: Proceedings of the International Conference on Theory Groups, Canberra, 1965, New York, 1967, pp. 209–215 |

[12] | Kegel O.H.: Produkte nilpotenter gruppen. Arch. Math. (Basel) 12, 90–93 (1961) · Zbl 0099.01401 |

[13] | Li D., Guo X.: The influence of c-normality of subgroups on the structure of finite groups II. Comm. Algebra 26, 1913–1922 (1998) · Zbl 0907.20052 |

[14] | Robinson D.J.S.: A Course in the Theory of Groups. Springer, New York (1982) · Zbl 0483.20001 |

[15] | Shemetkov, L.A.: Formations of finite groups, Moscow, Nauka, Main Editorial Board for Physical and Mathematical Literature (1978) · Zbl 0496.20014 |

[16] | Shemetkov, L.A., Skiba, A.N.: Formations of Algebraic Systems, Moscow, Nauka, Main Editorial Board for Physical and Mathematical Literature (1989) · Zbl 0667.08001 |

[17] | Skiba A.N.: On weakly S-permutable subgroups of finite groups. J. Algebra 315, 192–209 (2007) · Zbl 1130.20019 |

[18] | Srinivasan S.: Two sufficient conditions for supersolubility of finite groups. Israel J. Math. 3(35), 210–214 (1980) · Zbl 0437.20012 |

[19] | Wang Y.: c-normality of groups and its properties. J. Algebra 180, 954–965 (1996) · Zbl 0847.20010 |

[20] | Wang Y.: Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra 224, 467–478 (2000) · Zbl 0953.20010 |

[21] | Wielandt H.: Subnormal subgroups and permutation groups. Lectures given at the Ohio State University, Columbus, OH (1971) |

[22] | Xu M.: An Introduction to Finite Groups. Science Press, Beijing (1999) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.