On \(\mathfrak F_h\)-normal subgroups of finite groups.

*(English)*Zbl 1226.20011Only finite groups are considered. Let \(G\) be a group and let \(\mathfrak F\) be a formation. A subgroup \(H\) of \(G\) is said to be \(\mathfrak F_h\)-normal in \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(HT\) is a normal Hall subgroup of \(G\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathfrak F\)-hypercenter \(Z_\infty^{\mathfrak F}(G/H_G)\) of \(G/H_G\). We recall that the \(\mathfrak F\)-hypercenter is the largest normal subgroup of \(G\) whose \(G\)-chief factors are \(\mathfrak F\)-central (a chief factor \(H/K\) of \(G\) is \(\mathfrak F\)-central if \([H/K](G/C_G(H/K))\in\mathfrak F\)). This concept is an extension of the so-called \(\mathfrak F_n\)-supplemented subgroups considered by N. Yang and the second author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)]. (Notice that A. Y. Alsheik Ahmad, J. J. Jaraden and A. N. Skiba called them \(\mathfrak U_{\text{c}}\)-normal subgroups, for the class \(\mathfrak U\) of supersoluble groups, [in Algebra Colloq. 14, No. 1, 25-36 (2007; Zbl 1126.20012)].)

In the paper under review, the authors obtain some results about the influence of \(\mathfrak F_h\)-normality on the structure of groups.

As a sample, they prove: Theorem 3.1. Let \(\mathfrak F\) be a saturated formation containing \(\mathfrak U\), the class of supersoluble groups. Then \(G\in\mathfrak F\) if and only if there exists a soluble normal subgroup \(H\)of \(G\) such that \(G/H\in\mathfrak F\) and every maximal subgroup of every Sylow subgroup of \(F(H)\) not having a supersoluble supplement in \(G\) is \(\mathfrak U_h\)-normal in \(G\).

Some characterizations of supersoluble groups, soluble groups and \(p\)-nilpotent groups are also obtained by using this subgroup embedding property.

In the paper under review, the authors obtain some results about the influence of \(\mathfrak F_h\)-normality on the structure of groups.

As a sample, they prove: Theorem 3.1. Let \(\mathfrak F\) be a saturated formation containing \(\mathfrak U\), the class of supersoluble groups. Then \(G\in\mathfrak F\) if and only if there exists a soluble normal subgroup \(H\)of \(G\) such that \(G/H\in\mathfrak F\) and every maximal subgroup of every Sylow subgroup of \(F(H)\) not having a supersoluble supplement in \(G\) is \(\mathfrak U_h\)-normal in \(G\).

Some characterizations of supersoluble groups, soluble groups and \(p\)-nilpotent groups are also obtained by using this subgroup embedding property.

Reviewer: Ana MartĂnez-Pastor (Valencia)

##### MSC:

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

20D15 | Finite nilpotent groups, \(p\)-groups |

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

20D30 | Series and lattices of subgroups |

20D40 | Products of subgroups of abstract finite groups |

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\textit{X. Feng} and \textit{W. Guo}, Front. Math. China 5, No. 4, 653--664 (2010; Zbl 1226.20011)

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