On \(\mathfrak F_h\)-normal subgroups of finite groups.
Only 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 \textit{N. Yang} and the second author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)]. (Notice that \textit{A. Y. Alsheik Ahmad, J. J. Jaraden} and \textit{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.
scientific article; zbMATH DE number 5825459
scientific article