On conjugate-permutable subgroups of finite groups. (Q2847708)

The groups in the paper under review and in this report are assumed to be finite. The study of the influence of the maximal subgroups of the Sylow subgroups of a distinguished normal subgroup on the structure of a group has motivated a lot of interest during the last years in the research in abstract group theory. This paper is a new contribution in this research line. Here a complete set \(\mathfrak Z\) of Sylow subgroups of a finite group \(G\) is considered, that is, for each prime \(p\) dividing the order of \(G\), \(\mathfrak Z\) contains exactly one Sylow \(p\)-subgroup \(G_p\) of \(G\). Given a non-empty subset \(C\) of \(G\), a subgroup \(H\) of \(G\) is said to be \(C\)-\(\mathfrak Z\)-permutable (conjugate-\(\mathfrak Z\)-permutable) subgroup of \(G\) if there exists some \(x\in C\) such that \(H^xG_p=G_pH^x\) for all \(G_p\in\mathfrak Z\). In all this report, \(\mathfrak Z\) will denote a complete set of Sylow subgroups of a group \(G\) and \(C\) will be a soluble normal subgroup of \(G\). In addition, \(\mathfrak F\) will denote a saturated formation containing the class \(\mathfrak U\) of all supersoluble groups. Recall that a saturated formation is a class of groups closed under quotients, subdirect products, and Frattini extensions.NEWLINENEWLINE Theorem~3.1 states that if \(p\) is the smallest prime number dividing the order of \(G\) and the maximal subgroups of \(G_p\in\mathfrak Z\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\), then \(G\) is \(p\)-nilpotent. In Theorem~3.2, it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) in \(G\) such that \(G/H\in\mathfrak F\) and the maximal subgroups of \(G_p\cap H\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\) for all \(G_p\in\mathfrak Z\). In Theorem~3.5, it is shown that \(G\in\mathfrak F\) if and only if there exists a soluble normal subgroup \(H\) of \(G\) such that \(G/H\in\mathfrak F\) and the maximal subgroups of the Sylow subgroups of the Fitting subgroup \(F(H)\) of \(H\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\). In Theorem~3.9, it is shown that if \(H\) is a normal subgroup of \(G\) such that \(G/H\) is supersoluble and \(C\) is a soluble normal subgroup of the generalised Fitting subgroup \(F^*(H)\) and the maximal subgroups of \(G_p\cap F^*(H)\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\) for all \(G_p\in\mathfrak F\), then \(G\) is supersoluble. In Theorem~3.11 it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) of \(G\) and a soluble normal subgroup \(C\) of \(F^*(H)\) such that \(G/H\in\mathfrak F\) and the maximal subgroups of \(G_p\cap F^*(H)\) are \(C\)-\(\mathfrak Z\)-permutable subgroups of \(G\) for all \(G_p\in\mathfrak Z\). Theorem~3.12 shows that in Theorem~3.11 we can replace the maximal subgroups of \(G_p\cap F^*(H)\) by the maximal subgroups of \(G_p\cap F_n^*(H)\), where \(F_n^*(H)\) denotes the \(n\)-th term of the generalised Fitting series of \(H\).NEWLINENEWLINE Other known results based on other embedding properties are generalised with these theorems.