\(\mathfrak{F}_{\tau}\)-embedded and \(\mathfrak{F}_{\tau\phi}\)-embedded subgroups of finite groups
In the paper and in this review, all groups mentioned are finite and \(G\) stands for a finite group. Recall that a formation \(\mathfrak{F}\) is a class of groups such that it is closed under taking quotients and subdirect products. A formation is said to be saturated (respectively, solubly saturated) if \(G\in \mathfrak{F}\) whenever \(G/\Phi(G)\in\mathfrak{F}\) (respectively, \(G/\Phi(N)\in\mathfrak{F}\) for a soluble normal subgroup \(N\) of \(G\)). A chief factor \(L/K\) of \(G\) is \textit{\(\mathfrak{F}\)-central} if \((L/K)\rtimes (G/\mathrm{C}_G(L/K))\in\mathfrak{F}\). A normal subgroup \(N\) of \(G\) is said to be \(\mathfrak{F}\)-hypercentral (respectively, \(\mathfrak{F}\Phi\)-hypercentral) in \(G\) if either \(N=1\) or every chief factor (respectively, every non-Frattini chief factor) of \(G\) below \(N\) is \(\mathfrak{F}\)-central in \(G\). The \(\mathfrak{F}\)-hypercentre \(\mathrm{Z}_{\scriptstyle \mathfrak{F}}(G)\) (respectively, the \(\mathfrak{F}\Phi\)-hypercentre \(\mathrm{Z}_{\mathfrak{\scriptstyle F}\scriptstyle\Phi}(G)\)) of \(G\) is the product of all \(\mathfrak{F}\)-hypercentral (respectively, \(\mathfrak{F}\Phi\)-hypercentral) normal subgroups of~\(G\). A subgroup \(H\) of a group \(G\) is said to be quasinormal or permutable (respectively, S-quasinormal or S-permutable) in \(G\) if \(H\) permutes with all subgroups (respectively, all Sylow subgroups) of~\(G\). The influence on the structure of a group of the fact that some \(p\)-subgroups of a group satisfy a certain subgroup embedding property has been one of the major areas of interest in abstract group theory during the last years. Several properties have been introduced and studied. The aim of this paper is to give a common presentation of some known subgroup embedding properties and to unify some result that have parallel statements for the different embedding properties. Given a nonempty formation \(\mathfrak{F}\), a subgroup embedding functor \(\tau\) and a \(p\)-subgroup \(H\) of \(G\), we say that \(H\) is \(\mathfrak{F}_\tau\)-embedded (respectively, \(\mathfrak{F}_{\tau\Phi}\)-embedded) in \(G\) if, for some quasinormal subgroup \(\bar T\) of \(\bar G=G/H_G\) and some \(\tau\)-subgroup \(\bar S\) of \(\bar G\) contained in \(\bar H=H/H_G\), \(\bar H\bar T\) is S-quasinormal in \(\bar G\) and \(\bar H\cap \bar T\leq \bar S{\mathrm{Z}_{\mathfrak{{\scriptstyle F}}}(\bar G)}\) (respectively, \(\bar H\cap \bar T\leq \bar S{\mathrm{Z}_{\mathfrak{\scriptstyle F\scriptstyle\Phi}}(\bar G)}\)). A subgroup embedding property \(\tau\) is said to be inductive if for every group \(G\) and every \(N\trianglelefteq G\), if \(H\in \tau(G)\), then \(HN/N\in\tau(G/N)\); hereditary if when \(H\in \tau(G)\) and \(H\leq E\leq G\), then \(H\in \tau(E)\); regular (respectively, quasiregular) if, for every group \(G\), if \(H\in\tau(G)\) is a \(p\)-group and \(N\) is a minimal normal subgroup (respectively, an abelian minimal normal subgroup) of \(G\), then \(|G:\mathrm{N}_G(H\cap N)|\) is a power of \(p\); and \(\Phi\)-regular (respectively, \(\Phi\)-quasiregular) if the above property holds for every primitive group \(G\). One of the main results of the paper is the following one. Theorem 1.3. Let \(\mathfrak{F}\) be a solubly saturated formation containing all supersoluble groups and \(\tau\) a regular inductive subgroup functor. Let \(E\) be a normal subgroup of \(G\) such that \(G/E\in \mathfrak{F}\) and either \(X=E\) or \(X=\mathrm{F}^*(E)\). Suppose that for every noncyclic Sylow subgroup \(P\) of \(X\), every maximal subgroup \(H\) of \(P\) i \(\mathfrak{U}_\tau\)-embedded in~\(G\). Then \(G\in\mathfrak{F}\). Similar results are obtained by considering saturated formations instead of solubly saturated formations, \(\Phi\)-regular inductive subgroup functors, \(X=E\) or \(X=\mathrm{F}(E)\) (whenever \(E\) is soluble), and \(\mathfrak{U}_{\tau\Phi}\)-embedded subgroups (Theorem~1.4), and, in this last case, for a \(\Phi\)-regular hereditary inductive subgroup functor, when every maximal subgroup \(H\) of every noncyclic Sylow subgroup \(P\) of \(\mathrm{F}^*(E)\) is \(\mathfrak{U}_\tau\)-embedded in~\(G\) (Theorem~1.5). For minimal subgroups and subgroups of order~\(4\) of suitable normal subgroups, similar results are also obtained: Theorem~1.6. Let \(\mathfrak{F}\) be a solubly saturated formation containing all supersoluble groups and \(\tau\) either a quasiregular hereditary inductive subgroup functor or a regular inductive subgroup functor. Let \(E\) be a normal subgroup of \(G\) such that \(G/E\in\mathfrak{F}\). Suppose that for every noncyclic Sylow subgroup \(P\) of \(\mathrm{F}^*(E)\), every cyclic subgroup of \(P\) of order \(p\) or \(4\) (whenever \(P\) is not quaternion-free) is \(\mathfrak{U}_\tau\)-embedded in~\(G\). Then \(G\in\mathfrak{F}\). A similar result is obtained for saturated formations containing all supersoluble groups, a \(\Phi\)-quasiregular hereditary inductive subgroup functor \(\tau\) and assuming that every subgroup of order \(p\) or \(4\) (whenever \(P\) is not quaternion-free) of a non-cyclic Sylow subgroup \(P\) of \(G\) is \(\mathfrak{U}_\tau\)-embedded in~\(G\) (Theorem~1.7). Some applications of the main results are shown in Section~4 of the paper, because some known subgroup embedded properties satisfy the conditions of Theorems~1.3 and~1.6. These properties include normality, quasinormality, S-quasinormality, the \(\mathfrak{F}\)-hypercentral embedding, modularity, the cover and avoidance property (CAP-subgroups), SS-quasinormality, S-semipermutability, \(\tau\)-quasinormality, nearly S-permutable subgroups, the \(\Pi\)-property, \(\{1\leq G\}\)-embeddings, CAP\(^*\)-subgroups, S-quasinormal embeddings, complete c-permutability, complete c-semipermutability\dots.
\(\mathfrak{F}_{\tau\Phi}\)-embedded subgroup
scientific article; zbMATH DE number 6526401
scientific article