Finite groups with hereditarily \(g\)-permutable Schmidt subgroups (Q6564345)

It is well-known that if \(A\) is a normal subgroup of a group \(G\), then \(AB\) is a subgroup of \(G\) for every subgroup \(B\) of \(G\). This implies that \(AB=BA\). This property of normal subgroups has many generalizations in different directions. The authors of the paper use the following:\N\NDefinition. Let \(K\) be a subgroup of a group \(G\). \(K\) is said to be \textit{hereditarily \(G\)-permutable} (or \textit{\(G\)-\(h\)-permutable}, for short) in \(G\) if for every subgroup \(B\) of \(G\) there exists \(g\in\langle K,B\rangle\) such that \(KB^g=B^gK\).\N\NRecall that a group \(G\) is said to be a minimal non-nilpotent group or Schmidt group if \(G\) is not nilpotent and every proper subgroup of \(G\) is nilpotent. For example, it is clear that every finite non-nilpotent group contains Schmidt subgroups.\N\NThe authors earlier proved [Mediterr. J. Math. 20, No. 3, Paper No. 174, 12 pp. (2023; Zbl 1514.20059)] that if every Schmidt subgroup of a finite group \(G\) is \(G\)-\(h\)-permutable in \(G\), then \(G\) is soluble. The main goal of this paper is to complete the structural study of finite groups in which every Schmidt subgroup of a group \(G\) is \(G\)-\(h\)-permutable under the restriction that the Frattini subgroup is trivial.\N\NTo formulate the result we need two more definitions. Let \(p_1>p_2>\cdots>p_r\) be the primes dividing \(|G|\) and let \(P_i\) be a Sylow \(p_i\)-subgroup of \(G\), for each \(i=1,2,\ldots,r\). The finite group \(G\) is called \textit{a Sylow tower group of supersoluble type} if all subgroups \(P_1, P_1P_2,\ldots, P_1P_2\cdots P_{r-1}\) are normal in \(G\). The class of all Sylow tower groups of supersoluble type is denoted by \(\mathfrak{D}\). If \(\mathfrak{F}\) is a nonempty class of groups and \(\pi\) is a set of primes, then \(\mathfrak{F}_\pi\) is the class of all \(\pi\)-groups in \(\mathfrak{F}\). In particular, if \(p\) is a prime, then \(\mathfrak{N}_p\) is the class of all \(p\)-groups and \(\mathfrak{D}_{\pi(p-1)}\) is the class of all Sylow tower groups \(G\) of supersoluble type such that every prime dividing \(|G|\) also divides \(p-1\).\N\NTheorem. Let \(G\) be a finite group with \(\Phi(G)=1\). Assume that \(\mathfrak{F}=LF(F)\) is the saturated formation locally defined by the canonical local definition \(F\) such that \(F(p)=\mathfrak{N}_p\mathfrak{D}_{\pi(p-1)}\) for every prime \(p\). If every Schmidt subgroup of \(G\) is \(G\)-\(h\)-permutable in \(G\), then the following statements hold:\N\N(1) \(G=[\mathrm{Soc}(G)]M\) is the semidirect product of \(\mathrm{Soc}(G)\) with an \(\mathfrak{F}\)-group \(M\); \N\N(2) if \(\Phi(M)=1\), then \(M\) is supersoluble.