Finite groups with given properties of basic subgroups of fans of Sylow subgroups (Q6650386)

Let \(G\) be a finite group and \(\mathfrak{F}\) a formation of groups. A subgroup \(H \leq G\) is \(\mathfrak{F}\)-subnormal if either \(H=G\) or there exists a maximal chain of subgroups \(H=H_{0} \leq H_{1} \leq \ldots \leq H_{n}=G\) such that \(H_{i}/\mathrm{Core}_{G}(H_{i-1}) \in \mathfrak{F}\) for \(i\in \{ 1, \ldots, n\}\). Let \(\mathfrak{T}_{\prec}\) be the class of all groups with a Sylow tower of type \(\prec\), where \(\prec\) is an arbitrary linear ordering on the set of all primes, and let \(\mathfrak{N}^{2}\) be the class of all metanilpotent groups.\N\NThe main results in the paper under review are the following.\N\NTheorem 1.1. Let \(\mathfrak{F}\) be a hereditary saturated formation such that \(\mathfrak{F} \subseteq \mathfrak{T}_{\prec} \cap \mathfrak{N}^{2}\). A group \(G\) is in \(\mathfrak{F}\) if and only if \(\pi(G) \subseteq \pi(\mathfrak{F})\) and every basic subgroup of the fan of every Sylow subgroup in \(G\) is \(\mathfrak{F}\)-subnormal.\N\NTheorem 1.2. Let \(\mathfrak{F}\) be a hereditary saturated formation such that \(\mathfrak{F} \subseteq \mathfrak{T}_{\prec} \cap \mathfrak{N}^{2}\). A group \(G\) is in \(\mathfrak{F}\) if and only if every basic subgroup of the fan of every Sylow subgroup of \(G\) belongs to \(\mathfrak{F}\).