On Shemetkov's question about the \(\mathfrak{F} \)-hypercenter (Q6576762)

Let \(G\) be a finite group and let \(\mathfrak{F}\) be a class of groups. A chief factor \(H/K\) of \(G\) is said to be \(\mathfrak{F}\)-central if \((H/K) \rtimes G/C_{G}(H/K) \in \mathfrak{F}\). The \(\mathfrak{F}\)-hypercenter \(Z_{\mathfrak{F}}(G)\) of \(G\) is the maximal normal subgroup of \(G\) such that all \(G\)-composition factors below it are \(\mathfrak{F}\)-central.\N\NIn 1995, at the Gomel algebraic seminar, L. A. Shemetkov formulated the problem of describing formations of finite groups \(\mathfrak{F}\) for which, in any finite group, the intersection of \(\mathfrak{F}\)-maximal subgroups coincides with the \(\mathfrak{F}\)-hypercenter. A formation \(\mathfrak{F}\) is a Baer-Shemetkov formation in a class \(\mathfrak{X}\) of groups if in any \(\mathfrak{X}\)-group the intersection of all \(\mathfrak{F}\)-maximal subgroups coincides with the \(\mathfrak{F}\)-hypercenter. If \(\mathfrak{X}\) is the class of all groups, then \(\mathfrak{F}\) is a Baer-Shemetkov formation. \textit{A. N. Skiba} [J. Pure Appl. Algebra 216, No. 4, 789--799 (2012; Zbl 1261.20015)] solved Shemetkov's problem for hereditary saturated formations using the canonical local definition and the author [J. Group Theory 21, No. 3, 463--473 (2018; Zbl 1402.20031)] showed that there exist nonsaturated nonhereditary Baer-Shemetkov formations (in particular, the class of all quasinilpotent groups is such a formation).\N\NA formation \(\mathfrak{F}\) is said to be \(Z\)-saturated if \(\mathfrak{F} = (G \mid G = Z_{\mathfrak{F}}(G))\). The main result of this paper is Theorem 1: Let \(\mathfrak{F} \not = (1)\) be a hereditary formation. The following conditions are equivalent. (1) The formation \(\mathfrak{F}\) is a Baer-Shemetkov formation. (2) The following conditions are satisfied: (2.a) \(\mathfrak{F}\) is a hereditary \(Z\)-saturated formation; (2.b) There exists a partition \(\sigma = \{\pi_{i} \mid i\in I \}\) of the set \(\mathbb{P}\) of all primes such that \(\Gamma_{Nc}(\mathfrak{F})\) is the union of complete directed graphs on the vertex sets \(\pi_{i}\), \(i\in I\); (2.c) \(\mathfrak{F}\) is a Baer-Shemetkov formation in the class of all \(\pi_{i}\)-groups for any \(i \in I\).