On Shemetkov’s Question about the $$\mathfrak{F}$$ -Hypercenter

Abstract

The chief factor \(H/K\) of a group \(G\) is said to be \(\mathfrak{F}\)-central if

$$(H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}.$$

The \(\mathfrak{F}\)-hypercenter of a group \(G\) is defined to be a maximal normal subgroup of \(G\) such that all \(G\)-composition factors below it are \(\mathfrak{F}\)-central in \(G\). In 1995, at the Gomel algebraic seminar, L. A. Shemetkov formulated the problem of describing formations of finite groups \(\mathfrak{F}\) for which, in any group, the intersection of \(\mathfrak{F}\)-maximal subgroups coincides with the \(\mathfrak{F}\)-hypercenter. In the present paper, new properties of such formations are obtained. In particular, a series of hereditary nonsaturated formations of soluble groups is constructed, which answer Shemetkov’s problem.