\(\mathfrak{F}\)-projectors and \(\mathfrak{F}\)-covering subgroups of finite groups (Q515483)

All groups considered in the paper under review are finite. Given a non-empty set of primes \(\omega\) and a non-empty class of groups \(\mathfrak{F}\), the authors introduce the following notions: A subgroup \(H\) of a group \(G\) is an \(\mathfrak{F}^{\omega}\)-projector provided that \(HN/N\) is an \(\mathfrak{F}\)-maximal subgroup of \(G\), for every normal \(\omega\)-subgroup \(N\) of \(G\). Moreover, a subgroup \(H\) is called an \(\mathfrak{F}^{\omega}\)-covering subgroup of \(G\) if \(H \in \mathfrak{F}\), and if \(U=HV\) whenever \(H \leq U \leq G\), \(V\) is a normal \(\omega\)-subgroup of \(U\), and \(U/V \in \mathfrak{F}\). When \(\omega=\pi(G)\), the set of all prime divisors of the order of \(G\), the classical concepts of \( \mathfrak{F}\)-projector and \(\mathfrak{F}\)-covering subgroup, respectively, are recovered. The aim of this paper is to analyse some properties of those subgroups for certain classes of groups. As a sample, conditions for the existence of \(\mathfrak{F}\)-projectors are given (Theorem 3.1). Also it is studied when the set of all \(\mathfrak{F}\)-projectors coincides with the set of all \(\mathfrak{F}\)-covering subgroups for a given group (Theorem 3.3). The exact formulation of the results requires a large number of definitions and therefore is not given here. Some classical results of \textit{W. Gaschütz} [Math. Z. 80, 300--305 (1963; Zbl 0111.24402)], \textit{H. Schunck} [ibid. 97, 326--330 (1967; Zbl 0158.02802)], or \textit{R. P. Erickson} [Commun. Algebra 10, 1919--1938 (1982; Zbl 0498.20018)] are obtained as particular cases.