Generalized Frattini and Fitting-groups. (Q1070334)

\textit{K. Doerk} and \textit{P. Hauck} [Arch. Math. 35, 218-227 (1980; Zbl 0465.20017)] have introduced a \(\tau\)-Frattini subgroup of a finite group correspoding to an arbitrary closure operation \(\tau\), that generalizes the ``usual'' Frattini subgroup in the following sense: \(\Phi_{\tau}(G)\) is the largest normal subgroup of \(G\) such that \(G\) still belongs to \(\tau(G/\Phi_{\tau}(G))\). In my paper I continue their investigations, especially for the case \(\tau=\langle E_{\Phi},Q,R\rangle\) which means the closure operation that assigns to each class the smallest saturated formation it is contained in. Among other things I show that \(\Phi_{\tau}(G)\) is always soluble (it may be of any nilpotent length, however). Furthermore, I construct a generalized Fitting subgroup \(F_{\tau}(G)\) ``fitting'' to that particular \(\Phi_{\tau}(G)\) in as much as it allows the generalization of certain properties shared by \(\Phi(G)\) and \(F(G)\) in the soluble case to the universe of all finite groups. Finally, the class of ``\(\tau\)-nilpotent'' groups (that is \((G:G)=F_{\tau}(G)\)) is shown to be the saturation of the class of quasinilpotent groups.