Radical distributive functors (Q5930773)

Following \textit{H. Wielandt} [Abh. Math. Semin. Univ. Hamb. 21, 55-62 (1957; Zbl 0077.02802)] a distributive functor is a mapping \(\Theta\) that sends any finite group \(G\) to one of its subgroups \(\Theta(G)\) and has the following properties: (1) \(f(\Theta(G))=\Theta(f(G))\) for any isomorphism \(f\) from \(G\) onto another group. (2) \(\Theta(\langle X,Y\rangle)=\langle\Theta(X),\Theta(Y)\rangle\) for \(X,Y\) any subnormal subgroups of \(G\). A distributive functor is said to be radical if it satisfies: (3) \(\Theta(X)=X\cap\Theta(G)\). Let \(\Theta\) be a radical distributive functor. The authors prove that \({\mathcal F}=\{H\mid\Theta(H)=H\}\) is a Fitting class and that for any group \(G\), \(\Theta(G)\) coincides with the \(\mathcal F\)-radical \(G_{\mathcal F}\) of \(G\) (we denote \(\text{rad}_{\mathcal F}(G)=G_{\mathcal F}\)). As a consequence, each radical distributive functor is of the form \(\Theta=\text{rad}_{\mathcal F}\) for some Fitting class \(\mathcal F\) and the problem of describing the radical distributive functors is reduced to that of describing the Fitting classes \(\mathcal F\) such that \(\langle H,K\rangle_{\mathcal F}=\langle H_{\mathcal F},K _{\mathcal F}\rangle\), for \(H,K\) any subnormal subgroups of \(G\). The main result in this paper is the following: Let \(\mathcal F\) be a Fitting class. The functor \(\text{rad}_{\mathcal F}\) is a radical distributive functor if and only if the Fitting class \(\mathcal F\) is either the class of all finite groups or each group in \(\mathcal F\) has no Abelian composition factors.