Finite groups with Hall \(\pi\)-subgroups. (Q2901885)

Generalizing the known properties of Sylow subgroups, Hall introduced the classes \(E_\pi\), \(C_\pi\) and \(D_\pi\) of finite groups, [for a survey see \textit{E. P. Vdovin} and \textit{D. O. Revin}, Russ. Math. Surv. 66, No. 5, 829-870 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 3-46 (2011; Zbl 1243.20027)]. The author substantially generalizes these classes.NEWLINENEWLINE Let \(\mathcal F\) be a nonempty class of finite groups. Let \(E_\pi(\mathcal F)\) be the class of finite groups which contain an \(S_\pi(\mathcal F)\)-subgroup, \(C_\pi(\mathcal F)\) be the class of finite groups which contain precisely one conjugacy class of \(E_\pi(\mathcal F)\)-subgroups, and \(D_\pi(\mathcal F)\) be the class of finite groups in which every \(\mathcal F_\pi\)-subgroup is contained in an \(S_\pi(\mathcal F)\)-subgroup.NEWLINENEWLINE Theorems generalizing classical results of P. Hall, Wielandt, Baer, and Hartley are obtained. Few excerpts. Let \(\mathcal F\) be a (Q,S,Ext)-closed class of finite groups. The finite group \(G\in E_\pi(\mathcal F)\) (\(G\in D_\pi(\mathcal F)\)) if and only if \(\Aut_G(H/K)\in E_\pi(\mathcal F)\) (\(\Aut_G(H/K)\in D_\pi(\mathcal F)\)) for every composition factor \(H/K\) of \(G\). \(E_\pi(\mathcal F)\) and \(C_\pi(\mathcal F)\) are local formations. An \(E_\pi(\mathcal F)\)-by-\(D_\pi(\mathcal F)\) group is a \(D_\pi(\mathcal F)\)-group.