On Hall subnormally embedded subgroups of finite groups (Q344210)

Let \(G\) be a finite group. A subgroup \(H\) is said to be Hall subnormally (normally) embedded in \(G\) if there is a subnormal (normal) subgroup \(N\) of \(G\) such that \(H\) is a Hall subgroup of \(N\). \textit{S. Li} and \textit{J. Liu} [J. Algebra 388, 1--9 (2013; Zbl 1286.20018)] characterized all groups \(G\) which have a Hall normally embedded subgroup \(H\) of order \(d\) for any \(d\) dividing \(|G|\), and posed the problem of characterizing all groups \(G\) which have a Hall normally embedded subgroup \(H\) of order \(|B|\) for any subgroup \(B\) of \(G\). The question was answered by \textit{A. Ballester-Bolinches} and \textit{S. Qiao} [Arch. Math. 102, No. 2, 109--111 (2014; Zbl 1295.20015)]. The present paper answers the analogous question for Hall subnormally embedded subgroups. The main result is as follows. \(G\) has a Hall subnormally embedded subgroup \(H\) of order \(|B|\) for any subgroup \(B\) of \(G\) if and only if (1) \(G\) is a Sylow tower; (2) the supersolvable residual \(R\) is the direct product of non-cyclic Sylow-subgroups of \(G\) which are minimal normal subgroups in \(G\); (3) Hall \(\pi(R)'\)-subgroups of \(G\) are exactly the supersolvable projectors \(E\) of \(G\) which have nilpotent residuals \(S_E\) cyclic of square-free order and (4) if \(H\) is a subgroup of \(G\) then there exists a subgroup \(H_1\) of order \(|H|\) in \(G\) such that if \(p\in\pi(H)\) then a Hall \(p'\)-subgroup of \(H_1\) centralizes a complement \(K\) of the \(p\)-Sylow subgroup of \(H_1\) in \(R_p\), and if a Hall \(\pi'\)-subgroup \(H_{\pi'}\) of \(H_1\) is contained in the supersolvable projector \(E\) then the nilpotent residual of \(S_EH_{\pi'}\) normalizes \(H_{\pi'}\) and centralizes \(K\). This reproves the main theorem of the above-cited paper by Ballester-Bolinches and Qiao [loc. cit.].