On the supersolvability of finite groups (Q5906569)

Improving a previous result of \textit{M. Asaad} and \textit{A. Shaalan} [Arch. Math. 53, 318-326 (1989; Zbl 0685.20018)], the authors prove the following theorem. Let \(G = HK\) be a finite group, where \(H\) and \(K\) are subgroups of \(G\) such that \(H\) permutes with every subgroup of \(K\) and \(K\) permutes with every subgroup of \(H\). If \(K\) is supersoluble and all maximal subgroups of Sylow subgroups of \(H\) are normal in \(H\), then \(G\) is also supersoluble.