New characterizations of \(p\)-soluble and \(p\)-supersoluble finite groups. (Q2859347)

A finite group is called \(p\)-supersoluble if every chief factor of \(G\) is either a \(p'\)-group or it is of order \(p\). Unlike similar concepts in group theory, such as soluble, \(p\)-soluble or supersoluble groups, the class of \(p\)-supersoluble groups has not been extensively studied.NEWLINENEWLINE The authors of the present paper give new characterisations of \(p\)-supersoluble and \(p\)-soluble groups by using the concept of \(S\)-quasinormal subgroups. A subgroup of a finite group \(G\) is called \(S\)-quasinormal if it commutes with every Sylow subgroup of \(G\). For every subgroup \(H\) of \(G\), let \(H_{sG}\) be the largest \(S\)-quasinormal subgroup in \(H\) and let \(H^{sG}\) be the smallest \(S\)-quasinormal subgroup of \(G\) containing \(H\). (These subgroups are called the \(S\)-quasinormal core of \(H\) and the \(S\)-quasinormal closure of \(H\), respectively.)NEWLINENEWLINE Among other results, the authors prove that \(G\) is \(p\)-supersoluble if and only if for every cyclic subgroup \(H\) of \(\overline G=G/O_{p'}(G)\) of prime order or of order \(4\) (in case of \(p=2\)), there is a normal subgroup \(T\) of \(\overline G\) such that \(HT=H^{s\overline G}\) and \(H\cap T=H_{s\overline G}\cap T\). A number of previously known results are listed as corollaries of the theorems of the paper.