Finite groups with some ss-quasinormal and c-normal subgroups.
The paper under review is a further contribution to the research project consisting in studying the influence of the embedding of some distinguished families of subgroups on the structure of finite groups. In this case, the subgroup embedding properties are the c-normality and the \(ss\)-quasinormal embedding. Recall that a subgroup \(H\) of a finite group \(G\) is said to be \(ss\)-quasinormal in \(G\) if \(G\) has a subgroup \(B\) such that \(HB=G\) and \(HS=SH\) for any Sylow subgroup \(S\) of \(B\). A subgroup \(H\) of \(G\) is c-normal in \(G\) if there exists a normal subgroup \(K\) in \(G\) such that \(G=HK\) and \(H\cap K\subseteq\text{Core}(H)=\bigcap_{g\in G}H^g\). Let \(P\) be a Sylow \(p\)-subgroup of a group \(G\) for the smallest prime \(p\) that divides the order of \(G\). The authors prove that if each maximal subgroup in \(P\) is either c-normal or \(ss\)-quasinormal in \(G\), then \(G\) is \(p\)-nilpotent. Let \(\mathfrak F\) be a saturated formation that contains the formation of supersoluble groups. Consider \(N\) to be a normal soluble subgroup of a finite group \(G\). The authors then apply the above result to obtain that \(G\in\mathfrak F\) if and only if every maximal subgroup of each Sylow subgroup of the Fitting subgroup of \(N\) is either c-normal or \(ss\)-quasinormal.
scientific article; zbMATH DE number 5724865