On \(s\)-semipermutable subgroups of finite groups and \(p\)-nilpotency.
Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is `\(s\)-permutable' if \(H\) commutes with all Sylow subgroups of \(G\). The subgroup \(H\) is said to be `\(s\)-semipermutable' if \(HP=PH\) for each Sylow \(p\)-subgroup \(P\) of \(G\) such that the prime \(p\) does not divide \(|H|\). If \(H\) is an arbitrary subgroup of \(G\), the join of all subgroups of \(H\) which are \(s\)-permutable in \(G\) will be denoted by \(H_{sG}\). With this notation, a subgroup \(H\) of \(G\) is said to be `weakly \(s\)-permutable' if there exists a subnormal subgroup \(K\) of \(G\) such that \(HK=G\) and \(H\cap K\leq H_{sG}\). The paper under review deals with the structure of groups rich of \(s\)-semipermutable subgroups. Among other results, the author proves that if \(p\) is an odd prime and a finite group \(G\) contains a Sylow \(p\)-subgroup \(P\) such that the normalizer \(N_G(P)\) is \(p\)-nilpotent and there exists a subgroup \(D\) of \(P\) with \(1<|D|<|P|\) for which all subgroups of \(P\) of order \(|D|\) are \(s\)-semipermutable in \(G\), then \(G\) is \(p\)-nilpotent.
scientific article; zbMATH DE number 5810036
scientific article