Finite groups with given nearly \(s\)-embedded subgroups.
The paper considers only finite groups. A subgroup \(T\) of \(G\) is called \(s\)-permutable [\(s\)-semipermutable] if \(T\) permutes with all Sylow subgroups \(P\) of \(G\) [for which \((|P|,|T|)=1\)]. For a subgroup \(H\) let \(H^{sG}\) be the intersection of all \(s\)-permutable subgroups of \(G\) contained in \(H\), then \(H^{sG}\) is \(s\)-permutable in \(G\) as well. A subgroup \(H\) of \(G\) is \textit{nearly \(s\)-embedded} in \(G\) if \(G\) has an \(s\)-permutable subgroup \(T\) such that \(H^{sG}=HT\) and \(H\cap T\leq H_{ssG}\) for some \(s\)-semipermutable subgroup of \(G\) contained in \(H\). Nearly \(s\)-embedded subgroups are a generalization of \(s\)-embedded and \(n\)-embedded subgroups. The author proves the following characterizations of \(p\)-nilpotency and supersolvability: Theorem~1.4: Let \(G\) be a finite group and \(P\) be a Sylow \(p\)-subgroup of \(G\), where \(p\) is the smallest prime divisor of \(|G|\). Then \(G\) is \(p\)-nilpotent if and only if one of the following holds: (1) every maximal subgroup of \(P\) is nearly \(s\)-embedded in \(G\); (2) every cyclic subgroup of \(P\) of order \(p\) and (if \(P\) is a nonabelian \(2\)-group) of order 4 is nearly \(s\)-embedded in \(G\). Theorems~1.5-1.8: Let \(G\) be a finite group, \(N\triangleleft G\) such that \(G/N\) is supersolvable. If property (1) holds for every Sylow subgroup \(P\) of \(N\) or if property (2) holds for every Sylow subgroup \(P\) of \(N\), then \(G\) is supersolvable. In particular, choosing \(N=G'\) characterizes supersolvability. In Section~4 the author shows how these results generalize earlier characterizations of \(p\)-nilpotency and supersolvability using \(s\)-embedded, \(s\)-permutable and \(c\)-normal subgroups.
scientific article; zbMATH DE number 6384138
scientific article