New characterizations of finite supersoluble groups.
Let \(A\) be a subgroup of \(G\) and let \(X\) be a non-empty subset of \(G\). \(A\) is called an \(X\)-semipermutable subgroup of \(G\) if \(A\) has a supplement \(T\) in \(G\) such that for every subgroup \(T_1\) of \(T\) there exists an element \(x\in X\) such that \(AT_1^x=T_1^xA\). One of the most remarkable results in this paper is the following theorem: If every Sylow subgroup of a group \(G\) is \(F(G)\)-semipermutable in \(G\), then \(G\) is supersoluble and \(G/O_{p,p'}(G)\) is a cyclic group of order \(p-1\) for any prime \(p\). Also, the authors obtain that a group \(G\) is supersoluble if and only if every maximal subgroup of every Sylow subgroup of \(G\) not having a supersoluble supplement in \(G\) is \(F(G)\)-semipermutable in \(G\).
scientific article; zbMATH DE number 5323360
scientific article