New characterizations of some classes of finite groups.
Let \(G\) be a finite gorup. Let \(\mathcal F\) be a formation of finite groups. A subgroup \(H\) of \(G\) is defined to be \(\mathcal F_h\)-normal in \(G\) if \(G\) has a normal subgroup \(T\) such that \(HT\) is a normal Hall subgroup of \(G\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathcal F\)-hypercentre of \(G/H_G\). As a major application, the concept of \(\mathcal F_h\)-normality is used to characterize various classes of groups. Theorem 3.1: \(G\) is supersoluble if and only if there exists a normal subgroup \(E\) of \(G\) such that \(G/E\) is supersoluble and every maximal subgroup of every non-cyclic Sylow subgroup of \(G\) is \(\mathcal U_h\)-normal in \(G\), where \(\mathcal U\) is the class of supersoluble groups. Theorem 4.1: \(G\) is soluble if and only if every minimal subgroup of \(G\) is \(\mathcal S_h\)-normal in \(G\), where \(\mathcal S\) is the class of soluble groups. Theorem 5.1: If \(p\) is a prime dividing the order of \(G\), \((|G|,p-1)=1\) and \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(G\) is \(p\)-nilpotent if and only if every maximal subgroup of \(P\) is \(\mathcal U_h\)-normal in \(G\).
scientific article; zbMATH DE number 5976518