The influence of \(\mathcal M\)-supplemented subgroups on the structure of finite groups. (Q1041286)

From the author's summary: A subgroup \(H\) of a finite group \(G\) is said to be \(\mathcal M\)-supplemented in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=HB\) and \(TB<G\) for every maximal subgroup \(T\) of \(H\). Moreover, a subgroup \(H\) is called \(c\)-supplemented in \(G\) if there exists a subgroup \(K\) such that \(G=HK\) and \(H\cap K\leq H_G\) where \(H_G\) is the largest normal subgroup of \(G\) contained in \(H\). In this paper the author gives some conditions to guarantee the supersolubility of a finite group, under the assumption that some primary subgroups have some kinds of supplements. He also obtains some extensions by considering saturated formations containing the class \(\mathcal U\) of all supersoluble groups, which are generalizations of some recent results.