Finite groups with trivial Frattini subgroup.
The authors continue their investigations on \(xP\)-groups, that is groups in which each nontrivial \(x\)-subgroup has property \(P\), especially whenever \(x=a,n\), or \(c\) for arbitrary, normal, or characteristic subgroup respectively and \(P=C,S,PNS\), or \(CS\) for the respective types that each \(x\)-subgroup has a complement, proper supplement, a proper normal supplement, or a proper characteristic supplement. The investigation here focuses on \(nS\)-groups which of course includes the \(nC\)-groups about which much is known. Many analogous results are obtained due to the involvement of the Frattini subgroup. Among these results are characterizations of the supersoluble \(nS\)-groups, those soluble groups in which each proper epimorphic image is an \(nS\)-group, and \(nS\)-groups which contain an Abelian normal subgroup that has an Abelian quotient with trivial Frattini subgroup. The article concludes with an examination of conditions for normal subgroup supplementation, for example if a subgroup \(N\) normal in a group \(G\) is contained in a normal Sylow \(p\)-subgroup \(B\) of \(G\) such that \(N\) has a supplement in \(B\), then \(N\) has a supplement in \(G\). This paper supplements an earlier work of the authors [Glasg. Math. J. 42, No. 1, 37-50 (2000; Zbl 0945.20016)].
scientific article; zbMATH DE number 1966443