Finite p'-nilpotent groups. I (Q1820234)

A p'-nilpotent group is an extension of a p-group by a nilpotent group (which can, of course, be assumed to be a p'-group). The author observes that the p-Frattini subgroup of a group is p'-nilpotent and that the p'- nilpotent groups form a saturated formation. He generalizes some work of M. Torres on \(\Phi^*(G)\), the product of the p-Frattini subgroups of the group G for all the prime factors of the order of G, showing, for example, that the Fitting length of G is less than 3 if and only if \(G'\subseteq \Phi^*(G)\). He shows also that if the group G contains three p'-nilpotent subgroups with pairwise relatively prime indices then G is p'-nilpotent. In the final section of the paper the author investigates the \({\mathcal F}\)-hypercenter of certain solvable groups for the formation \({\mathcal F}\) of p'-nilpotent groups. Only finite groups are treated in the paper.