Extensions of finite nilpotent groups

Source abstract

If G is a finite group and P is a group-theoretic property, G will be called P -max-core if for every maximal subgroup M of G , M/M G has property P where M G = ∩ is the core of M in G . In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is ( p -nilpotent)-max-core, then G is p -solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p -nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize ( p -nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-( p -nilpotent) groups.