Powerfully nilpotent groups

Abstract

We introduce a special class of powerful p-groups that we call powerfully nilpotent groups that are finite p-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an ‘ancestry tree’ and powerful coclass. We show that there are finitely many powerfully nilpotent p-groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent p2 and order pn where p is odd. The number of these is f(n)=pαn3+o(n3) where α=9+42394. For the larger class of all powerful groups of exponent p2 and order pn, where p is odd, the number is p227n3+o(n3). Thus here the class of powerfully nilpotent p-groups is large while sparse within the larger class of powerful p-groups.