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 $p^{2}$ and order $p^{n}$ where $p$ is odd. The number of these is $f(n)=p^{\alpha n^{3}+o(n^{3})}$ where $\alpha=\frac{9+4\sqrt{2}}{394}$. For the larger class of all powerful groups of exponent $p^{2}$ and order $p^{n}$, where $p$ is odd, the number is $p^{\frac{2}{27}n^{3}+o(n^{3})}$. Thus here the class of powerfully nilpotent $p$-groups is large while sparse within the larger class of powerful $p$-groups.