Powerfully nilpotent groups of maximal powerful class

G Traustason,J Williams

doi:10.1007/s00605-019-01331-4

Abstract

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime p>r, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.

Highlights

In this paper we continue the study of powerfully nilpotent p-groups started in [6] and continued in [7]
The class of powerfully nilpotent groups is a special subclass of these, containing groups that possess a central series of a special kind
In [6] we showed that a strongly powerful p-group is powerfully nilpotent of powerful class at most e − 1 where pe is the exponent of the group

Summary

Introduction

In this paper we continue the study of powerfully nilpotent p-groups started in [6] and continued in [7]. Definition Let G be a powerfully nilpotent p-group of powerful class c and order pn. Two vertices G and H are joined by a directed edge from H to G if and only if H ∼= G/Z (G)p and G is not abelian Notice that this implies that Z (G)p = {1} and the powerful class of G is one more than that of H. Theorem [6] Let G be a powerfully nilpotent group of rank r ≥ 2 that has a maximal tail. A powerfully nilpotent group, for which this bound is attained, will be called a group of maximal powerful class. In the final section we use the Lazard correspondence to obtain the analogous results for powerfully nilpotent p-groups of maximal powerful class when p > r. We refer to [4] for an account of coclass theory for finite p-groups

Powerfully nilpotent Lie rings

Powerfully nilpotent Lie rings of maximal powerful class

Powerfully nilpotent groups with maximal powerful class