Powerfully nilpotent groups of maximal powerful class

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 $$p>r$$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.

Transporting cohomology in Lazard correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-[Formula: see text] groups of nilpotency class smaller than [Formula: see text] and finitely generated nilpotent [Formula: see text]-Lie algebras of nilpotency class smaller than [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for [Formula: see text], [Formula: see text] and [Formula: see text], and for a given category of modules the cohomology functors [Formula: see text] and [Formula: see text] are naturally equivalent. A similar result is proved for [Formula: see text] with the relative cohomology groups.