Powerfully nilpotent groups of maximal powerful class

Gunnar Traustason, James Williams

Research output: Contribution to journalArticle


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.
Original languageEnglish
Pages (from-to)779-799
Number of pages21
JournalMonatshefte fur Mathematik
Early online date13 Sep 2019
Publication statusPublished - 30 Apr 2020


  • Coclass
  • Maximal class
  • Nilpotent
  • Powerful
  • p-group

ASJC Scopus subject areas

  • Mathematics(all)

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