Powerfully nilpotent groups

Gunnar Traustason, James Williams

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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 αn 3+o(n 3) where α=[Formula presented]. For the larger class of all powerful groups of exponent p 2 and order p n, where p is odd, the number is p [Formula presented]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.

Original languageEnglish
Pages (from-to)80-100
Number of pages21
JournalJournal of Algebra
Early online date18 Dec 2018
Publication statusPublished - 15 Mar 2019


  • Nilpotent
  • Powerful
  • p-group

ASJC Scopus subject areas

  • Algebra and Number Theory


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