### 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
^{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 language | English |
---|---|

Pages (from-to) | 80-100 |

Number of pages | 21 |

Journal | Journal of Algebra |

Volume | 522 |

Early online date | 18 Dec 2018 |

DOIs | |

Publication status | Published - 15 Mar 2019 |

### Keywords

- Nilpotent
- Powerful
- p-group

### ASJC Scopus subject areas

- Algebra and Number Theory

## Cite this

*Journal of Algebra*,

*522*, 80-100. https://doi.org/10.1016/j.jalgebra.2018.12.007