# Restricted regular wreath product of group of prime order and quasicyclic group

From Groupprops

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Definition

Let be a prime number. The **restricted wreath product of group of prime order and quasicyclic group** is defined as the restricted external wreath product of the cyclic group of prime order (i.e., the cyclic group of order ) by the -quasicyclic group for the regular group action.

Note that the term *restricted* indicates that we use a restricted direct product rather than an unrestricted direct product in the base.

Equivalently, if denotes the -quasicyclic group, then this group is the external semidirect product of the additive group of the group ring by the action of by left multiplication.

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

p-group | Yes | |

abelian group | No | It is a semidirect product for a nontrivial action |

centerless group | Yes | |

nilpotent group | No | It is a nontrivial centerless group. |

hypercentral group | No | It is a nontrivial centerless group. |

metabelian group | Yes | |

solvable group | Yes | |

finite group | No | |

finitely generated group | No | Its quotient, the quasicyclic group, is not finitely generated. |

countable group | Yes |