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

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## Definition

Let $p$ 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 $p$) by the $p$-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 $H$ denotes the $p$-quasicyclic group, then this group is the external semidirect product of the additive group of the group ring $\mathbb{F}_p[H]$ by the action of $H$ 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