Restricted regular wreath product of group of prime order and quasicyclic group
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.
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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.
|abelian group||No||It is a semidirect product for a nontrivial action|
|nilpotent group||No||It is a nontrivial centerless group.|
|hypercentral group||No||It is a nontrivial centerless group.|
|finitely generated group||No||Its quotient, the quasicyclic group, is not finitely generated.|