Periodic nilpotent group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: nilpotent group and periodic group
View other group property conjunctions OR view all group properties
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: nilpotent group and locally finite group
View other group property conjunctions OR view all group properties

Definition

A periodic nilpotent group or locally finite nilpotent group is defined in the following equivalent ways:

  1. It is both locally finite (every finitely generated subgroup is finite) and nilpotent.
  2. It is both periodic (every element has finite order) and nilpotent.
  3. It is a nilpotent group and also a group generated by periodic elements: it has a generating set where all the elements have finite orders.
  4. Its abelianization is a periodic abelian group.
  5. It is a nilpotent group and all the quotient groups between successive members of its lower central series are periodic abelian groups.
  6. It is a restricted direct product of nilpotent p-groups, with a common bound on their nilpotency class.


Equivalence of definitions

For full proof, refer: Equivalence of definitions of periodic nilpotent group

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite nilpotent group nilpotent and finite |FULL LIST, MORE INFO
nilpotent p-group nilpotent and every element has order a power of a fixed prime p |FULL LIST, MORE INFO
periodic abelian group abelian and every element has finite order |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
periodic group Locally finite group, Periodic solvable group|FULL LIST, MORE INFO
nilpotent group |FULL LIST, MORE INFO
locally finite group Periodic solvable group|FULL LIST, MORE INFO