Equivalence of definitions of periodic nilpotent group
This article gives a proof/explanation of the equivalence of multiple definitions for the term periodic nilpotent group
View a complete list of pages giving proofs of equivalence of definitions
The following are equivalent for a group :
- It is both locally finite (every finitely generated subgroup is finite) and nilpotent.
- It is both periodic (every element has finite order) and nilpotent.
- 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.
- Its abelianization is a periodic abelian group.
- It is a nilpotent group and all the quotient groups between successive members of its lower central series are periodic abelian groups.
- It is a restricted direct product of nilpotent p-groups, with a common bound on their nilpotency class.
Variant for a set of primes
If is a set of primes, the following are equivalent for a group :
- It is nilpotent and every finitely generated subgroup of is a finite -group (i.e., all the prime factors of its order are in .
- It is nilpotent and every element has finite order and the order is a -number.
- It is nilpotent and generated by -elements: elements whose orders are all -numbers.
- It is nilpotent and its abelianization is a -group.
- It is nilpotent and all the quotient groups between successive members of its lower central series are abelian -groups.
- It is a restricted direct product of nilpotent p-groups, , with a common bound on their nilpotency class.
- All successive quotients between lower central series members of a group are homomomorphic images of tensor powers of the abelianization.
- Equivalence of definitions of periodic abelian group: This states in particular that for an abelian group, being periodic is equivalent to being locally finite.
- Local finiteness is extension-closed
(1) implies (2)
This is obvious from the general fact that locally finite implies periodic. We do not use that is nilpotent for this implication.
(2) implies (3)
This is obvious from the general fact that any periodic group is a group generated by periodic elements -- in fact, we could take the generating set as the set of all elements. Note that this step again does not use anything about being nilpotent.
(3) implies (4)
Suppose is a generating set for that comprises only elements of finite order. Under the quotient map from to its abelianization, the image of is a generating set for the abelianization of . Since the abelianization is an abelian group, being generated by a set of elements all of finite order is equivalent to the whole group being a periodic group, and hence a periodic abelian group.
(4) implies (5)
Denote by the member of the lower central series of . Each of the successive quotients is a homomorphic image of a tensor power of the abelianization of by Fact (1). Since the abelianization of is periodic, each is periodic.
(5) implies (1)
We use Fact (2) to note that all the quotients between successive members of the lower central series are in fact locally finite. Fact (3) now completes the proof.