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

Statement

The following are equivalent for a group $G$

1. It is both periodic (every element has finite order) and nilpotent.
2. It is both locally finite (every finitely generated subgroup is finite) and nilpotent.
3. It is a nilpotent group and has a generating set where all the elements have finite orders.
4. Its abelianization is a periodic abelian group.
5. It is a restricted direct product of nilpotent p-groups, with a common bound on their nilpotency class.

Related facts

• Root set of a subgroup for a set of primes in a nilpotent group is a subgroup