Nilpotent variety of groups

Definition

Definition without universal algebra

A nilpotent variety of groups is a collection of groups (closed under isomorphism) that is closed under taking subgroups, quotients and arbitrary direct products, such that the following two equivalent conditions are satisfied:

1. Every group in the collection is a nilpotent group
2. There exists a constant $c$ such that every group in the collection is nilpotent of class at most $c$

Definition using universal algebra

A nilpotent variety of groups is a subvariety of the variety of groups satisfying the following equivalent conditions:

1. Every group in it is nilpotent
2. There exists a nonnegative integer $c$ such that every group in the collection is nilpotent with nilpotency class at most $c$

Equivalence of definitions

For full proof, refer: Equivalence of definitions of nilpotent variety