Equivalence of definitions of nilpotent variety

This article gives a proof/explanation of the equivalence of multiple definitions for the term nilpotent variety of groups
View a complete list of pages giving proofs of equivalence of definitions

The two definitions that we have to prove as equivalent

Consider a subvariety of the variety of groups, i.e., a collection of groups (closed upto isomorphism) that is closed under taking subgroups, quotients, and arbitrary direct products. Then, the following are equivalent:

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

Proof

Clearly, (2) implies (1). We need to show that (1) implies (2). We do this by contradiction: suppose every group in the variety is nilpotent, and suppose that, for every $c$, there exists a $G_{c}$ in the variety with nilpotence class strictly greater than $c$. Then, let $G$ be the direct product of all the $G_{c}$s.

Suppose the group $G$ is nilpotent of nilpotence class $d$. Then, every subgroup of $G$ has nilpotence class at most $d$. But, $G$ has a direct factor $G_{d}$ of nilpotence class more than $d$, leading to a contradiction.