Equivalence of definitions of nilpotent variety

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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_cs.

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.