Equivalence of definitions of solvable variety

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

The 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 direct products. Then, the following are equivalent:

1. Every group in the variety is a solvable group
2. There exists a nonnegative integer $l$ such that every group in the variety is solvable with solvable length at most $l$

Proof

Clearly (2) implies (1). We need to show that (1) implies (2), which we do by contradiction. Suppose there exists a variety of groups where every group is solvable, but where, for every $l$, there exists a member $G_{l}$ of the variety that has solvable length strictly greater than $l$.

Now, consider $G$ to be the direct product of the $G_{l}$s. Then, $G$ is in the variety (since a variety is closed under arbitrary direct products). Suppose $G$ is solvable. Let $m$ be the solvable length of $G$. Then, every subgroup of $G$ has solvable length at most $m$. But $G$ has as a direct factor (and hence as a subgroup) the subgroup $G_{m}$, whose solvable length is strictly greater than $m$, a contradiction.