# 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:

- Every group in the variety is a solvable group
- There exists a nonnegative integer such that every group in the variety is solvable with solvable length at most

## 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 , there exists a member of the variety that has solvable length strictly greater than .

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