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
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.