Equivalence of definitions of solvable variety

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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_ls. 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.