Solvability testing problem

Given data
Our universe is some group $$U$$ (such as a linear group or a permutation group) in which products and inverses can be readily computed.

A group $$G$$ in $$U$$ is specified by a generating set $$A$$.

Goal
We need to determine whether $$G$$ is a solvable group.

Problems it reduces to

 * Derived series computation problem: If we can find thederived series of the group, wecan also check whether it terminates at the identity.
 * Commutator subgroup-finding problem: If we can find the commutator subgroup, we can iterate this process to compute the derived series, and check whether it terminates at the identity.
 * Normal closure-finding problem: This is because the commutator subgroup-finding problem reduces to the normal closure-finding problem.

Solution
The idea is to simply appeal to the commutator subgroup-finding problem.