Solvability testing problem

This article describes the group property testing problem for the group property: solvable group

Description

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.

Relation with other problems

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.