# Subgroup testing problem

From Groupprops

## Definition

We are working inside a universe group described by means of an encoding .

We are given subgroups and of , specified by generating sets ( for and for ). The goal is to determine whether .

## Relation with other problems

### Problems it reduces to

Problem | Description | Description of reduction |
---|---|---|

membership testing problem | given a generating set for a subgroup, find a membership test for the subgroup. | black-box reduction of subgroup testing problem to membership testing problem: First, use the membership testing problem to obtain a membership test for the group . Then, test whether all the elements of satisfy the test for membership in . |

order-finding problem | find the order of a subgroup given a generating set. | via membership testing problem |