# Subgroup testing problem

## Definition

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

We are given subgroups $G$ and $H$ of $U$, specified by generating sets ($A$ for $G$ and $B$ for $H$). The goal is to determine whether $H \le G$.

## 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 $G$. Then, test whether all the elements of $B$ satisfy the test for membership in $G$.
order-finding problem find the order of a subgroup given a generating set. via membership testing problem