# Membership testing problem

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## Definition

### Given data

$G$ is a finite group equipped with an encoding. $H$ is a subgroup of $G$ and we are given a generating set $B$ for $H$.

### Goal

We need to describe a test for membership in $H$, i.e., we need to construct an algorithm that can take as input the code-word for any $g \in G$ and outputs whether or not $g \in H$.

## Relation with other problems

### Problems it reduces to

Problem Nature of problem Description of reduction of membership testing to the problem
Order-finding problem Suppose $G$ is a finite group described by means of an encoding. Consider a subset $B$ of $G$. The goal is to find the order of the subgroup $\langle B \rangle$. Compute the orders of the subgroups $H = \langle B \rangle$ and $\langle H, g \rangle = \langle B \cup \{ g \} \rangle$. If the orders are the same, then $g \in H$. If the orders are different, then $g \notin H$.

### Problems that are solved using it

• Subgroup testing problem: For this problem, we are given sets $A$ and $B$ inside $U$ and we are asked whether the group $G$ generated by $A$ contains the group $H$ generated by $B$. The subgroup testing problem reduces to the membership testing problem via a positive truth-table reduction. The idea of the reduction is to check, for each element in $B$, whether it is a member of $G$.
• Normality testing problem: Given generating sets $A$ for $G$ and $B$ for $H$, the problem asks whether $H$ is a normal subgroup of $G$. The normality testing problem reduces to the membership testing problem via a positive truth-table reduction. The idea is to first use the subgroup testing problem and to then check whether every conjugate of an element in $B$ by an element in $A$ must be in $B$.

## Algorithms

### Black-box group algorithms

These work for a group specified by means of an encoding.

Algorithm Additional information needed for algorithm, if any Time taken, where $N$ is the order of the group and $s$ is the size of the generating set Type of algorithm
Black-box group algorithm for finding the subgroup generated by a subset $O(Ns)$ times the time for the group operations. deterministic
Nondeterministic black-box group algorithm for membership testing nondeterministic