Membership testing problem

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 $⟨ B ⟩$ .	Compute the orders of the subgroups $H = ⟨ B ⟩$ and $⟨ H, g ⟩ = ⟨ B \cup {g} ⟩$ . 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 $B$ .

Normal closure-finding: This is solved using the normality testing problem

Subnormality testing problem: This is solved using the normal closure-finding algorithm

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		$O (N s)$ times the time for the group operations.	deterministic
Nondeterministic black-box group algorithm for membership testing			nondeterministic

Permutation group algorithms

Permutation group algorithm for membership testing

Linear group algorithms

Linear group algorithm for membership testing