Membership testing problem: Difference between revisions

Revision as of 19:26, 25 June 2013

Description

Given data

$G$ is a group equipped with an encoding $C$ . $H$ 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
Order-finding problem	Compute the order of the subgroup generated by a subset inside a group specified by means of an encoding.

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

@@ Line 8: / Line 8: @@
 We need to describe a test for membership in <math>H</math>, i.e., we need to construct an algorithm that can take as input the code-word for any <math>g \in G</math> and outputs whether or not <math>g \in H</math>.
+==Relation with other problems==
+===Problems it reduces to===
+{| class="sortable" border="1="
+! Problem !! Nature of problem !! Description of reduction
+|-
+| [[Order-finding problem]] || Compute the order of the subgroup generated by a subset inside a group specified by means of an encoding. ||
+|}
+===Problems that are solved using it===
+* [[Subgroup testing problem]]: For this problem, we are given sets <math>A</math> and <math>B</math> inside <math>U</math> and we are asked whether the group <math>G</math> generated by <math>A</math> contains the group <math>H</math> generated by <math>B</math>. 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 <math>B</math>, whether it is a member of <math>G</math>.
+* [[Normality testing problem]]: Given generating sets <math>A</math> for <math>G</math> and <math>B</math> for <math>H</math>, the problem asks whether <math>H</math> is a [[normal subgroup]] of <math>G</math>. 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 <math>B</math> by an element in <math>A</math> must be in <math>B</math>.
+* [[Normal closure-finding]]: This is solved using the normality testing problem
+* [[Subnormality testing problem]]: This is solved using the normal closure-finding algorithm
 ==Algorithms==