Coset intersection problem

This article describes a problem in the setup where the group(s) involved is/are defined by means of an embedding in a suitable universe group (such as a linear or a permutation group) -- viz in terms of generators described as elements sitting inside this universe group

History

The coset intersection problem was introduced by Hoffmann in his paper Group-theoretic methods in graph isomorphism published in 1982. Hoffmann showed that graph isomorphism was a special case of a problem called the double coset membership testing problem and studied a whole class of problems (including the coset intersection problem) that are Turing-equivalent to the double coset membership testing problem.

Description

Given data

Our universe is some group $U$ (such as a linear group or a permutation group) in which products and inverses can be readily computed.

A group $G$ in $U$ is specified by a generating set $A$, and subgroups $H$ and $K$ of $G$ are specified by means of generating sets $B$ and $C$ respectively. An elements $x$ in $G$ is given (described as an element of $U$).

Goal

Determine whether $Hx$ intersects $K$.

Relation with other problems

Equivalent decision problems

• Group factorization problem: This asks whether $x\in HK$. The group factorization problem is clearly equivalent to the coset intersection problem, because $x\in HK\iff Hx\cap K$ is nonempty.
• Double coset membership testing problem: Here, two subgroups $H$ and $K$ are specified by means of generating sets, and elements $g$ and $h$ are given. We need to check whether $h$ is in $HgK$.

Clearly, the coset intersection problem reduces to the double coset membership problem, because asking if $Hx\cap K$ is nonempty is the same as asking whether the double coset $HK$ contains $x$.

Conversely, given a double coset membership testing problem, wherein we want to know if $x_{2}\in Hx_{2}K$, the corresponding coset equality problem is the problem of asking whether $H^{x_{2}^{-1}}(x_{2}^{-1}x_{1})\cap K$ is nonempty.