Coset intersection problem

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.

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 $$K$$.

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 $$x_2 \in Hx_2K$$, the corresponding coset equality problem is the problem of asking whether $$H^{x_2^{-1}}(x_2^{-1}x_1) \cap K$$ is nonempty.