Group factorization problem

History
The group factorization 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 group factorization 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 $$h$$ in $$G$$ is given (described as an element of $$U$$).

Goal
Determine whether $$h$$ is in $$HK$$.

Equivalent decision problems

 * Coset intersection problem: Here, two subgroups $$H$$ and $$K$$ are specified by means of generating sets. An element $$x$$ in $$G$$ is given, and we need to determine whether $$Hx$$ intersects $$K$$ nontrivially.

The coset equality problem is equivalent to the group factorization problem because saying that $$Hx$$ intersects $$K$$ nontrivially is equivalent to saying that $$x^{-1}$$ is in $$KH$$.
 * 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 $$HgK$$.

The group factorization problem reduces to double coset membership testing simply by setting $$g$$ to be the identity element. The reduction the other way is a little more tricky: it uses the fact that $$x_1 \in Hx_2K$$ if and only iff $$x_2^{-1}x_1 \in H^{x_2}K$$, which is a group factorization.