Group intersection 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.

Two groups $$G_1$$ and $$G_2$$ in $$U$$ are specified by means of their generating sets $$A_1$$ and $$A_2$$.

Goal
We are required to determine a generating set for $$G_1$$ &cap; $$G_2$$.

Problems that can be solved using it

 * Set-stabilizer problem: This problem asks us to compute, for a group $$G$$ acting faithfully on a set $$S$$, the set-stabilizer of a subset $$T$$ of $$S$$ within $$G$$. Here, $$G$$ is given by a generating set $$A$$ of permutations.

The set-stabilizer problem reduces many-one to the group intersection problem as follows: the set stabilizer of $$T$$ in $$G$$ is the intersection of $$G$$ with the set-stabilizer of $$T$$ in $$Sym(S)$$. The latter group is simply $$Sym(T)$$ &times; $$Sym(S- T)$$.


 * Graph-automorphism finding: This problem asks us to compute the automorphism group of a graph. Graph-automorphism finding has a many-one reduction to the set-stabilizer problem, and hence, by the transitivity of many-one reductions, it has a many-one reduction to the group intersection problem.


 * Graph isomorphism: The decision problem of graph isomorphism can be solved using graph-automorphism finding, and hence, using the group intersection problem.

Easier problems

 * Normalizing group intersection problem: This can be solved for permutation groups
 * Subnormalizing group intersection problem: This can also be solved for permutation groups