Group intersection PTIME equals set-stabilizer

Statement
The group intersection problem for subgroups of the permutation group is polynomial-time equivalent to the set stabilizer problem.

Group intersection problem
Here, two subgroups $$G_1$$ and $$G_2$$ of the symmetric group on a set $$S$$ are specified by means of generating sets $$A_1$$ and $$A_2$$, and we need to find a small generating set for $$G_1$$ &cap; $$G_2$$.

Set-stabilizer problem
Here, we have a set $$S$$ and a group $$G$$ acting on $$S$$. Let $$T$$ be a subset of $$S$$. We need to find the subgroup of $$G$$ comprising those elements which stabilize $$T$$ as a set.

Set stabilizer reduces to group intersection
Consider the data $$(G,S,T)$$ as given for the set stabilizer problem. Now, send the following data to the group intersection problem:


 * Set $$S$$ to be the same as the $$S$$ for the set stabilizer problem
 * Set $$G_1 = G$$
 * Set $$G_2 = Sym(T)$$ &times; $$Sym(T^c)$$ viz $$G_2$$ is the stabilizer of $$S$$ in the whole of the symmetric group

Note that $$G_2$$ is actually a group with a very small generating set (a generating set of size 4).

The output given by this problem is precisely the same as the set stabilizer of $$S$$ in $$G$$.

Note that this is a many-one reduction.

Group intersection reduces to set stabilizer
Consider the data $$(G_1,G_2,S)$$ as given for the group intersection problem. Consider the set $$S^2$$ with $$G_1$$ acting on the first coordinate and $$G_2$$ acting on the second coordinate.

Then send the following data to the set stabilizer problem:


 * Set $$S$$ of the set stabilizer problem to $$S^2$$
 * Set $$G$$ of the set stabilizer problem to be $$G_1$$ &times; $$G_2$$
 * Set $$T$$ of the set stabilizer problem as the set of diagonal elements in $$S^2$$

The set stabilizer of $$T$$ is thus precisely those pairs of elements $$(g,h)$$ such that for every $$s$$ in $$S$$, $$g.s = h.s$$. This forces the set stabilizer of $$T$$ to comprise only elements of the form $$(g,h)$$ where $$g = h$$, which forces the set stabilizer to be the diagonal subgroup of $$G_1$$ &cap; $$G_2$$ &times; $$G_1$$ &times; $$G_2$$. Finding a generating set for this diagonal subgroup is the same as finding a generating set for $$G_1$$ &cap; $$G_2$$.