Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair of subgroups H and H' of G and G' whose action on S coincide?



Two groups acting on a set.
[This paper](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CDIQFjAC&url=http%3A%2F%2Fwww.cs.trincoll.edu%2F~miyazaki%2Fbeijing.ps&ei=k_xjT863JI_NtgfZ5rT-DQ&usg=AFQjCNF0S6ua5ZOz2GBOutjjsxOVBdP0KQ) studies the (easier) problem of checking if the intersection of two subgroups of $S\_n$ is trivial. In particular, it is shown that the graph isomorphism problem polynomially reduces to that problem. The converse reduction is not known. Thus the problem of checking that the intersection is trivial is at least as hard as the  [graph isomorphism problem](http://en.wikipedia.org/wiki/Graph_isomorphism_problem). It is difficult but not known to be NP-hard. 


**Update** Also look at [this](http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1479.1490.ocr.pdf) ICM talk by Babai.


