A Strong Generating Test and Short Presentation for Permutation Groups

The group membership problem for permutation groups is perhaps the most important problem of computational group theory. Solution of this problem seems to depend intrinsically on constructing a strong generating set. Until now, recognizing if a set of generators is strong has been thought to be as hard as constructing a strong generating set from an arbitrary generating set. This paper shows how to verify a strong generating set in O(n) time, where n is the size of the set on which the group acts. This is faster than the best known algorithms in the literature. The work also leads to related algorithms for discovering all orbit information contained in an arbitrary set of generators S in O(n|S|+ n log n) time, and, if S is strong, for finding a presentation with no more than |S|(n− 1) relations. Refinements in the analysis are given for the case in which a small base exists.