Subcomplete generalizations of graph isomorphism

We present eight group-theoretic problems in NP one of which is a reformulation of graph isomorphism. We give technical evidence that none of the problems is NP-complete, and give polynomial time reductions among the problems. There is a good possibility that seven of these problems are harder than graph isomorphism (relative to polynomial time reduction), so that they might be examples of natural problems of intermediate difficulty, situated properly between the class of NP-complete problems and the class P of problems decidable in deterministic polynomial time. Because of strong structural similarity, two of the apparently harder problems can be interpreted as generalized isomorphism and generalized automorphism, respectively. Whether these problems ultimately prove to be harder than graph isomorphism seems to depend, in part, on the open problem whether every permutation group of degree n arises as the automorphism group of a combinatorial structure of size polynomial in n. Finally, we give an O(n² · k) algorithm for constructing the centralizer of a permutation group of degree n presented by a generating set of k permutations. Note that we may assume that k is O(n · log n), without loss of generality. This problem is a special case of one of the harder group-theoretic problems. From the method of constructing the centralizer, we recover results about the group-theoretic structure of the centralizer. Furthermore, applying our algorithm for intersecting with a normalizing permutation group, we show how to find the center of a permutation group of degree n in O(n⁶) steps, having constructed the centralizer of the group first.