Graph isomorphism problem and 2-closed permutation groups

Abstract

We demonstrate a number of hard permutation group algorithmic problems, such as Group Intersection and Setwise Stabilizer, which become polynomially equivalent to the Graph Isomorphism Problem after restricting them to 2-closed permutation groups. It is shown that these problems are closely related to problems concerning coherent configurations. We also present a polynomial-time procedure for determining the 2-closure of a nilpotent group. This procedure provides a polynomial-time reduction of the 2-closure problem for odd order groups to the same problem for primitive odd order groups.