Fast computation of the centralizer of a permutation group in the symmetric group

Abstract

Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G-orbit Δ of size n, compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G-orbits Δ and ${\Delta }^{\prime }$ each of size n, find an equivalence between the action of G restricted to Δ and the action of G restricted to ${\Delta }^{\prime }$ when one exists. If G is given by a generating set X, previous solutions to each of these two problems take $O\left(\right|X\left|n^2\right)$ time.

In this paper, we first solve each fundamental problem in $O(\delta n+|X|n\log n)$ time, where δ is the depth of the breadth-first-search Schreier tree for X rooted at some fixed vertex. For the important class of small-base groups G, we improve the theoretical worst-case time bound of our solutions to $O(n{\log }^{c}n+|X|n\log n)$ for some constant c. Moreover, if $\lceil 20{\log }_{2}n\rceil$ uniformly distributed random elements of G are given in advance, our solutions have, with probability at least $1-1/n$, a running time of $O(n{\log }^{2}n+|X|n\log n)$. We then apply our solutions to obtain a more efficient algorithm for computing the centralizer of G in the symmetric group on Ω. In an experimental evaluation we demonstrate that it is substantially faster than previously known algorithms.