An O(n3log n) deterministic and an O(n3) Las Vegs isomorphism test for trivalent graphs
Journal of the ACM (JACM)
Reduction of group constructions to point stabilizers
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Fast Monte Carlo algorithms for permutation groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Nearly linear time algorithms for permutation groups with a small base
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Computation with permutation groups
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
An elementary algorithm for computing the composition factors of a permutation group
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Finding blocks of imprimitivity in small-base groups in nearly linear time
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
A nearly linear algorithm for Sylow subgroups in small-base groups
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
The growth rate of vertex-transitive planar graphs
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
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A base of a permutation group G is a subset B of the permutation domain such that only the identity of G fixes B pointwise. The permutation representations of important classes of groups, including all finite simple groups other than the alternating groups, admit O(log n) size bases, where n is the size of the permutation domain. Groups with very small bases dominate the work on permutation groups within computational group theory.We use the “soft” version of the big-O notation introduced by [BLSI]: for two functions f(n), g(n), we write f(n)=O˜(g(n)) if for some constants c, C 0, we have f(n) ≤ Cg(n) logcn.We address the problems of finding structure trees and composition factors for permutation groups with small (O˜(1) size) bases. For general permutation groups, a method of Atkinson will find a structure tree in O(n2) time. We give an O˜(n) algorithm for the small base case. The composition factor problem was first shown to have a polynomial time solution by Luks [Lu], and recently Babai, Luks, Seress [BLS2] gave an O˜(n3) algorithm. The [BLS2] algorithm takes &THgr;(n3) time even in the small base case. We overcome several quadratic and cubic bottlenecks in the [BLS2] algorithm to give an O˜(n) Monte Carlo algorithm for the small base case. In addition, we show that the center of a small base group can be found in time O˜(n).