Fast management of permutation groups

Novel algorithms for computation in permutation groups are presented. They provide an order-of-magnitude improvement in the worst-case analysis of the basic permutation-group problems, including membership testing and computing the order of the group. For deeper questions about the group, including finding composition factors, an improvement of up to four orders of magnitude is realized. These and other essential investigations are all accomplished in O(n/sup 4/log/sup c/n) time. The approach is distinguished by its recognition and use of the intrinsic structure of the group at hand.<<ETX>>

[1]  Christoph M. Hoffmann,et al.  Group-Theoretic Algorithms and Graph Isomorphism , 1982, Lecture Notes in Computer Science.

[2]  Zvi Galil,et al.  An O(n3log n) deterministic and an O(n3) Las Vegs isomorphism test for trivalent graphs , 1987, JACM.

[3]  William M. Kantor.  Permutation Representations of the Finite Classical Groups of Small Degree or Rank , 1979 .

[4]  Paul Walton Purdom,et al.  A New Base Change Algorithm for Permutation Groups , 1989, SIAM J. Comput..

[5]  M. Hall.  The Theory Of Groups , 1959 .

[6]  Mike D. Atkinson,et al.  An algorithm for finding the blocks of a permutation group , 1975 .

[7]  E. Landau.  Handbuch der Lehre von der Verteilung der Primzahlen , 1974 .

[8]  Martin W. Liebeck.  On minimal degrees and base sizes of primitive permutation groups , 1984 .

[9]  László Babai,et al.  On the degree of transitivity of permutation groups: A short proof , 1987, J. Comb. Theory, Ser. A.

[10]  E. Landau,et al.  Handbuch der Lehre von der Verteilung der Primzahlen , 1974 .

[11]  Eugene M. Luks,et al.  Parallel algorithms for permutation groups and graph isomorphism , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[12]  Charles C. Sims,et al.  Graphs and finite permutation groups , 1967 .

[13]  P. Cameron.  FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS , 1981 .

[14]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[15]  László Babai,et al.  Permutation groups in NC , 1987, STOC '87.

[16]  Mark Jerrum,et al.  A compact representation for permutation groups , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[17]  R. Carmichael,et al.  Introduction to the theory of groups of finite order , 1908 .

[18]  Pierre McKenzie,et al.  Fast parallel computation with permutation groups , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[19]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[20]  John E. Hopcroft,et al.  Polynomial-time algorithms for permutation groups , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[21]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[22]  Eugene M. Luks.  Computing the composition factors of a permutation group in polynomial time , 1987, Comb..

[23]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .

[24]  John H. Reif,et al.  Probabilistic algorithms in group theory , 1985, FCT.

[25]  W. Kantor,et al.  The 2-transitive permutation representa-tions of the finite Chevalley groups , 1976 .

[26]  L. Babai.  On the Order of Uniprimitive Permutation Groups , 1981 .

[27]  László Babai,et al.  Computational complexity and the classification of finite simple groups , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[28]  J. Leon.  On an algorithm for finding a base and a strong generating set for a group given by generating permutations , 1980 .

[29]  C. Sims.  Computational methods in the study of permutation groups , 1970 .

[30]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[31]  László Babai,et al.  Bounds on the diameter of Cayley graphs of the symmetric group , 1988, Journal of Algebraic Combinatorics.

[32]  László Babai,et al.  A Las Vegas - NC algorithm for isomorphism of graphs with bounded multiplicity of eigenvalues , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[33]  László Babai.  On the length of subgroup chains in the symmetric group , 1986 .

[34]  D. Gorenstein.  Finite simple groups and their classification , 1974 .

[35]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[36]  R. C. Bose.  Strongly regular graphs, partial geometries and partially balanced designs. , 1963 .

[37]  Dexter Kozen,et al.  Lower bounds for natural proof systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[38]  László Babai,et al.  Canonical labeling of graphs , 1983, STOC.