Computing in quotient groups

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients.

[1]  Charles C. Sims,et al.  Some group-theoretic algorithms , 1978 .

[2]  László Babai,et al.  Moderately Exponential Bound for Graph Isomorphism , 1981, FCT.

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

[4]  William M. Kantor,et al.  Polynomial-Time Versions of Sylow's Theorem , 1988, J. Algorithms.

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

[6]  Gene Cooperman,et al.  Reduction of group constructions to point stabilizers , 1989, ISSAC '89.

[7]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[8]  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).

[9]  Peter J. Cameron,et al.  On the orders of primitive groups with restricted nonabelian composition factors , 1982 .

[10]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

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

[12]  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).

[13]  Gary L. Miller,et al.  Isomorphism of k-Contractible Graphs. A Generalization of Bounded Valence and Bounded Genus , 1983, Inf. Control..

[14]  Gary L. Miller.  Isomorphism of Graphs Which are Pairwise k-separable , 1983, Inf. Control..

[15]  William M. Kantor.  Finding Sylow Normalizers in Polynomial Time , 1990, J. Algorithms.

[16]  William M. Kantor,et al.  Sylow's Theorem in Polynomial Time , 1985, J. Comput. Syst. Sci..

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

[18]  William M. Kantor.  Polynomial-Time Algorithms for Finding Elements of Prime Order and Sylow Subgroups , 1985, J. Algorithms.

[19]  Endre Szemerédi,et al.  On the Complexity of Matrix Group Problems I , 1984, FOCS.

[20]  Silvio Micali,et al.  Proofs that yield nothing but their validity and a methodology of cryptographic protocol design , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

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

[22]  R. Heller,et al.  © Macmillan , 1977 .

[23]  B. McKay.  nauty User ’ s Guide ( Version 2 . 4 ) , 1990 .

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

[25]  Lajos Rónyai,et al.  Computing the Structure of Finite Algebras , 1990, J. Symb. Comput..

[26]  Ludek Kucera,et al.  Canonical labeling of regular graphs in linear average time , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[27]  P. M. Neumann.  Proceedings of Groups – St Andrews 1985: Some algorithms for computing with finite permutation groups , 1987 .

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

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

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

[31]  Martin Fürer,et al.  Normal forms for trivalent graphs and graphs of bounded valence , 1983, STOC.