ABSTRACT
Group membership is a fundamental algorithm, upon which most other algorithms of computational group theory depend. Until now, group membership for permutation groups has been limited to ten million points or less. We extend the applicability of group membership algorithms to permutation groups acting on more than 100,000,000 points. As an example, we experimentally construct a group membership data structure for Thompson's group, acting on 143,127,000 points, in 36 minutes. More significantly, we require approximately 10 GB of RAM for the computation --- even though a single permutation of Thompson's group already requires half a gigabyte of storage.In addition, we propose a disk-based group membership algorithm with the promise of extending group membership to well over one billion (1,000,000,000) points. Such a disk-based algorithm has formerly been impossible, due in part to the lack of a practical disk-based algorithm for multiplying and taking inverses of such large permutations. Random access to disk is prohibitively expensive. We demonstrate the first practical disk-based implementation of the basic permutation operations. We also propose a disk-based architecture for group membership data structures.
- L. Babai. Local expansion of vertex-transitive graphs and random generation in finite groups. In Theory of Computing, pages 164--174, New York, 1991. (Los Angeles, 1991), Association for Computing Machinery. Google ScholarDigital Library
- L. Babai, G. Cooperman, L. Finkelstein, E. M. Luks, and A. Seress. Fast Monte Carlo algorithms for permutation groups. J. Comp. Syst. Sci., 50:296--308, 1995. Google ScholarDigital Library
- L. Babai, G. Cooperman, L. Finkelstein, and A. Seress. Nearly linear time algorithms for permutation groups with a small base. In Proc. of International Symposium on Symbolic and Algebraic Computation ISSAC '91, pages 200--209. (Bonn), ACM Press, 1991. Google ScholarDigital Library
- L. Babai, E. M. Luks, and A. Seress. Fast management of permutation groups I. SIAM J. Computing, 26:1310--1342, 1997. Google ScholarDigital Library
- W. Bosma, J. Cannon, and C. Playoust. The magma algebra system i: The user language. J. Symbolic Comput., 24:235--265, 1997. Google ScholarDigital Library
- G. Butler and J. J. Cannon. Computing in permutation and matrix groups I: Normal closure, commutator subgroups, series. Math. Comp., 39:663--670, 1982.Google Scholar
- F. Celler, C. R. Leedham-Green, S. H. Murray, A. C. Niemeyer, and E. O'Brien. Generating random elements of a finite group. Comm. Algebra, 23:4931--4948, 1995.Google ScholarCross Ref
- G. Cooperman. Towards a practical, theoretically sound algorithm for random generation in finite groups. arXiv:math.PR/0205203, http://arxiv.org/abs/math.PR/0205203.Google Scholar
- G. Cooperman and L. Finkelstein. New methods for using cayley graphs in interconnection networks. Discrete Applied Mathematics, 37/38:95--118, 1992. (special issue on Interconnection Networks). Google ScholarDigital Library
- G. Cooperman and L. Finkelstein. Randomized algorithms for permutation groups. Centrum Wissenschaft Institut Quarterly (CWI), pages 107--125, June 1992.Google Scholar
- G. Cooperman and L. Finkelstein. Combinatorial tools for computational group theory. In Groups and Computation, volume 11 of Amer. Math. Soc. DIMACS Series, pages 53--86. (DIMACS, 1991), 1993.Google Scholar
- G. Cooperman and L. Finkelstein. A random base change algorithm for permutation groups. J. Symbolic Comput., 17:513--528, 1994. Google ScholarDigital Library
- G. Cooperman, L. Finkelstein, and N. Sarawagi. A random base change algorithm for permutation groups. In Proc. of International Symposium on Symbolic and Algebraic Computation ISSAC '90, pages 161--168, Tokyo, Japan, 1990. Google ScholarDigital Library
- G. Cooperman, L. Finkelstein, M. Tselman, and B. York. Constructing permutation representations for matrix groups. J. Symbolic Comput., 1997. Google ScholarDigital Library
- G. Cooperman, L. Finkelstein, B. York, and M. Tselman. Constructing permutation representations for large matrix groups. In Proceedings of International Symposium on Symbolic and Algebraic Computation ISSAC '94, pages 134--138, New York, 1994. (Oxford), ACM Press. Google ScholarDigital Library
- G. Cooperman and V. Grinberg. Scalable parallel coset enumeration: Bulk definition and the memory wall. J. Symbolic Comput., 33:563--585, 2002. Google ScholarDigital Library
- G. Cooperman and G. Havas. Practical parallel coset enumeration. In Workshop on High Performance Computing and Gigabit Local Area Networks, volume 226 of Lecture Notes in Control and Information Sciences, pages 15--27, 1997.Google ScholarCross Ref
- G. Cooperman, G. Hiss, K. Lux, and J. Müller. The Brauer tree of the principal $19$-block of the sporadic simple thompson group. J. Experimental Math., 6:293--300, 1997.Google ScholarCross Ref
- G. Cooperman, W. Lempken, G. Michler, and M. Weller. A new existence proof of Janko's simple group J4. In Computational Methods for Representations of Groups and Algebras, volume 173 of Progress in Mathematics, pages 161--175, 1999.Google ScholarCross Ref
- G. Cooperman and X. Ma. Overcoming the memory wall in symbolic algebra: A faster permutation algorithm (formally reviewed communication). SIGSAM Bulletin, 36:1--4, Dec. 2002. Google ScholarDigital Library
- G. Cooperman and M. Tselman. New sequential and parallel algorithms for generating high dimension Hecke algebras using the condensation technique. In Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC '96), pages 155--160. ACM Press, 1996. Google ScholarDigital Library
- The GAP Group. GAP --- Groups, Algorithms, and Programming, Version 4.3, 2002. http://www.gap-system.org.Google Scholar
- H. Gollan. A new existence proof for Ly, the sporadic simple group of R. Lyons. Preprint 30, 1995.Google Scholar
- H. Gollan. A contribution to the revision project of the sporadic groups: Lyons' simple group Ly. Vorlesungen aus dem FB Mathematik, 26, 1997.Google Scholar
- H. Gollan. A new existence proof for Ly, the sporadic simple group of R. Lyons. J. Symbolic Comput., 31:203--209, 2001. Google ScholarDigital Library
- H. Gollan and G. Havas. On Sims' presentation for Lyons' simple group. In Computational Methods for Representations of Groups and Algebras, volume 173 of Progress in Mathematics, pages 235--240, 1999.Google ScholarCross Ref
- G. Havas and C. Sims. A presentation for the Lyons simple group. In Computational Methods for Representations of Groups and Algebras, volume 173 of Progress in Mathematics, pages 241--249, 1999.Google ScholarCross Ref
- G. Havas, L. Soicher, and R. Wilson. A presentation for the Thompson sporadic simple group. In Groups and Computation III, pages 193--200, New York, 2001. (Ohio, 1999), de Gruyter.Google Scholar
- W. M. Kantor. Sylow's theorem in polynomial time. J. Comp. Syst. Sci., 30:359--394, 1985.Google ScholarCross Ref
- C. Leedham-Green. The computational matrix group project. In Groups and Computation rm III, pages 229--248, New York, 2001. (Ohio, 1999), de Gruyter.Google Scholar
- C. Leedham-Green, E. O'Brien, and C. Praeger. Recognising matrix groups. In J. Grabmeier, E. Kaltofen, and V. Weispfenning, editors, Computer Algebra Handbook, pages 474--475, 2003.Google Scholar
- E. M. Luks. Computing the composition factors of a permutation group in polynomial time. Combinatorica, 7:87--99, 1987. Google ScholarDigital Library
- I. Pak. The product replacement algorithm is polynomial. In Proc. 41st IEEE Symposium on Foundations of Computer Science (FOCS), pages 476--485. IEEE Press, 2000. Google ScholarDigital Library
- R. Ramakrishnan and J. Gehrke. Database Management Systems. McGrawHill, second edition, 2000. Google ScholarDigital Library
- C. C. Sims. Computation with permutation groups. In Proc. Second Symp. on Symbolic and Algebraic Manipulation. ACM Press, 1971. Google ScholarDigital Library
- M. Weller. Construction of large permutation representations for matrix groups. In W. J. E. Krause, editor, High Performance Computing in Science and Engineering '98, pages 430--. Springer, 1999.Google Scholar
- M. Weller. Construction of large permutation representations for matrix groups ii. Applicable Algebra in Engineering, Communication and Computing, 11:463--488, 2001.Google Scholar
- M. Weller. Computer aided existence proof of Thompson's sporadic simple group. manuscript, 2003.Google Scholar
- R. Wilson. Atlas of finite group representations. http://www.mat.bham.ac.uk/atlas.Google Scholar
Index Terms
- Memory-based and disk-based algorithms for very high degree permutation groups
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