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On the orbit-stabilizer problem for integral matrix actions of polycyclic groups

Authors: Bettina Eick and Gretchen Ostheimer
Journal: Math. Comp. 72 (2003), 1511-1529
MSC (2000): Primary 20F16, 20-04; Secondary 68W30
Published electronically: February 3, 2003
MathSciNet review: 1972750
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Abstract | References | Similar Articles | Additional Information

Abstract: We present an algorithm to solve the orbit-stabilizer problem for a polycyclic group $G$ acting as a subgroup of $GL(d, \mathbb Z)$ on the elements of $\mathbb Q^d$. We report on an implementation of our method and use this to observe that the algorithm is practical.

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  • 1. Gilbert Baumslag, Frank B. Cannonito, Derek J. S. Robinson, and Dan Segal, The algorithmic theory of polycyclic-by-finite groups, J. Algebra 142 (1991), 118-149. MR 92i:20036
  • 2. Gregory Butler, Fundamental algorithms for permutation groups, Lecture Notes in Comput. Sci., vol. 559, Springer-Verlag, New York, Heidelberg, Berlin, 1991. MR 94d:68049
  • 3. Henri Cohen, A course in computational algebraic number theory, Graduate texts in mathematics, vol. 138, Springer-Verlag, New York, 1995. MR 94i:11105
  • 4. M. Daberkow, C.Fieker, J. Klüners, M. Pohst, K.Roegner, and K. Wildanger, Kant V4, J. Symb. Comput. 24 (1997), 267-283. MR 99g:11150
  • 5. Karel Dekimpe, Almost-bieberbach groups: Affine and polynomial structures, Lecture notes in Math., vol. 1639, Springer, 1996. MR 2000b:20066
  • 6. Karel Dekimpe and Bettina Eick, Aclib, 2000, A GAP share package, see [15].
  • 7. John D. Dixon, The orbit-stabilizer problem for linear groups, Can. J. Math. 37 (1985), 238-259. MR 86m:20039
  • 8. Bettina Eick, Algorithms for polycyclic groups, Habilitationsschrift, Universität Kassel, 2001.
  • 9. Bettina Eick and Werner Nickel, Polycyclic, 2000, A GAP share package, see [15].
  • 10. R. Laue, J. Neubüser, and U. Schoenwaelder, Algorithms for finite soluble groups and the SOGOS system, Computational Group Theory, Durham, 1982, Academic Press, 1984, pp. 105-135. MR 86h:20023
  • 11. Eddie H. Lo, A polycyclic quotient algorithm, J. Symbolic Computation 25 (1998), 61-97. MR 99c:20040
  • 12. Gretchen Ostheimer, Practical algorithms for polycyclic matrix groups, J. Symbolic Computation 28 (1999), 361-379. MR 2000h:20004
  • 13. Michael E. Pohst, Computational algebraic number theory, DMV Seminar, vol. 21, Birkäuser, 1993. MR 94j:11132
  • 14. Charles C. Sims, Computation with finitely presented groups, Encyclopedia of mathematics and its applications, Cambridge University Press, New York, 1994. MR 95f:20053
  • 15. The GAP Group, GAP--Groups, Algorithms and Programming,, 2000.

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Additional Information

Bettina Eick
Affiliation: Institut für Geometrie, Universität Braunschweig, 38106 Braunschweig, Germany

Gretchen Ostheimer
Affiliation: Department of Computer Science, 103 Hofstra University, Hempstead, New York 11549

Received by editor(s): July 9, 2001
Published electronically: February 3, 2003
Additional Notes: The authors thank Werner Nickel for useful discussions.
Article copyright: © Copyright 2003 American Mathematical Society

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