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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Testing polycyclicity of finitely generated rational matrix groups
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by Björn Assmann and Bettina Eick PDF
Math. Comp. 76 (2007), 1669-1682 Request permission


We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of $\mathrm {GL}(d,\mathbb {Q})$ and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of $\mathrm {GL}(d,\mathbb {Q})$.
  • B. Assmann. Polenta - Polycyclic presentations for matrix groups, 2006. A refereed GAP 4 package, see [14].
  • Björn Assmann and Bettina Eick, Computing polycyclic presentations for polycyclic rational matrix groups, J. Symbolic Comput. 40 (2005), no. 6, 1269–1284. MR 2178086, DOI 10.1016/j.jsc.2005.05.003
  • L. Babai, R. Beals, and D. Rockmore. Deciding finiteness of matrix groups in deterministic polynomial time. In Proc. of International Symposium on Symbolic and Algebraic Computation ISSAC ’93, pages 117–126. (Kiev), ACM Press, 1993.
  • Gilbert Baumslag, Lecture notes on nilpotent groups, Regional Conference Series in Mathematics, No. 2, American Mathematical Society, Providence, R.I., 1971. MR 0283082
  • Gilbert Baumslag, Frank B. Cannonito, Derek J. Robinson, and Dan Segal, The algorithmic theory of polycyclic-by-finite groups, J. Algebra 142 (1991), no. 1, 118–149. MR 1125209, DOI 10.1016/0021-8693(91)90221-S
  • Robert Beals, Improved algorithms for the Tits alternative, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 63–77. MR 1829471
  • L. E. Dickson. Algebras and their arithmetics. University of Chicago, 1923.
  • B. Eick. Algorithms for polycyclic groups. Habilitationsschrift, Universität Kassel, 2001.
  • Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2129747, DOI 10.1201/9781420035216
  • E. I. Khukhro, $p$-automorphisms of finite $p$-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819, DOI 10.1017/CBO9780511526008
  • A. I. Mal′cev, On some classes of infinite soluble groups, Mat. Sbornik N.S. 28(70) (1951), 567–588 (Russian). MR 0043088
  • Gretchen Ostheimer, Practical algorithms for polycyclic matrix groups, J. Symbolic Comput. 28 (1999), no. 3, 361–379. MR 1716421, DOI 10.1006/jsco.1999.0287
  • Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953
  • The GAP Group. GAP–Groups, Algorithms and Programming., 2006.
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Additional Information
  • Björn Assmann
  • Affiliation: Centre for Interdisciplinary Research in Computational Algebra (CIRCA), University of St Andrews, North Haugh, St Andrews, KY16 9SS Fife, Scotland
  • Email:
  • Bettina Eick
  • Affiliation: Institut Computational Mathematics, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig, Braunschweig, Germany
  • MR Author ID: 614875
  • Email:
  • Received by editor(s): February 21, 2006
  • Received by editor(s) in revised form: August 3, 2006
  • Published electronically: March 9, 2007
  • Additional Notes: The first author was supported by a Ph.D. fellowship of the “Gottlieb Daimler- und Karl Benz-Stiftung" and the UK Engineering and Physical Science Research Council (EPSRC)
    The second author was supported by a Feodor Lynen Fellowship from the Alexander von Humboldt Foundation and by the Marsden Fund of New Zealand via grant UOA412
  • © Copyright 2007 American Mathematical Society
  • Journal: Math. Comp. 76 (2007), 1669-1682
  • MSC (2000): Primary 20F16, 20-04; Secondary 68W30
  • DOI:
  • MathSciNet review: 2299794