Testing polycyclicity of finitely generated rational matrix groups

Assmann, Björn; Eick, Bettina

doi:10.1090/S0025-5718-07-01979-5

Testing polycyclicity of finitely generated rational matrix groups
HTML articles powered by AMS MathViewer

by Björn Assmann and Bettina Eick PDF

Math. Comp. 76 (2007), 1669-1682 Request permission

Abstract:

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 $\mathrm {GL}(d,\mathbb {Q})$.

References

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. www.gap-system.org, 2006.