Computer Algebra



Bettina Eick




Algorithms to compute with groups, associative algebras and Lie algebras has been one of my main research projects throughout many years. Central topics are algorithms for finite or infinite polycyclic groups, classifications of various types of groups and rings and algorithms to compute with associative algebras and Lie algebras. Below are my main publications in these areas.


Book and Survey
  • Handbook of Computational Group Theory. Derek Holt with Bettina Eick and Eamonn O'Brien. CRC Press, ISBN 1-58488-372-3 (2005)
  • Computational Group Theory. (Survey) Jahresberichte der DMV 107, Heft 3, 155 - 170 (2005)
Algorithms in Group Theory
  • The conjugacy problem in GL(n,Z). With T. Hofmann and E.A. O'Brien. J. London Math. Soc. (to appear)
  • Polynomials describing the multiplication in finitely generated torsion-free nilpotent groups. With A. Cant. J. Symb. Comput.
  • The automorphism group of a finitely generated virtually abelian group. Groups Complex. Cryptol. 8, 35 -- 45 (2016)
  • The constructive membership problem for discrete free subgroups of rank 2 of SL(2,R). With M. Kirschmer and C.R. Leedham-Green. LMS J. Comput. Math. 17, 345 -- 359 (2014), 166--182.
  • Computing Hall subgroups of finite groups. With A. Hulpke. LMS J. Comput. Math. 15, 205 -- 218 (2012).
  • Collection by polynomials in finite p-groups. Contemp. Math. 582, 95 -- 103 (2012).
  • Computing with matrix groups over infinite fields. With A. Detinko and D. Flannery. In Proceedings of Groups St. Andrews 2009, 256--270 (2011).
  • Periodicities and p-groups of given rank, width and obliquity. With T. Rossmann. In Proceedings of the AMS Special Session 'Computational Group Theory' II, 11--24 (2010)
  • Computing $p$-groups with trivial Schur multiplicator. J. Algebra 322, 741 -- 751 (2009)
  • Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group. With W. Nickel. J. Algebra 320, 927 -- 944 (2008)
  • Testing polycyclicity of finitely generated rational matrix groups. With B. Assmann. Math. Comp. 76, 1669 -- 1682 (2007)
  • Computing polycyclic presentations for polycyclic rational matrix groups. With Björn Assmann. J. Symb. Comput. 40, 1269 -- 1284 (2005)
  • Special polycyclic generating sequences for finite soluble groups. With John J. Cannon and Charles R. Leedham-Green. J. Symb. Comput. 38, 1445 - 1460 (2004)
  • Computations with almost-crystallographic groups. With Karel Dekimpe. Proceedings "Groups St. Andrews in Oxford", 2001, C.M. Campbell, E.F. Robertson, G.C. Smith, eds., London Math. Soc. 110 - 119 (2003)
  • On the orbit-stabilizer problem for integral matrix actions of polycyclic groups. With Gretchen Ostheimer. Math. Comp. 72, (Number 243) 1511 - 1529 (2003)
  • The solvable primitive permutation groups of degree at most 6560. With Burkhard Höfling. Electronic J. LMS 6, 29--39 (2003)
  • Orbit-stabilizer problems and computing normalizers for polycyclic groups. J. Symb. Comput. 24, 1 - 19 (2002)
  • Constructing automorphism groups of $p$-groups. With C. R. Leedham-Green and E. A. O'Brien. Comm. Alg. 30(5), 2271 - 2295 (2002)
  • Computational aspects of group extensions and their applications in topology. With Karel Dekimpe. Exp. math. 11, 183 - 199 (2002)
  • On the Fitting subgroup of a polycyclic-by-finite group and its applications. J. Algebra. 242, 176 - 187 (2001)
  • Computing with infinite polycyclic groups. Proceedings "Groups and Computation III" 1999, W. M. Kantor and A. Seress, ed., de Gruyter, 139 - 153 (2001)
  • Computing the maximal subgroups of a permutation group I. With Alexander Hulpke. Proceedings "Groups and Computation III" 1999, W. M. Kantor and A. Seress, ed., de Gruyter, 155 - 168 (2001)
  • Special presentations for finite soluble groups and computing (Pre-) Frattini subgroups. Proceedings "Groups and Computation II" 1995, L. Finkelstein and W. M. Kantor, ed., Amer. Math. Soc. 101 - 112 (1997)
Algorithms for Associative Algebras
  • Computing nilpotent quotients of associative algebras and algebras satisfying a polynomial identity. IJAC 21 (8), 1339--1355 (2011).
  • Computing automorphism groups and testing isomorphisms for modular group algebras. J. Algebra 320, 3895--3910 (2008)
Algorithms for Lie Algebras
  • Some new simple Lie algebras in characteristic 2. J.Symbol. Comput. 45, 943 -- 951 (2010)
  • Computing faithful representations for nilpotent Lie algebras. With D. Burde and W. de Graaf. J. Algebra 322, 602 -- 612 (2009)
  • Computing the automorphism group of a solvable Lie algebra. Lin. Alg. Appl. 382, 195 - 209 (2004)