Enumerations and Classifications



Bettina Eick




The classification of groups of `small' order is another of my main research projects. One milestone in this project was the classification of all groups of order at most 2000 (except 1024). The result is available in the well-known Small Groups Library. In recent years, the classification of groups of order at most 20.000 (with few exceptions) has been an interesting project. The enumeration of groups of certain types of orders is a highly interesting more recent research project. This includes questions whose solution is PORC (polynomial on residue classes) as initiated by Graham Higman.


Small Groups and Group Classifications
  • Constructing groups of 'small' order: Recent results and open problems. With M. Horn and A. Hulpke. In Proceedings: Algorithmic and experimental methods in algebra, geometry, and number theory. Springer (ISBN 978-3-319-70565-1 and 978-3-319-70566-8), 199 -- 211 (2017).
  • The construction of finite solvable groups revised. With M. Horn. J. Algebra 408, 166 -- 182 (2014).
  • On the groups of cube-free order. With Heiko Dietrich. J. Algebra 292, 122 - 137 (2005)
  • A millennium project: constructing small groups. With Hans Ulrich Besche and E. A. O'Brien. International Journal of Algebra and Computation 12, 623 - 644 (2002)
  • The groups of order at most 2000. With Hans Ulrich Besche and E. A. O'Brien. Electronic Research Announcemen Amer. Math. Soc. 7, 1-4 (2001)
  • The groups of order q^n * p. With Hans Ulrich Besche. Comm. Alg. 29(4), 1759 - 1772 (2001)
  • Enumerating p-groups. With E. A. O'Brien. J. Austral. Math. Soc. (Series A) 67, 191 - 205 (1999)
  • Construction of finite groups. With Hans Ulrich Besche. J. Symb. Comput. 27 (4), 387 - 404 (1999)
  • The groups of order at most 1000 except 512 and 768. With Hans Ulrich Besche. J. Symb. Comput. 27 (4), 405 - 413 (1999)
  • The groups of order 512. With E. A. O'Brien. Proceedings "Abschlusstagung des DFG Schwerpunktes Algorithmische Algebra und Zahlentheorie" 1997, B.H. Matzat, G.-M. Greuel and G. Hiss, eds, Springer 379 - 380 (1998)
Classification of Torsion Free Nilpotent Groups
  • The torsion free nilpotent groups of Hirsch length at most 5. With A.-K. Engel. Groups Complex. Cryptol. 9, 55 -- 75 (2017)
Enumerations and PORC
  • Counting p-groups and Lie algebras using PORC formulae With M. Vaughan-Lee. J. Algebra (to appear)
  • Enumeration of groups of order p^n q for n at most 5. With T. Moede. J. Algebra 507, 571 -- 591 (2018).
  • Enumeration of groups whose order factorises in at most 4 primes. ArXiv 1702.02616
  • Enumeration of nilpotent associative algebras of class 2 over arbitrary finite fields. With M. Wesche. J. Algebra 503, 573 -- 589 (2018).