The Small Groups library provides access to descriptions of the
groups of ``small" order. The groups are listed up to isomorphism.
At present, the library contains the following groups:
The Small Groups library has been prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien. The groups in the Small Groups library have been been constructed by the authors of the library with the help of Mike Newman (orders p^4, p^5 and p^6), Boris Girnat (order p^5),Mike Vaughan-Lee (orders p^6 and p^7) and Heiko Dietrich (squarefree and cubefree orders).
The construction of all groups of a given order is one of the oldest
problems in finite group theory. It was initiated by Cayley in 1854
when he determined all groups of order 4 and 6. |
Historically, the approaches to this problem involved a large number of hand-computations and case distinctions, and focused on very specific properties of the groups. They were consequently ad-hoc in nature, and many contained significant errors.
For the determination of the Small Groups library, we developed practical algorithms to construct the groups of a given order. While these methods rely on group-theoretic properties, they are inherently general-purpose and their results appear reliable.
A survey of the history of group construction is available in ``A millennium project: constructing Small Groups'', Internat. J. Algebra Comput. 12, 623 - 644 (2002) by Besche, Eick and O'Brien.
The library was developed in various stages. It incorporates the 2-and
3-group libraries determined and distributed by Newman and O'Brien. These
and all other p-groups of order at most 2000 were constructed using the
p-group generation algorithm. They were used to obtain all nilpotent
groups of order at most 2000 in the catalogue.
The non-nilpotent groups of order at most 2000 in the library have been constructed by Besche and Eick using the methods of the GrpConst package of GAP 4. They rely on the catalogue of p-groups, the catalogue of perfect groups by Holt and Plesken, and the catalogue of irreducible soluble groups by Short and Höfling.
The groups of cubefree order at most 50000 have been computed by Besche and Eick based on the work by Dietrich and Eick. They rely on the catalogue of irreducible soluble groups by Short and Höfling.
The groups of order p^5 have been determined by Girnat and the groups of order p^6 have been classified by Newman, O'Brien and Vaughan-Lee. The groups of order p^7 have been classified by O'Brien and Vaughan-Lee. The Small Groups Library includes them for p = 3,5,7,11.
The groups of order q^n * p for q and n fixed and p ≠ q have been constructed by Besche and Eick.
The groups of squarefree order, of an order which factorises in at most 3 primes and of order dividing $p^4$ have been known for a long time.
|The following is intended to act as a reference point for further study of the algorithms employed and does not purport to be comprehensive; our algorithms rely on many other contributions to the field.|
|The Small Groups library is available as part of the distribution of the computer algebra systems Gap and Magma. Both systems provide search facilities for the library and, for most available orders, an identification function corresponding to the groups in the library.|