smallgroup

SmallGroup, in the context of computational group theory, refers to a standardized library of finite groups used by algebra researchers and software systems to facilitate group-theoretic computations. The Small Groups Library provides representatives for many isomorphism classes of finite groups and serves as a reference catalog for testing conjectures, benchmarking algorithms, and performing explicit calculations.

In practice, each group in the library is identified by a pair of integers, denoted SmallGroup(n, i). Here n is the order of the group and i is an index that distinguishes the different isomorphism classes of that order included in the library. For a given n, the library lists the distinct groups it contains and supplies concrete representations, such as presentations or permutation/matrix representations, that can be used directly in computations. This structure allows users to access a group by its order and position in the catalog and to study its structural properties computationally.

The library is widely maintained and distributed with major computer algebra systems, notably GAP, and is associated with the work of Besche, Eick, and O’Brien. It is designed to support researchers by providing readily available, explicitly described examples of finite groups, enabling reproducible experiments and algorithm testing. Updates and expansions are published to increase coverage, improve accuracy, and incorporate new classifications as they become available.

Limitations include that the library targets finite groups of relatively small order and does not include every possible finite group, especially as orders grow large. For many larger orders or for non-listed isomorphism classes, researchers must construct groups by other means. Nevertheless, the SmallGroup library remains a foundational resource in computational group theory.