Computational group theory
In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is impractical to perform calculations by hand.
Important algorithms in computational group theory include:
- the Schreier-Sims algorithm for finding the order of a permutation group
- the Todd-Coxeter algorithm and Knuth-Bendix algorithm for coset enumeration
- the product-replacement algorithm for finding random elements of a group
Two important computer algebra systems (CAS) used for group theory are GAP and MAGMA. Historically, other systems such as CAS (for character theory) and CAYLEY (a predecessor of MAGMA) were important.
Some achievements of the field include:
- complete enumeration of all finite groups of order less than 2000
- computation of representations for all the sporadic groups
Computational Group Theory References
An excellent survey of the subject by Akos Seress of the Ohio State University, expanded from an article that appeared in the Notices of the American Mathematical Society is available on-line. There are also three books covering various parts of the subject: the Handbook of Computational Group Theory, by Holt, Eick and O'Brien ISBN 1584883723; Computation with Finitely-presented Groups by Sims ISBN 0521432138; and Algorithms for Permutation Groups by Seress ISBN 052166103X.