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

Contents

References

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 1-58488-372-3; Computation with Finitely-presented Groups by Sims ISBN 0-521-43213-8; and Algorithms for Permutation Groups by Seress ISBN 0-521-66103-X.