Computational group theory

In mathematics , computational group theory is the study of groups by means of computers. It is concerned with designing and analyzing 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 the 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 (predecessor of Magma) have been important.

Some achievements of the field include:

  • complete enumeration of all finite groups of order less than 2000
  • computation of representations for the sporadic groups

See also

  • Black box group

References

  • A survey of the subject by Ákos Seress from Ohio State University , expanded from an article that appeared in the Notices of the American Mathematical Society is available online. There is also a survey by Charles Simsfrom Rutgers University and an older survey by Joachim Neubüser from RWTH Aachen .