In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s and

data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...

s 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 The Schreier–Sims algorithm is an algorithm in computational group theory, named after the mathematicians Otto Schreier and Charles Sims. This algorithm can find the order of a finite permutation group, test membership (is a given permutation ...

for finding the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

of a

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

* the

Todd–Coxeter algorithm In group theory, the Todd–Coxeter algorithm, created by J. A. Todd and H. S. M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem. Given a presentation of a group ''G'' by generators and relations and a subgroup ''H'' of ...

and Knuth–Bendix algorithm for coset enumeration * the product-replacement algorithm for finding random elements of a group Two important

computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...

s (CAS) used for group theory are GAP and

Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

. Historically, other systems such as CAS (for

character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...

) 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 ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...

for all the sporadic groups

References

*
survey
of the subject by Ákos Seress from

Ohio State University The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best pu ...

, expanded from an article that appeared in the

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...

is available online. There is also
survey
by

Charles Sims Charles Sims may refer to: * Charles Sims (painter) (1873–1928), British painter * Charles Sims (mathematician) (1938–2017), American mathematician * Charles Sims (aviator) (1899–1929), British World War I flying ace * Charles Sims (America ...

from

Rutgers University Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's College, and wa ...

and a
older survey
by Joachim Neubüser from

RWTH Aachen RWTH Aachen University (), also known as North Rhine-Westphalia Technical University of Aachen, Rhine-Westphalia Technical University of Aachen, Technical University of Aachen, University of Aachen, or ''Rheinisch-Westfälische Technische Hoch ...

. There are three books covering various parts of the subject: * Derek F. Holt, Bettina Eick, Eamonn A. O'Brien, "Handbook of computational group theory", Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, Florida, 2005. * Charles C. Sims, "Computation with Finitely-presented Groups", Encyclopedia of Mathematics and its Applications, vol 48,

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

, Cambridge, 1994. * Ákos Seress, "Permutation group algorithms", Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. {{ISBN, 0-521-66103-X. Computational fields of study

See also

References