Editing Computational group theory

{{Short description|Study of mathematical groups by means of computers}}
{{No footnotes|date=January 2020}}
In [[mathematics]], '''computational group theory''' is the study of
[[group (mathematics)|group]]s by means of computers. It is concerned
with designing and analysing [[algorithm]]s and
[[data structure]]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]] for finding the [[order (group theory)|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 system]]s (CAS) used for group theory are
[[GAP computer algebra system|GAP]] and [[Magma computer algebra system|Magma]]. Historically, other systems such as CAS (for [[character theory]]) and [[Cayley computer algebra system|Cayley]] (a predecessor of Magma) were important.

Some achievements of the field include:
* complete enumeration of [[List of small groups|all finite groups of order less than 2000]]
* computation of [[group representation|representations]] for all the [[sporadic groups]]

== See also ==
* [[Black box group]]

== References ==
* A [https://web.archive.org/web/20070208012642/http://www.math.ohio-state.edu/~akos/notices.ps 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 [http://www.math.rutgers.edu/~sims/publications/survey.pdf survey] by [[Charles Sims (mathematician)|Charles Sims]] from [[Rutgers University]] and an [http://www.math.rwth-aachen.de/~Joachim.Neubueser/preprint.html older survey] by Joachim Neubüser from [[RWTH Aachen]].
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.   {{ISBN|1-58488-372-3}}
* [[Charles C. Sims]], "Computation with Finitely-presented Groups", Encyclopedia of Mathematics and its Applications, vol 48, [[Cambridge University Press]], Cambridge, 1994.  {{ISBN|0-521-43213-8}}
*  Ákos Seress, "Permutation group algorithms", Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003.  {{ISBN|0-521-66103-X}}.

[[Category:Computational group theory| ]]
[[Category:Computational fields of study]]