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{{Short description|Study of mathematical groups by means of computers}} |
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{{No footnotes|date=January 2020}} |
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In [[mathematics]], '''computational group theory''' is the study of |
In [[mathematics]], '''computational group theory''' is the study of |
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[[group (mathematics)|group]]s by means of computers. It is concerned |
[[group (mathematics)|group]]s by means of computers. It is concerned |
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with designing and analysing [[algorithm]]s and |
with designing and analysing [[algorithm]]s and |
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[[data structure]]s to compute information about groups. The subject |
[[data structure]]s to compute information about groups. The subject |
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has attracted interest |
has attracted interest because for many interesting groups |
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(including most of the [[sporadic groups]]) |
(including most of the [[sporadic groups]]) it is impractical |
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to perform calculations by hand. |
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Important algorithms in computational group theory include: |
Important algorithms in computational group theory include: |
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* the [[ |
* the [[Schreier–Sims algorithm]] for finding the [[order (group theory)|order]] of a [[permutation group]] |
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* the [[ |
* the [[Todd–Coxeter algorithm]] and [[Knuth–Bendix algorithm]] for [[coset enumeration]] |
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* the [[product-replacement algorithm]] for finding random elements of a group |
* the [[product-replacement algorithm]] for finding random elements of a group |
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Two important [[computer algebra system]]s |
Two important [[computer algebra system]]s (CAS) used for group theory are |
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[[GAP]] and [[ |
[[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. |
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Some achievements of the field include: |
Some achievements of the field include: |
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* complete enumeration of all finite groups of order less than 2000 |
* complete enumeration of [[List of small groups|all finite groups of order less than 2000]] |
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* computation of representations for all the [[sporadic groups]] |
* computation of [[group representation|representations]] for all the [[sporadic groups]] |
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== See also == |
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* [[Black box group]] |
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== References == |
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* 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]]. |
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There are three books covering various parts of the subject: |
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* 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}} |
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* [[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}} |
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* Ákos Seress, "Permutation group algorithms", Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. {{ISBN|0-521-66103-X}}. |
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[[Category:Computational group theory| ]] |
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[[Category:Computational fields of study]] |
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Latest revision as of 18:19, 23 September 2023
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (January 2020) |
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
See also
[edit]References
[edit]- 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 Sims from Rutgers University and an 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.