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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Flowcharts are often used to represent algorithms. ... A binary tree, a simple type of branching linked data structure. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

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. The Schreier-Sims algorithm is an efficient method of computing a strong generating set (SGS) of a permutation group. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... The Todd-Coxeter algorithm, discovered by Todd and Coxeter in 1936, is a procedure that can solve the coset enumeration problem. ... The Knuth-Bendix completion algorithm is an algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. ... In mathematics, coset enumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra similar to Mathematica with particular emphasis on, but not restricted to, computational group theory. ... Magma is molten rock often located inside a magma chamber beneath the surface of the Earth. ... In mathematics, the character of a group representation ρ : G → GLn is the function χ : G -> C which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). ...

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

The following list in mathematics contains the finite groups of small order up to group isomorphism. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

Computational Group Theory 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 1584883723; Computation with Finitely-presented Groups by Sims ISBN 0521432138; and Algorithms for Permutation Groups by Seress ISBN 052166103X. The Ohio State University The Ohio State University (formal name), also known as Ohio State or OSU, is currently the third largest public university in the United States and ranked by US News as the best public university in Ohio and the twenty-first best public university in the nation. ... The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ...