The Wolfram Language offers a coherent collection of algorithms and data structures for working with permutation groups. Building upon the Wolfram Language's proven symbolic architecture, permutations can operate on group-theoretical data structures, as well as on arbitrary symbolic Wolfram Language expressions. State-of-the-art algorithms enable the efficient manipulation of very large groups. Commonly used groups are conveniently represented as built-in objects.

Group Representations

PermutationGroup — permutation group object

Named Groups »

SymmetricGroup  ▪  AlternatingGroup  ▪  DihedralGroup  ▪  CyclicGroup  ▪  AbelianGroup

MathieuGroupM24  ▪  HigmanSimsGroupHS  ▪  ConwayGroupCo1  ▪  ...

FiniteGroupData — precomputed information for named groups

Enumeration of Elements

GroupOrder — number of elements in a group

GroupElements — list of elements of a group

GroupElementQ — test membership of an element in a group

GroupElementPosition — locate element in a group

GroupElementToWord, GroupElementFromWord — element as product of generators

Description of a Group

GroupGenerators — list of generators of a group

GroupMultiplicationTable — all products of elements in a group

CayleyGraph — graph representation of a group and a set of generators

GroupStabilizerChain — strong generators and base for a group

GroupActionBase — option to specify a base for a (permutation) group

Computations with a Group

GroupOrbits — orbits of points under a group

GroupStabilizer — pointwise stabilizer subgroup

GroupSetwiseStabilizer — setwise stabilizer subgroup

CycleIndexPolynomial — cycle index polynomial of a permutation group

RightCosetRepresentative — smallest element in a coset

GroupCentralizer — centralizer of an element in a group