represents the group generated by multiplication of the permutations perm1,…,permn.
- The generating permutations permi must be given in disjoint cyclic form, with head Cycles.
- Properties of a permutation group are typically computed by constructing a strong generating set representation of the group using the Schreier–Sims algorithm.
Examplesopen allclose all
An empty list of generators represents the identity (or trivial, or neutral) group:
Find the order of a group generated by two permutations:
Test the equality of permutation groups with the same support but possibly generated by different permutations:
This is the group of all rotations and reflections of a regular -sided polygon, the dihedral group, for . It can be generated by a rotation of an angle and a reflection along an axis through a vertex:
Construct the octagon corresponding to each group element:
This is the original polygon and its seven rotations. Numbers increase counterclockwise:
This is the polygon reflected along the bisection 1–5 and its seven rotations. Numbers increase clockwise:
Properties & Relations (3)
Neat Examples (1)
The moves of a Rubik's cube form a group. Number the moving facelets from 1 to 48:
These are the six basic rotations:
Swapping two neighbor edge facelets is not allowed:
Simultaneous swaps of two edge pairs is allowed:
This is the superflip move, which switches all edge pairs simultaneously without changing any corner:
Edges and corners cannot be mixed (as the action of the group on the cube is not transitive), but any two corners or any two edges can be swapped:
Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.
Wolfram Language. 2010. "PermutationGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationGroup.html.
Wolfram Language. (2010). PermutationGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationGroup.html