BUILT-IN MATHEMATICA SYMBOL

PermutationGroup

represents the group generated by multiplication of the permutations

.

The generating permutations 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.

A permutation group defined by two generators:

Compute its order:

A permutation group defined by two generators:

Out[1]=

Compute its order:

Out[2]=

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:

They are different as Mathematica expressions:

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:

Explicit representation of a named group:

Generate the symmetric group of degree

using

transpositions:

Generate the alternating group of degree

using

generators:

The moves of a Rubik's cube form a group. Number the moving facelets from 1 to 48:

These are the six basic rotations:

Group order:

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: