This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# PermutationGroup

 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]=
 Scope   (3)
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:
 Applications   (1)
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:
New in 8