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

# GroupElements

 GroupElements[group] returns the list of all elements of group. GroupElementsreturns the elements numbered in group in the standard order.
• The elements of a permutation group are found by constructing a strong generating set representation of the group.
• The order of elements returned by GroupElements depends on the base of the strong generating set. An explicit base can be chosen by setting GroupActionBase.
• GroupElements gives the identity element for any choice of the group base.
• Negative positions are assumed to count from the end.
Elements of a cyclic permutation group:
First three elements:
Last element:
Elements of a cyclic permutation group:
 Out[1]=

First three elements:
 Out[1]=

Last element:
 Out[1]=
 Scope   (2)
A permutation group:
The first permutation is always the identity. Then we have permutations moving the last points of the support:
Alternating groups can be generated with 3-cycles:
 Options   (1)
Take the symmetric group of degree 5, generated by a transposition and a shift:
By default the permutations are generated in standard ordering:
Generate the same permutations, but in a different order:
The role of the base can be understood as conjugation under the permutation relating the bases:
 Applications   (1)
We can generate uniformly distributed random permutations in a group by generating uniform ranks and then constructing those permutations:
A permutation group:
It is still a small subgroup of :
Take some permutations in the group:
Find the positions of permutations:
Position zero is not defined:
Positions must not be larger than the group order:
Permutations are sorted by images, not by Mathematica's canonical order:
These are generators of a permutation representation of the largest Mathieu group, :
Find a strong generating set for the group, relative to a sorted base:
A subgroup of order 960:
Construct its permutations using a non-sorted base:
Find their positions in the group:
They are not sorted:
Different bases produce different reordering patterns:
By default the base is taken sorted:
New in 8