BUILT-IN MATHEMATICA SYMBOL

GroupElements

GroupElements[group]
returns the list of all elements of group.

GroupElements

returns the elements numbered

in group in the standard order.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (3)

Elements of a cyclic permutation group:

First three elements:

Last element:

Elements of a cyclic permutation group:

In[1]:=

Out[1]=

First three elements:

In[1]:=

Out[1]=

Last element:

In[1]:=

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:

Properties & Relations (1)

A permutation group:

It is still a small subgroup of

:

Take some permutations in the group:

Find the positions of permutations:

Possible Issues (3)

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:

Neat Examples (1)

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: