# GroupElements

GroupElements[group]

returns the list of all elements of group.

GroupElements[group,{r1,,rk}]

returns the elements numbered r1,,rk in group in the standard order.

# Details and Options

• 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->{p1,p2,}.
• GroupElements[group,{1}] gives the identity element for any choice of the group base.
• Negative positions are assumed to count from the end.

# Examples

open allclose all

## Basic Examples(3)

Elements of a cyclic permutation group:

First three elements:

Last element:

## 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)

### GroupActionBase(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 the Wolfram Language'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:

Wolfram Research (2010), GroupElements, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupElements.html.

#### Text

Wolfram Research (2010), GroupElements, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupElements.html.

#### CMS

Wolfram Language. 2010. "GroupElements." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupElements.html.

#### APA

Wolfram Language. (2010). GroupElements. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupElements.html

#### BibTeX

@misc{reference.wolfram_2024_groupelements, author="Wolfram Research", title="{GroupElements}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupElements.html}", note=[Accessed: 09-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_groupelements, organization={Wolfram Research}, title={GroupElements}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupElements.html}, note=[Accessed: 09-September-2024 ]}