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.
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 all close allBasic 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:
Tech Notes
Related Guides
History
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_2025_groupelements, author="Wolfram Research", title="{GroupElements}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupElements.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_groupelements, organization={Wolfram Research}, title={GroupElements}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupElements.html}, note=[Accessed: 17-November-2025]}