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GroupElementPosition[group,g]

returns the position of the element g in the list of elements of group.

GroupElementPosition[group,{g1,…,gn}]

returns the list of positions of the elements g1,…,gn in group.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
GroupActionBase  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
History
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GroupElementPosition

GroupElementPosition[group,g]

returns the position of the element g in the list of elements of group.

GroupElementPosition[group,{g1,…,gn}]

returns the list of positions of the elements g1,…,gn in group.

Details and Options

  • Each position is an integer between 1 and the order of the group.
  • The positions of elements in a permutation group are by default based on the standard ordering of their images.
  • The ordering of elements in a permutation group can be controlled by setting the GroupActionBase option.

Examples

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Basic Examples  (1)

Find positions of group elements in a group. The identity is always the first element:

Wolfram Language code: GroupElementPosition[MathieuGroupM12[], {Cycles[{}], Cycles[{{1, 4}, {3, 10}, {5, 11}, {6, 12}}], Cycles[{{1, 8, 9}, {2, 3, 4}, {5, 12, 11}, {6, 10, 7}}]}]

Scope  (2)

Position of a permutation in the list of elements of a group:

Wolfram Language code: GroupElementPosition[MathieuGroupM11[], Cycles[{{1, 3, 7, 8, 6}, {2, 11, 9, 5, 4}}]]
Wolfram Language code: Position[GroupElements[MathieuGroupM11[]], Cycles[{{1, 3, 7, 8, 6}, {2, 11, 9, 5, 4}}]]

List of positions of several elements in a group:

Wolfram Language code: GroupOrder[group = PermutationGroup[{Cycles[{{1, 2, 5}, {3, 9}}], Cycles[{{1, 6, 7, 8}}]}]]
Wolfram Language code: GroupElementPosition[group, {Cycles[{}], Cycles[{{1, 7, 5, 6, 2}}], Cycles[{{1, 2, 7, 8, 5}}], Cycles[{{1, 8}, {2, 7}, {3, 9}, {5, 6}}]}]

Options  (1)

GroupActionBase  (1)

Change the ordering of the permutations in a group by specifying a base:

Wolfram Language code: GroupElementPosition[MathieuGroupM12[], {Cycles[{}], Cycles[{{1, 4}, {3, 10}, {5, 11}, {6, 12}}], Cycles[{{1, 8, 9}, {2, 3, 4}, {5, 12, 11}, {6, 10, 7}}]}]
Wolfram Language code: GroupElementPosition[MathieuGroupM12[], {Cycles[{}], Cycles[{{1, 4}, {3, 10}, {5, 11}, {6, 12}}], Cycles[{{1, 8, 9}, {2, 3, 4}, {5, 12, 11}, {6, 10, 7}}]}, GroupActionBase -> {4, 3, 2, 1}]

Properties & Relations  (1)

GroupElements performs the inverse operation for a list of elements:

Wolfram Language code: GroupElements[MathieuGroupM12[], {40000}]
Wolfram Language code: GroupElementPosition[MathieuGroupM12[], %]

Possible Issues  (1)

For an element not present in the group the input is returned unevaluated:

Wolfram Language code: GroupElementPosition[AlternatingGroup[3], Cycles[{{1, 2}}]]
Wolfram Language code: GroupElementQ[AlternatingGroup[3], Cycles[{{1, 2}}]]
Wolfram Research (2010), GroupElementPosition, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupElementPosition.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_groupelementposition, author="Wolfram Research", title="{GroupElementPosition}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupElementPosition.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_groupelementposition, organization={Wolfram Research}, title={GroupElementPosition}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupElementPosition.html}, note=[Accessed: 13-June-2026]}