({{, (RefLink[GroupElementPosition, paclet:ref/GroupElementPosition])[group,g] 
returns the position of the element g in the list of elements of group.}, {, (RefLink[GroupElementPosition, paclet:ref/GroupElementPosition])[group,{(Subscript[g, 1]),…,(Subscript[g, n])}]
returns the list of positions of the elements (Subscript[g, 1]),…,(Subscript[g, n]) in group.}})

OpenersNotesSection(Spacer1)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 (RefLink[GroupActionBase, paclet:ref/GroupActionBase]) option.

OpenersPrimaryExamplesSection(Spacer1)Examples  (6)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

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

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}}]}]

OpenersExampleSection1(Spacer1)Scope  (2)

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

GroupElementPosition[MathieuGroupM11[],Cycles[{{1,3,7,8,6},{2,11,9,5,4}}]]

Position[GroupElements[MathieuGroupM11[]],Cycles[{{1,3,7,8,6},{2,11,9,5,4}}]]

List of positions of several elements in a group:

GroupOrder[group=PermutationGroup[{Cycles[{{1,2,5},{3,9}}],Cycles[{{1,6,7,8}}]}]]

GroupElementPosition[group,{Cycles[{}],Cycles[{{1,7,5,6,2}}],Cycles[{{1,2,7,8,5}}],Cycles[{{1,8},{2,7},{3,9},{5,6}}]}]

OpenersExampleSection2(Spacer1)Options  (1)

OpenersExampleSubsection0(Spacer1)GroupActionBase  (1)

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

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}}]}]

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}]

OpenersExampleSection3(Spacer1)Properties & Relations  (1)

(RefLink[GroupElements, paclet:ref/GroupElements]) performs the inverse operation for a list of elements:

GroupElements[MathieuGroupM12[],{40000}]

GroupElementPosition[MathieuGroupM12[],%]

OpenersExampleSection4(Spacer1)Possible Issues  (1)

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

GroupElementPosition[AlternatingGroup[3],Cycles[{{1,2}}]]

GroupElementQ[AlternatingGroup[3],Cycles[{{1,2}}]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Cycles, paclet:ref/Cycles])  ▪  (RefLink[PermutationGroup, paclet:ref/PermutationGroup])  ▪  (RefLink[GroupElements, paclet:ref/GroupElements])  ▪  (RefLink[GroupElementQ, paclet:ref/GroupElementQ])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Permutations, paclet:tutorial/Permutations])}, {▪(RefLinkPlain[Permutation Groups, paclet:tutorial/PermutationGroups])}, {▪(RefLinkPlain[Named Groups, paclet:tutorial/NamedGroups])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Group Theory, paclet:guide/GroupTheory])}})}})