({{, (RefLink[GroupActionBase, paclet:ref/GroupActionBase]) 
is an option to specify a base for a group.}})

OpenersNotesSection(Spacer1)Details

A base of a group is a list of points of its domain of action such that the only element in the group fixing them all is the identity.

Reordering the points in a base gives a different base.

The choice of base affects the ordering of group elements in the functions (RefLink[GroupElements, paclet:ref/GroupElements]) and (RefLink[GroupElementPosition, paclet:ref/GroupElementPosition]).

If the list of points given is not a base, the Wolfram Language will add more points until obtaining a base. In particular, (RefLink[GroupActionBase, paclet:ref/GroupActionBase])->{} will make the Wolfram Language choose an appropriate base.

OpenersPrimaryExamplesSection(Spacer1)Examples  (1)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Elements of a group in standard order:

GroupElements[PermutationGroup[{Cycles[{{1,2},{3,4}}],Cycles[{{1,3}}]}]]

OrderedQ[%,Less]

Sort them, specifying different priorities with a base:

GroupElements[PermutationGroup[{Cycles[{{1,2},{3,4}}],Cycles[{{1,3}}]}],GroupActionBase->{4,2}]

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

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Permutation Groups, paclet:tutorial/PermutationGroups])}, {▪(RefLinkPlain[Group Theory Algorithms, paclet:tutorial/GroupTheoryAlgorithms])}})}})

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