({{, (RefLink[GroupSetwiseStabilizer, paclet:ref/GroupSetwiseStabilizer])[group,{(Subscript[p, 1]),…,(Subscript[p, n])}] 
returns the subgroup of group for which the images of the points (Subscript[p, i]) are still in the list {(Subscript[p, 1]),…,(Subscript[p, n])}.}, {, (RefLink[GroupSetwiseStabilizer, paclet:ref/GroupSetwiseStabilizer])[group,{(Subscript[p, 1]),…,(Subscript[p, n])},f]
returns the setwise stabilizer subgroup under the action given by the function f.}})

OpenersNotesSection(Spacer1)Details

Group elements in the setwise stabilizer do not necessarily fix the points (Subscript[p, i]).

The pointwise stabilizer of a list of points, computed with (RefLink[GroupStabilizer, paclet:ref/GroupStabilizer]), is a subgroup of the setwise stabilizer of the same list of points.

The output is a subgroup of group defined by generators, but possibly using different generators.

Evaluation of f[p,g] for an action function f, a point p and a permutation g of the given group, is assumed to return another point p'.

For permutation groups, the default group action is taken to be (RefLink[PermutationReplace, paclet:ref/PermutationReplace]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (6)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Setwise stabilizer of four points:

GroupSetwiseStabilizer[PermutationGroup[{Cycles[{{1,7,2},{3,10,5,4},{6,8,9}}],
Cycles[{{1,4,2,5,3}}]}],{3,7,1,9}]

Take an element of the stabilizer:

First@GroupElements[%,{100}]

It moves the points of the list among them:

PermutationReplace[{3,7,1,9},%]

OpenersExampleSection1(Spacer1)Scope  (3)

Compute the setwise stabilizer of a permutation group defined by generators:

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

Possible results of the action of the elements of the setwise stabilizer:

GroupOrbits[%,{{1,3}}]

Compute the setwise stabilizer of a named permutation group:

GroupSetwiseStabilizer[AlternatingGroup[8],{1,2,3}]

Possible results of the action of the elements of the setwise stabilizer:

GroupOrbits[%,{{1,2,3}}]

Subgroup of permutations that leave invariant a set of lists of objects under (RefLink[Permute, paclet:ref/Permute]) action:

setwisestab=GroupSetwiseStabilizer[SymmetricGroup[5],{{a,a,b,a,b},{b,b,a,a,a}},Permute]

Check that such a set does indeed form a single orbit under (RefLink[Permute, paclet:ref/Permute]) action:

GroupOrbits[setwisestab,{{a,a,b,a,b},{b,b,a,a,a}},Permute]

OpenersExampleSection3(Spacer1)Properties & Relations  (2)

Take the group:

GroupOrder[group=PermutationGroup[{Cycles[{{1,7,2},{3,10,5,4},{6,8,9}}],
Cycles[{{1,4,2,5,3}}]}]]

And the list of points to stabilize:

list={3,7,1,9};

Compute the setwise stabilizer:

setstab=GroupSetwiseStabilizer[group,list]

GroupOrder[setstab]

And the pointwise stabilizer:

stab=GroupStabilizer[group,list]

GroupOrder[stab]

Check that the pointwise stabilizer is a subgroup of the setwise stabilizer:

GroupElementQ[setstab,#]&/@GroupGenerators[stab]

Compare the possible reorderings of the list in both cases. The six reorderings correspond to the six cosets of the stabilizer in the setwise stabilizer:

GroupOrbits[setstab,{list}]

GroupOrbits[stab,{list}]

Subgroup of permutations that leave invariant a set of lists of objects under (RefLink[Permute, paclet:ref/Permute]) action:

setwisestab=GroupSetwiseStabilizer[SymmetricGroup[5],{{a,a,b,a,b},{b,b,a,a,a}},Permute]

Check that such a set does indeed form a single orbit under (RefLink[Permute, paclet:ref/Permute]) action:

GroupOrbits[setwisestab,{{a,a,b,a,b},{b,b,a,a,a}},Permute]

Compare with the result of (RefLink[GroupStabilizer, paclet:ref/GroupStabilizer]), giving a smaller subgroup:

stab=GroupStabilizer[SymmetricGroup[5],{{a,a,b,a,b},{b,b,a,a,a}},Permute]

Now each list of objects forms its own orbit:

GroupOrbits[stab,{{a,a,b,a,b},{b,b,a,a,a}},Permute]

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