# GroupSetwiseStabilizer

GroupSetwiseStabilizer[group,{p1,,pn}]

returns the subgroup of group for which the images of the points pi are still in the list {p1,,pn}.

GroupSetwiseStabilizer[group,{p1,,pn},f]

returns the setwise stabilizer subgroup under the action given by the function f.

# Details

• Group elements in the setwise stabilizer do not necessarily fix the points pi.
• The pointwise stabilizer of a list of points, computed with 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 PermutationReplace.

# Examples

open allclose all

## Basic Examples(1)

Setwise stabilizer of four points:

Take an element of the stabilizer:

It moves the points of the list among them:

## Scope(3)

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

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

Compute the setwise stabilizer of a named permutation group:

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

Subgroup of permutations that leave invariant a set of lists of objects under Permute action:

Check that such a set does indeed form a single orbit under Permute action:

## Properties & Relations(2)

Take the group:

And the list of points to stabilize:

Compute the setwise stabilizer:

And the pointwise stabilizer:

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

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:

Subgroup of permutations that leave invariant a set of lists of objects under Permute action:

Check that such a set does indeed form a single orbit under Permute action:

Compare with the result of GroupStabilizer, giving a smaller subgroup:

Now each list of objects forms its own orbit:

Wolfram Research (2010), GroupSetwiseStabilizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupSetwiseStabilizer.html (updated 2012).

#### Text

Wolfram Research (2010), GroupSetwiseStabilizer, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupSetwiseStabilizer.html (updated 2012).

#### CMS

Wolfram Language. 2010. "GroupSetwiseStabilizer." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/GroupSetwiseStabilizer.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_groupsetwisestabilizer, author="Wolfram Research", title="{GroupSetwiseStabilizer}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GroupSetwiseStabilizer.html}", note=[Accessed: 14-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_groupsetwisestabilizer, organization={Wolfram Research}, title={GroupSetwiseStabilizer}, year={2012}, url={https://reference.wolfram.com/language/ref/GroupSetwiseStabilizer.html}, note=[Accessed: 14-July-2024 ]}