# GroupStabilizer

GroupStabilizer[group,{p1,,pn}]

returns the subgroup of elements of group that move none of the points p1, , pn.

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

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

# Details

• The output is a subgroup of group defined by generators, but possibly using different generators.
• The stabilizer group is also known as the little group or isotropy group.
• The stabilizer of a list of points is a subgroup of the setwise stabilizer of the same list of points.
• 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

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## Basic Examples(1)

A stabilizer subgroup of an alternating group:

None of the permutations move any of the points {1,5}:

## Scope(4)

Compute the stabilizer of a permutation group defined by generators:

Compute the stabilizer of a named permutation group:

The stabilizer of a group can be trivial:

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

Check that the corresponding orbit under Permute action contains only that list:

## Applications(1)

The symmetric group is -transitive and the alternating group is -transitive. It is known that any other group can be at most 5-transitive. The Mathieu group is 5-transitive:

There is just one orbit, and hence it is transitive:

The stabilizer of 1 acts transitively on the remaining 23 points, and hence is 2-transitive:

It is also 3-transitive, 4-transitive, and 5-transitive:

But it is not 6-transitive, because there are two nontrivial orbits now:

## Properties & Relations(3)

The orbit-stabilizer theorem states that the size of the orbit of a point p under a group equals the number of cosets of the stabilizer of p in group.

Take the 3×3×3 Rubik group and compute the stabilizer of point 20:

The number of cosets of the stabilizer in the full group, using the Lagrange theorem:

The orbit of point 20 has length 24:

A stabilizer subgroup computed with GroupStabilizer might be described using more generators than the original group:

The stabilizer of a permutation under conjugation action coincides with the centralizer of that permutation:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_groupstabilizer, author="Wolfram Research", title="{GroupStabilizer}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GroupStabilizer.html}", note=[Accessed: 14-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_groupstabilizer, organization={Wolfram Research}, title={GroupStabilizer}, year={2012}, url={https://reference.wolfram.com/language/ref/GroupStabilizer.html}, note=[Accessed: 14-June-2024 ]}