returns the subgroup of elements of group that move none of the points p1, …, pn.
returns the stabilizer subgroup under the action given by the function f.
- 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.
Examplesopen allclose all
Basic Examples (1)
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:
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).
Wolfram Language. 2010. "GroupStabilizer." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/GroupStabilizer.html.
Wolfram Language. (2010). GroupStabilizer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupStabilizer.html