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GroupStabilizer

GroupStabilizer
returns the subgroup of elements of group that move none of the points .
  • 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.
A stabilizer subgroup of an alternating group:
None of the permutations move any of the points :
A stabilizer subgroup of an alternating group:
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None of the permutations move any of the points :
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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:
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:
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:
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