This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# GroupSetwiseStabilizer

 GroupSetwiseStabilizer returns the subgroup of group for which the images of the points are still in the list .
• Group elements in the setwise stabilizer do not necessarily fix the points .
• 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.
Setwise stabilizer of four points:
Take an element of the stabilizer:
It moves the points of the list among them:
Setwise stabilizer of four points:
 Out[1]=
Take an element of the stabilizer:
 Out[2]=
It moves the points of the list among them:
 Out[3]=
 Scope   (2)
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:
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:
New in 8