BUILT-IN MATHEMATICA SYMBOL

GroupStabilizer

returns the subgroup of elements of group that move none of the points

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (1)

A stabilizer subgroup of an alternating group:

None of the permutations move any of the points

:

A stabilizer subgroup of an alternating group:

In[1]:=

Out[1]=

None of the permutations move any of the points

:

In[2]:=

Out[2]=

Scope (3)

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:

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 (2)

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: