Group Theory Algorithms

Coset Representatives	Setwise Stabilizer
Centralizer

This tutorial introduces some basic algorithms for computing with finite permutation groups, other than those introduced in "Permutation Groups".

Coset Representatives

A subgroup

of a group

partitions the list of elements of

into disjoint subsets, called cosets of

, such that

is one of them and the rest are of the form

for some element

, with ⊙ denoting the product law. A way of identifying the coset is by selecting a representative from each coset, for example the smallest element in the coset.

RightCosetRepresentative

compute smallest group element in a coset

Computation of a canonical coset representative.

Take two permutations and the group generated by them:

This group partitions

into 210 disjoint cosets:

Given any other permutation, it is possible to list the elements of its associated right coset:

The canonical representative of the right coset is taken to be the least permutation in the order defined by images:

Construct the canonical representative directly, without listing the permutations in the coset:

Check that this also agrees with the minimum-rank permutation (in the group

) in the coset:

For bigger groups, it is not possible to list or rank all permutations, but you can still use RightCosetRepresentative.

Take two permutations and construct the group generated by them:

This is the number of cosets induced by this group on

For any permutation belonging to the group itself, the canonical representative is always the identity permutation:

Now take some random permutation in

This is its coset representative:

Check that the representative indeed belongs to the same coset. This will be the case if there exists a permutation h in the original group such that permPermutationProduct[h,rep]:

Centralizer

The centralizer

of an element

in a group

is the subgroup of elements of

that commute with

GroupCentralizer

compute the centralizer subgroup of some group element

Computation of centralizers.

Take the group:

Choose a permutation:

This is its centralizer in the group:

Check the result by direct computation of all commutators in the group:

Setwise Stabilizer

The (pointwise) stabilizer of a group

is the subgroup of elements of

fixing a set of one or several points of the domain of action. This concept can be extended to the setwise stabilizer, which is the subgroup of elements either fixing those points or moving them among themselves.

GroupSetwiseStabilizer

compute the setwise stabilizer subgroup of a list of points

Computation of setwise stabilizers.

Take the group again:

This is the list of points to be stabilized:

The action of the permutations of the setwise stabilizer changes the individual elements of the list, but always to other elements of the list:

Compare with the (pointwise) stabilizer of the same list of elements. It only contains permutations stabilizing all elements of the list:

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Group Theory Algorithms

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