Group Theory Algorithms
This tutorial introduces some basic algorithms for computing with finite permutation groups, other than those introduced in "Permutation Groups".
A subgroup of a group partitions the list of elements of into disjoint subsets, called cosets of in , such that is one of them and the rest are of the form for some element of , 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.
compute smallest group element in a coset
The canonical representative of the right coset is taken to be the least permutation in the order defined by images:
For bigger groups, it is not possible to list or rank all permutations, but you can still use RightCosetRepresentative.
For any permutation belonging to the group itself, the canonical representative is always the identity permutation:
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 permPermutationProduct[h,rep]:
compute the centralizer subgroup of some group element
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.
compute the setwise stabilizer subgroup of a list of points
The action of the permutations of the setwise stabilizer changes the individual elements of the list, but always to other elements of the list: