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 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:
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.