Schreier–Sims algorithm: Difference between revisions

Revision as of 04:47, 10 April 2017

Revision as of 15:52, 10 April 2017

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Notable details left out here include the growing of the orbit tree and the calculation of each new Schreier generator. In place of the orbit tree, a [[Schreier vector]] can be used, but the idea is essentially the same. The tree is rooted at the identity element, which fixes the point stabilized by the subgroup. Each node of the tree can represent a permutation that, when combined with all permutations in the path from the root to it, takes that point to some new point not visited by any other node of the tree. By the [[orbit-stabilizer theorem]], these form a [[transversal]] of the subgroup of our group that stabilizes the point whose entire orbit is maintained by the tree. Calculating a Schreier generator is a simple matter of applying [[Schreier's subgroup lemma]].

Another detail left out is the membership test. This test is abased simpleupon matterthe ofsifting process. A permutation is sifted down the chain at each step by finding the containing coset, then using that containscoset's representative to find a permutation in the givensubgroup, and the process is repeated in the subgroup with that found permutation. AIf linearthe end of the searchchain is usuallyreached fine(i.e., aswe reach the trivial subgroup), indexthen isthe usuallysifted notpermutation was a member of the group at the top of the highchain.

==References==