![]() ![]() GAP can also detect whether a given permutation group is a symmetric The commands SymmetricGroup and AlternatingGroup (see Basic Groups)Ĭonstruct symmetric and alternating permutation groups. Gap> small:= SmallerDegreePermutationRepresentation( image ) Gap> image:= Image( iso ) NrMovedPoints( image ) Not guaranteed to be the same for different calls of Permutation representation (or even the degree of the representation) is The methods used might involve the use of random elements and the The degree of which may be smaller than the original degree. In this case, the actions on the cosets of these subgroups give rise toĪn intransitive permutation representation Of small index for which the cores intersect trivially Using GAP interactively, one might be able to choose subgroups Or of smallest degree among the transitive permutation representations Note that the result is not guaranteed to be a faithful permutation In the worst case this is the identity mapping on G. ![]() The result is a group homomorphism onto a permutation group, Permutation representation of smaller degree. SmallerDegreePermutationRepresentation tries to find a faithful Let G be a permutation group that acts transitively SmallerDegreePermutationRepresentation( G ) F.In many cases the permutation representation constructed by The method will select a suitable permutation representation. Returns an isomorphism j from the group G ontoĪ permutation group P which is isomorphic to G. Rather than permutation groups themselves, for example BaseStabChainĤ1.2 Computing a Permutation Representation Permutation groups address stabilizer chains (see Stabilizer Chains) Similarly, several functions concerning the natural action of Returns the list of orbits of the positive integers in the list D Under the group generated by the permutations in the list perms. Returns the orbit of the positive integer pnt Via ^, the following functions are provided for this purpose. (see above) or orbits, it may be useful to avoid the explicit constructionįor the special case of the action of permutations on positive integers If one has a list of group generators and is interested in the moved points Therefore all action functions can be appliedįor example Orbit, Stabilizer, Blocks, IsTransitive, IsPrimitive. The action of a permutation group on the positive integers is a groupĪction (via the acting function OnPoints). Gap> LargestMovedPoint( g ) SmallestMovedPoint( g ) Gap> MovedPoints( g ) NrMovedPoints( g ) In particular they can be applied to permutation groups. Permutations (see Moved Points of Permutations), The functions MovedPoints, NrMovedPoints, LargestMovedPoint,Īnd SmallestMovedPoint are defined for arbitrary collections of Installed for permutation groups that make computations more effective. Permutation groups are groups and therefore all operations for groups (seeĬhapter Groups) can be applied to them. Specify the operation domain W when a permutation group is Working with large degree permutation groupsĪ permutation group is a group of permutations on a finite set.Low Level Routines to Modify and Create Stabilizer Chains. ![]() Randomized Methods for Permutation Groups.On Wolfram|Alpha Permutation Cite this as: Skiena,ĭiscrete Mathematics: Combinatorics and Graph Theory with Mathematica. "Permutations: Johnson's' Algorithm."įor Mathematicians. "Permutation Generation Methods." Comput. Knuth,Īrt of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. "Generation of Permutations byĪdjacent Transpositions." Math. "Permutations by Interchanges." Computer J. "Arrangement Numbers." In Theīook of Numbers. The permutation which switches elements 1 and 2 and fixes 3 would be written as (2)(143) all describe the same permutation.Īnother notation that explicitly identifies the positions occupied by elements before and after application of a permutation on elements uses a matrix, where the first row is and the second row is the new arrangement. There is a great deal of freedom in picking the representation of a cyclicĭecomposition since (1) the cycles are disjoint and can therefore be specified inĪny order, and (2) any rotation of a given cycle specifies the same cycle (Skienaġ990, p. 20). This is denoted, corresponding to the disjoint permutation cycles (2)Īnd (143). The unordered subsets containing elements are known as the k-subsetsĪ representation of a permutation as a product of permutation cycles is unique (up to the ordering of the cycles). (Uspensky 1937, p. 18), where is a factorial. ![]()
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