Permutation groups of large degree it is still not good enough. I.e., an element of G (i) that takes b i to p.įor permutation groups of small degree this might be possible, but for Representative of the coset corresponding to the point p ∈ O (i), Represent a transversal as a list T such that T is a G (i) with the points in O (i) := b i G (i). Theory tells us that we can identify the cosets of G (i+1) in The most useful tools for permutation groups. This is one reason why stabilizer chains are one of With the knowledge of the transversals R (i) we know each element of Then each element g of G has a unique representation of Set of right coset representatives of the cosets of G (i+1) in Let R (i) be a right transversal of G (i+1) in G (i), i.e., a G 2(4) possesses reduced bases of length 5 and 7. Note that different reduced bases for one group The chain relative to B is trivial, i.e., there exists no i such that A base B is called reduced if no stabilizer in The number of points in a base is called the The moved points of G, certainly is a base for G there exists a baseįor each permutation group. Since, where n is the degree of G and b i are The chain of subgroups of G itself is called the That a set S that satisfies the conditions above is a SGS of G relative to B. A set S of generators for G satisfying the condition , b n] is called a base of G, the points b i are calledīasepoints. If is a list of points, G (1) = G and G (i+1) The concept of stabilizer chains was introduced by Charles Most of the algorithms for permutation groups need a stabilizer chain Gap> PermGroupOps.NrMovedPoints( Group( () ) ) PermGroupOps.NrMovedPoints returns the number of moved points of theĮlement of G (also see PermGroupOps.MovedPoints). Gap> PermGroupOps.LargestMovedPoint( s4 ) PermGroupOps.LargestMovedPoint returns the largest positive integer Gap> PermGroupOps.SmallestMovedPoint( s3b ) Which is moved by the permutation group G (see also PermGroupOps.SmallestMovedPoint returns the smallest positive integer Gap> PermGroupOps.MovedPoints( Group( () ) ) Permutation group G, i.e., points which are moved by at least oneĮlement of G (also see PermGroupOps.NrMovedPoints). PermGroupOps.MovedPoints returns the set of moved points of the Gap> f := FactorGroup( s4, Subgroup( s4, ) ) Will signal an error if obj is an unbound variable. Of an arbitrary type, is a permutation group, and false otherwise. IsPermGroup returns true if the object obj, which may be an object Permutation Group Records 21.1 IsPermGroup.The external functions are in the file LIBNAME/"permgrp.g". The last section in this chapterĭescribes the representation of permutation groups (see Permutation Group operates naturally on the positive integers, so all operationsįunctions can be applied (see chapter Operations of Groups and Group all group functions can be applied to it (see chapter Groups and Also because each permutation group is after all a This is explained in section Random Methods for Permutation Groups.īecause each permutation group is a domain all set theoretic functionsĬan be applied to it (see chapter Domains and Set Functions for In manyĬases it is preferable then, to use random methods for computation. If the permutation groups become bigger, computations become slower. Permutation group with a property (see PermGroupOps.ElementProperty and Two sections describe the functions that find elements or subgroups of a PermGroupOps.Indices, and PermGroupOps.StrongGenerators). (see Base for Permutation Groups, ListStabChain, The next sectionsĭescribe the functions that extract information from stabilizer chains ReduceStabChain, MakeStabChainStrongGenerators). Stabilizer chain (see StabChain, ExtendStabChain, The following sections describe the functions that compute or change a Used by most functions for permutation groups (see Stabilizer Chains). Theįollowing section describes the concept of stabilizer chains, which are PermGroupOps.LargestMovedPoint, and PermGroupOps.NrMovedPoints). PermGroupOps.MovedPoints, PermGroupOps.SmallestMovedPoint, Related to the set of points moved by a permutation group (see The next sections describe the functions that are That tests whether an object is a permutation group or not (see section This introduction is followed by a section that describes the function Our standard example in this chapter will be the symmetric group ofĭegree 4, which is defined by the following GAP3 statements. Positive integers (see chapters Groups and Permutations). GAP3 Manual: 21 Permutation Groups 21 Permutation GroupsĪ permutation group is a group of permutations on a set Ω of
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