4. Orbits and Schreier vectors

An essential tool in the analysis of the structure of a group of permutations is the concept of orbit of a point. It is just the set of images of that point which can be obtained by repeated action of the members of the group.

The command Orbit gives the orbit of a point under a group generated by a generating set:

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We can compute all orbits of a group. This can be understood as a partition of a range of integers which by default has the length of the highest degree of the permutations

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but can supply the length as a second argument. Many functions in the package have an additional argument to specify the length of the underlying set of points. See the documentation of each command for details:

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Orbit                Orbit of a point under a GS
Orbits
All orbits of a group

Compute orbits of a group.

Given an orbit of a group it is very useful to be able to find a permutation of the group linking two points of the orbit, that is, such that the latter point is the image of the former under that permutation. Schreier introduced a nice way to do that: together with the points of the orbit we store two vectors:

Same orbit, now with its Schreier vectors. Point 2 is the first point of the orbit. Point 4 can be obtained from point 2 (see second vector) usign permutation Perm[{4,1,3,2,6,5}] (see first vector). Point 1 can be obtained from point 4 (see second vector) using the same permutation (see first vector). Again we

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The function TraceSchreier gives the permutation that links the first point of the orbit with a given point:

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SchreierOrbit            Calculate orbit with its Schreier vectors
Schreier
Head of an orbit with its Schreier vectors
SchreierOrbits            Calculate all orbits with a compound Schreier vector
TraceSchreier            Trace an orbit to find a permutation linking two points

Schreier vectors.

 Created by Mathematica  (May 16, 2008)