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:

In[97]:=

Out[97]=

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

In[98]:=

Out[98]=

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:

In[99]:=

Out[99]=

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

In[100]:=

Out[100]=

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

In[101]:=

Out[101]=

In[102]:=

Out[102]=

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) |