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A permutation group G on
X = {1, ..., n} can be conveniently represented by a
(small) generating set of
permutations. Most algorithms for permutation groups take such a
generating set as input.
Let B be a generating set for G. |
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Definition
A Schreier tree with root x for B is a tree rooted at x whose edges are labeled with the following properties:
- Its vertices are the elements of Gx.
- Each edge i, j with i closer to the root
x than j is labeled by a generator b
B such
that b(i) = j. We denote such a labeled edge by
[i, j; b].
The Schreier tree can be used to find generators for the stabilizer Gx of a point x. This is the content of the next theorem. First, however, we show how to find Schreier trees. It is a slight extension of the algorithm for determining orbits.
Schreier tree algorithm
- Input: a set B of generators of G and x in X.
- Output: a Schreier tree for B with root x.
- Put T = {x} and M = {x}.
- Put
N = {g(a) | g B,
a M}\T;
this is the set of new elements.
-
If N is nonempty, then
- choose, for each
n N, a vertex a T and
g G with g(a) = n, and add
vertex n to T with
labeled edge [a, n; g].
- Replace M by N and go to Step 2.
- choose, for each
n
- Return T.