Let B be a generating set for G
and T a Schreier tree for B with root x. If
a 
Gx, then a is a vertex
of T. Hence there is a unique path from x to a
in the tree. If the labels of the edges in this path are
bi1, ...,
bik, respectively, then the
element
of G satisfies ta(x) = a. We call ta a Schreier element.
Theorem
{tb(a)-1bta
| a T, b B}.
The stabilizer Gx is generated by
T, b B}.
The above theorem together with the algorithm to determine a Schreier tree yields a way to completely describe the action of G on X (see Step 4 below). In particular, we can determine the order of G by applying the order theorem (Step 3).
Order algorithm
- Input: a set B of generating permutations for G
Sn.
- Output: the order of G.
- Set X = {1, ..., n} and Order = 1.
-
Fix a point x X and determine the length l of the G-orbit of x.
- Replace Order by Order · l.
- Determine a generating set B' for Gx.
- If B' generates only the trivial group, then return Order, otherwise replace X by X\{x}, B by B', G by Gx and go to Step 2.