Section 8.2
Orbits
Let G -> Sym(X) be a permutation representation
of the group G on X. If x, y 
X and there is g G with
g(x) = y, then we say that x and y
are in the same orbit, and write
x itso y.
It is easily verified that being in the same orbit is an equivalence relation.
The equivalence classes of the relation `itso' are called orbits of G on the set X.
The group G is said to be transitive on X if there is only one orbit on X.
Thus, G is transitive on X if every two elements from X are mutually equivalent.
The fact that itso is an equivalence relation implies that a G-orbit is equal to
G}.
for any point x in this orbit.
This observation leads to the following algorithm for a permutation group G on a finite set X.
- Input: a set B of generators of G and an element x of X;
- Output: the G-orbit of x.
- Put Orbit = {x} and LastFound = {x}.
- Put
New = {g(a) | g B,
a LastFound}\Orbit.
-
If New is nonempty, then
replace Orbit by the union of Orbit and New,
and LastFound by New; go to Step 2.
If New is empty, then output Orbit.