Consider a permutation representation of G on a set X.
By a previous theorem,
the order of a permutation group G can be determined
once we know the order of
the stabilizer Gx of a point x
X.
Since
Gx is also a permutation
group, the result leads to a recursive computation.
Definition
A basis for G on X is a sequence B = [b1, ..., bt] of elements bi of X such that the stabilizer Gb1, ..., bt in G of each of b1, ..., bt is the trivial group.
If G is a subgroup of Sn and B
= [b1, ..., bt] is a basis
for G on X = {1, ..., n}, then the order of
G is equal to the size of the G-orbit of
B. Alternatively,
we can determine the order as follows,
where
Gx stands for the orbit of G on x.
A handicap in applying this theorem to a group
generated by a set of permutations is that
we have no way (yet) of determining the stabilizer. This is taken care
of on the next page.