Example

Let F be a finite field of size q and put V = F³ for the 3-dimensional vector space over F. Consider G = GL(V), the group of all invertible linear mappings on V, acting on the vectors of the vector space V.

By an earlier example, we know that the q³ - 1 nonzero vectors form a nonzero orbit. Fix a nonzero vector x, so that |G| = |G_x| · |Gx|.

We select a vector y outside the linear subspace of V spanned by x. There are q³ - q such vectors (as the linear subspace spanned by v has exactly q members).

Next we select a vector z outside the 2-dimensional subspace of V spanned by x and y (of size q²). There are q³ - q² such choices.

Now the triple x, y, z is a basis of the vector space V, and any ordered basis can be obtained by the above procedure. As discussed in the previous example, [x, y, z] is also a basis of G acting on V. Since G is transitive on bases (the matrix with column vectors x, y, z is invertible and maps the standard basis onto (x, y, z)), we find

|G| = (q³ - 1)(q³ - q)(q³ - q²).

Can you find the order of GL_n(F), the group of invertible n×n matrices with coefficients in F?