Remark

Let B be an orbit of the permutation representation f : G -> Sym(X) of the finite group G. If x is an element of B, then, by the theorem,

|G|/|G_x| = |B|.

In particular, the number of elements of B is a divisor of |G|, and the order |G_x| of the stabilizer of x does not depend on the choice of x in B.