A theorem on the orders of orbits of a finite permutation group

In this paper we give a proof of the following theorem. Let G be a permutation group on a finite set N. Then the order of G is odd if and only if the degrees of all transitive constituents of G and the degrees of all transitive constituents of each G a (a?N) are odd. The method we devised in proving the above theorem led to an important generalization.