1. The problem statement, all variables and given/known data If G is a finite group which acts transitively on X, and if H is a normal subgroup of G, show that the orbits of the induced action of H on X all have the same size. 3. The attempt at a solution By the Orbit-Stabilizer theorem the size of the orbit induced by H on X is a divisor of H. This could certainly help... And H is normal, therefore H is the stabilizer of the action of conjugation. Plus the fact that points in the same orbit have conjugate stabilizers... I don't know how to put the elements together.... Can anyone hint me?