In: Advanced Math
Let G act transitively on Ω and assume G is finite. Define an action of G on the set Ω × Ω by putting
(α,β) · g = (α · g, β · g).
Let α ∈ Ω Show that G has the same number of orbits on Ω × Ω That Gα has on Ω
For the proof, at first a lemma has been proved using the transitivity of the action of G on . Then this lemma has been used to exhibit a bijection between the two respective orbit sets, which proves the given result.