The Question is let acts transitively on a nonempty finite set A

let H be normal subgroup of G,Let Orbits of H on A be

prove that G acts transitively on

My work I want to prove that for any two oribts there exist such that

I will call elements with "a" and with "b"

pick

there exist g in G such that since G acts transitively on A

let

want to show that is an element of

but H is normal subgroup so b_s in O_i

b_s is arbitrary so for any b in O_j g.b is an element of O_i

ends

is it correct ?? is there any shorter way ? any ideas

Thanks