Suppose that a group $G$ acts transitively on a finite set $A$ and let $N$ be a normal subgroup of $G$. Let $\{O_i\}$ be the set of distinct orbits of $A$ under the action of $N$ on $A$.
(a) Show that the equation $g * O_i = \{gc \,|\, c \in O_i\}$ defines a transitive action of G on the set of orbits.
(b) Show that all the orbits have the cardinality $[N : N \cap \text{Stab } a]$ for some $a \in O_1$. (where Stab a means the stabilizer of $a$)
I have managed to prove (a) so it can be used as a result. May I please ask how to prove (b)? Any hints would be appreciated. Here is a written solution of a similar question which can be helpful: http://math.arizona.edu/~cais/594Page/soln/soln2.pdf Thanks so much.
If you have proved (a), then you know that $G$ permutes the orbits. As multiplication by $g\in G$ is a bijection (with inverse given by multiplication by $g^{-1}$), all orbits must be the same size. This means you only need to compute the size of the orbit $O_1$.
To compute the size of this orbit, use the orbit-stabilizer theorem which says that $|O_1|=[N:N_a]$ where $a\in O_1$ is any element and $N_a=\{n\in N\mid n.a=a\}$ is the stabilizer of $a$ in $N$. It is just left to observe that $N_a=N\cap G_a$, where $G_a$ is the stabilizer of $a$ in $G$.