Question: Assume the group G acts transitively on the finite set A and let $H\unlhd G$. If $\{\Theta_1 , ... , \Theta_n\}$ are the distinct orbits of H on A, prove that G acts transitively on these orbits.
Proposed Solution: Because H is normal, every $g\in G$ is contained in some coset of the quotient group, i.e. there exists $g',h'$ such that $g=g'h'$. Therefore, $g\cdot\Theta_i=\{g\cdot a_1,...,g\cdot a_n\}=\{g'h'\cdot a_1,...,g'h'\cdot a_n\}=\{g'\cdot a'_1,...,g'\cdot a'_n\}=g'\cdot\Theta_j$ and so $({g'}^{-1}g)\cdot\Theta_i=g''\cdot\Theta_i=\Theta_j$ which (I think) shows that all g send orbits to orbits. As for the transitivity, given $g\in G$, choosing 2 separate representatives of $gH$, i.e. $g_1h_1$ or $g_2h_2$ where $g_1h_1=g_2h_2=g$, and reproducing the above argument will be such that $({g_1}^{-1}g)\cdot\Theta_i\neq ({g_2}^{-1}g)\cdot\Theta_i$ and seeing as there are only finitely many orbits, I believe we are done.
Can somebody ensure my proof is correct?