Exercise 18, § 5

Let $\Gamma$ be a group of permutations of a set E. Let A and B be two subsets of E, complements one of the other and stable under $\Gamma$. Let $\Gamma_A$ and $\Gamma_B$ denote the groups consisting of the restrictions of the permutations in $\Gamma$ to A and B respectively and $\Delta_A$ and $\Delta_B$ the subgroups of $\Gamma$ which leave invariant respectively every element of A and every element of B. Show that $\Delta_A$ and $\Delta_B$ are normal subgroups of $\Gamma$ and that $\Gamma_A$ is isomorphic to $\Gamma/\Delta_A$ and $\Gamma_B$ to $\Gamma/\Delta_B$; if $\Delta_{AB}$ (resp. $\Delta_{BA}$) is the group consisting of the restrictions of the permutations in $\Delta_A$ (resp. $\Delta_B$) to B (resp. A), show that the quotient groups $\Gamma_A/\Delta_{BA}$, $\Gamma_B/\Delta_{AB}$ and $\Gamma/(\Delta_A \Delta_B)$ are isomorphic (use Exercise 8).