Exercise 40, § 6

Let $G_1, G_2$ be two groups and $G$ a subgroup of $G_1 \times G_2$ such that $\mathrm{pr}_1 G = G_1, \mathrm{pr}_2 G = G_2$; identifying $G_1$ and $G_2$ with $G_1 \times \{e_2\}$ and $\{e_1\} \times G_2$ respectively, $H_1 = G \cap G_1$ and $H_2 = G \cap G_2$ are normal subgroups of $G$ such that $G/H_1$ is isomorphic to $G_2$ and $G/H_2$ isomorphic to $G_1$.

(a) Show that for $G$ to be normal in $G_1 \times G_2$, it is necessary and sufficient that $G$ contain the commutator group $D(G_1 \times G_2) = D(G_1) \times D(G_2)$ (prove that $G$ contains $D(G_1)$ by writing $xzx^{-1}z^{-1} \in G$ for $x \in G_1$ and $z \in G$). Deduce that $G_1/H_1$ is commutative.

(b) If $G$ is normal in $G_1 \times G_2$, show that a homomorphism is defined of $G_1 \times G_2$ onto $G_1/H_1$ by associating with each ordered pair $(s_1, s_2)$ the coset of $s_1 t_1^{-1}$ modulo $H_1$, where $t_1 \in G_1$ is such that $(t_1, s_2) \in G$. Deduce that $(G_1 \times G_2)/G$ is isomorphic to $G/(H_1 \times H_2)$.