(1) Let A, B, and C be groups.
(a) Suppose $C \unlhd A$. Show that $C \times \{e_B\} \unlhd A \times B$, where $e_B$ is the identity in B.
(b) Use the First Isomorphism Theorem to prove that
$\frac{A \times B}{C \times \{e_B\}} \cong \frac{A}{C} \times B$.
(c) Prove that
$\frac{A \times B}{A \times \{e_B\}} \cong B$.