Exercise 25, § 2

Let $G$ be a topological group, $K$ a normal subgroup of $G$, $\varphi$ the canonical mapping $G \to G/K$. Let $f$ be a continuous homomorphism of a topological group $H$ into $G$, such that every element of $f(H)$ commutes with every element of $K$. Then $(y, z) \to f(y)z$ is a continuous homomorphism $g$ of $H \times K$ into $G$. $g$ is a strict morphism of $H \times K$ into $G$ if and only if $\varphi \circ f$ is a strict morphism of $H$ into $G/K$.