Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup

Theorem
Let $G$ and $H$ be groups.

Let $\map Z G$ be the center of $G$.

Let $Z \le \map Z G, W \le \map Z H$ such that $Z \cong W$.

Let such a group isomorphism be $\theta: Z \to W$.

Let $X$ be the set defined as:
 * $X = \set {\tuple {x, \map {\theta^{-1} } x}: x \in Z}$

Then $X$ is a central subgroup of $G \times H$.

Proof
The fact that the set $X = \set {\tuple {x, \map {\theta^{-1} } x}: x \in Z}$ is a subgroup of $G \times H$ follows from elementary properties of homomorphisms.

The fact that $X$ is a central subgroup of $G \times H$ follows from the definition of a central subgroup.