Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup

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

Let $Z \left({G}\right)$ be the center of $G$.

Let $Z \le Z \left({G}\right), W \le Z \left({H}\right)$ such that $Z \cong W$.

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

The set $X = \left\{{\left({x, \theta^{-1} \left({x}\right)}\right): x \in Z}\right\}$ is a central subgroup of $G \times H$.

The quotient group $\left({G \times H}\right) / X$ is denoted $\left({G \times_\theta H}\right)$ and is called the central product of $G$ and $H$ via $\theta$.

Proof
The fact that the set $X = \left\{{\left({x, \theta^{-1} \left({x}\right)}\right): x \in Z}\right\}$ 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.