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 an 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 Central Subgroup.