Direct_Product_of_Central_Subgroup_with_Inverse_Isomorphism_is_Central_Subgroup

Let $G$ and $H$ be groups. Let $\map Z G$ denote the center of $G$. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let $Z \cong W$ where $\cong$ denotes isomorphism. Let such a group isomorphism be $\theta: Z \to W$. Let $X$ be the set defined as X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z} Then $X$ is a central subgroup of $G \times H$.