Definition:Central Product

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

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

Let:
 * $Z \le \map Z G$
 * $W \le \map Z H$

where:
 * $Z \le \map Z G$ denotes that $Z$ is a subgroup of $\map Z G$.

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:
 * $X = \set {\tuple {x, \map {\theta^{-1} } x}: x \in Z}$

Then the quotient group $\struct {G \times H} / X$ is denoted $\struct {G \times_\theta H}$ and is called the central product of $G$ and $H$ via $\theta$.

Also see

 * Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup