# Central Product/Examples/Q with Q

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## Example of Central Product

Let $G$ be the quaternion group $Q$ whose group presentation is:

- $Q = \gen {x, y: x^4 = e_H, y^2 = x^2, x y = y x^{-1} }$

From Center of Quaternion Group, the center of $H$ is:

- $\map Z H = \set {e_G, x^2}$

Let:

- $Z = W = \set {e_G, x^2}$

Let $\theta: Z \to W$ be the mapping defined as:

- $\map \theta g = \begin{cases} e_G & : g = e_G \\ x^2 & : g = x^2 \end{cases}$

Let $X$ be the set defined as:

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

The central product of $G$ and $H$ via $\theta$ has $19$ elements.

## Proof

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $13$: Direct products: Exercise $4$