Central Product/Examples/D4 with Q

Example of Central Product
Let $G$ be the dihedral group $D_4$ whose group presentation is:
 * $G = \gen {a, b: a^4 = b^2 = e_G, a b = b a^{-1} }$

From Center of Dihedral Group $D_4$, the center of $G$ is:
 * $\map Z G = \set {e_G, a^2}$

Let $H$ 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:
 * $\map Z H = \set {e_H, x^2}$

Let:
 * $Z = \set {e_G, a^2}$
 * $W = \set {e_H, x^2}$

Let $\theta: Z \to W$ be the mapping defined as:
 * $\map \theta g = \begin{cases} e_H & : g = e_G \\ x^2 & : g = a^2 \end{cases}$

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 $32$ elements.

Proof
We have that every element $h$ of $H$ which is not in its center:
 * is of order $4$
 * is such that $h^2 = x^2$.

The set $X$ consists of:
 * $X = \set {\tuple {e_G, e_H}, \tuple {a^2, x^2} }$

The central product of $G$ and $H$ via $\theta$ is:
 * $\dfrac {G \times H} X$

Thus:

In the direct product $G \times H$:


 * $\tuple {e_G, e_H}$ has order $1$ and is in $X$

For all $h \in H$ such that $h$ is of order $4$:

For all $h \in H$ such that $h$ is of order $4$:

For all $g \in G$ such that $g$ is of order $2$:

For all $g \in G$ such that $g$ is of order $2$:

All remaining elements of $G \times H$ are of order $4$, and none of their squares is in $X$.