Central Product/Examples/D4 with D4

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:

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

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

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

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 $19$ elements of order $2$.

Proof

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Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Example $13.10$