Central Product/Examples/D4 with D4
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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
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Example $13.10$