Definition:Central Product
Definition
Let $G$ and $H$ be groups.
Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively.
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:
- $X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z}$
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 $\theta$.
Examples
Dihedral Group $D_4$ with Quaternion Group $Q$
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.
Dihedral Group $D_4$ with Itself
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 $2$.
Quaternion Group $Q$ with Itself
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.
Also see
- Results about central products can be found here.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Definition $13.9$