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

• Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup

• 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 $13.9$