# central product of groups

## 1 Definitions

A central decomposition is a set $\mathcal{H}$ of subgroups   of a group $G$ where

1. 1.

for $\mathcal{J}\subseteq\mathcal{H}$, $G=\langle\mathcal{J}\rangle$ if, and only if, $\mathcal{J}=\mathcal{H}$, and

2. 2.

$[H,\langle\mathcal{H}-\{H\}\rangle]=1$ for all $H\in\mathcal{H}$.

A group $G$ is centrally indecomposable  if its only central decomposition is $\{G\}$. A central decomposition is fully refined if its members are centrally indecomposable.

###### Remark 1.

Condition 1 is often relaxed to $G=\langle\mathcal{H}\rangle$ but this has the negative affect of allowing, for example, $\mathbb{R}^{2}$ to have the central decomposition such sets as $\{\langle(1,0)\rangle,\langle(0,1)\rangle,\langle(1,1)\rangle,0,\mathbb{R}^{2}\}$ and in general a decomposition of any possible size. By impossing 1, we then restrict the central decompositions of $\mathbb{R}^{2}$ to direct decompositions. Furthermore, with condition 1, the meaning of indecomposable is easily had.

A central product is a group $G=\left(\prod_{H\in\mathcal{H}}H\right)\large/N$ where $N$ is a normal subgroup  of $\prod_{H\in\mathcal{H}}H$ and $H\cap N=1$ for all $H\in\mathcal{H}$.

###### Proposition 2.

Every finite central decomposition $\mathcal{H}$ is a central product of the members in $\mathcal{H}$.

###### Proof.

Suppose that $\mathcal{H}$ is a a finite central decomposition of $G$. Then define $\pi:\prod_{H\in\mathcal{H}}H\to G$ by $(x_{H}:H\in\mathcal{H})\mapsto\prod_{H\in\mathcal{H}}x_{H}$. Then $G=\left(\prod_{H\in\mathcal{H}}H\right)/\ker\pi$. Furthermore, $H\cap\ker\pi=1$ for each direct factor  $H$ of $\prod_{K\in\mathcal{H}}K$. Thus, $G$ is a central product of $\mathcal{H}$. ∎

## 2 Examples

1. 1.
2. 2.

Let $E=\left\{\begin{bmatrix}1&\alpha&\gamma\\ 0&1&\beta\\ 0&0&1\end{bmatrix}:\alpha,\beta,\gamma\in K\right\}$, for a field $K$. Then $G$ is a centrally indecomposable group.

3. 3.

If

 $F=\left\{\begin{bmatrix}1&\alpha_{1}&\dots&\alpha_{n}&\gamma\\ 0&1&0&&\beta_{1}\\ \vdots&&\ddots&0&\vdots\\ &&&1&\beta_{n}\\ &&&&1\end{bmatrix}:\alpha_{i},\beta_{i},\gamma\in K,1\leq i\leq n\right\}$

and

 $H_{i}=\{A\in F:\alpha_{j}=0=\beta_{j}\forall j\neq i,1\leq j\leq n\}$

then $\{H_{1},\dots,H_{n}\}$ is a central decomposition of $G$. Furthermore, each $H_{i}$ is isomorphic   to $E$ and so $\mathcal{H}$ is a fully refined central decomposition.

4. 4.

If $D_{8}=\langle a,b|a^{4},b^{2},(ab)^{2}\rangle$ – the dihedral group  of order 8, and $Q_{8}=\langle i,j|i^{4},i^{2}=j^{2},i^{j}=i^{-1}\rangle$ – the quaternion group   of order $8$, then $D_{8}\circ D_{8}=D_{8}\times D_{8}/\langle(a^{2},a^{2})\rangle$ is isomorphic to $Q_{8}\circ Q_{8}=(Q_{8}\times Q_{8})/\langle(i^{2},i^{2})\rangle$; yet, $D_{8}$ and $Q_{8}$ are nonisomorphic and centrally indecomposable. In particular, central decompositions are not unique even up to automorphisms    . This is in contrast the well-known Krull-Remak-Schmidt theorem for direct products of groups.

## References

Title central product of groups CentralProductOfGroups 2013-03-22 18:49:45 2013-03-22 18:49:45 Algeboy (12884) Algeboy (12884) 6 Algeboy (12884) Definition msc 20E34 central decomposition central decomposition central product