central product of groups
1 Definitions
A central decomposition is a set $\mathscr{H}$ of subgroups^{} of a group $G$ where

1.
for $\mathcal{J}\subseteq \mathscr{H}$, $G=\u27e8\mathcal{J}\u27e9$ if, and only if, $\mathcal{J}=\mathscr{H}$, and

2.
$[H,\u27e8\mathscr{H}\{H\}\u27e9]=1$ for all $H\in \mathscr{H}$.
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 $G\mathrm{=}\mathrm{\u27e8}\mathrm{H}\mathrm{\u27e9}$ but this has the negative affect of allowing, for example, ${\mathrm{R}}^{\mathrm{2}}$ to have the central decomposition such sets as $\mathrm{\{}\mathrm{\u27e8}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{\u27e9}\mathrm{,}\mathrm{\u27e8}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{\u27e9}\mathrm{,}\mathrm{\u27e8}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{\u27e9}\mathrm{,}\mathrm{0}\mathrm{,}{\mathrm{R}}^{\mathrm{2}}\mathrm{\}}$ and in general a decomposition of any possible size. By impossing 1, we then restrict the central decompositions of ${\mathrm{R}}^{\mathrm{2}}$ to direct decompositions. Furthermore, with condition 1, the meaning of indecomposable is easily had.
A central product is a group $G=\left({\prod}_{H\in \mathscr{H}}H\right)/N$ where $N$ is a normal subgroup^{} of ${\prod}_{H\in \mathscr{H}}H$ and $H\cap N=1$ for all $H\in \mathscr{H}$.
Proposition 2.
Every finite central decomposition $\mathrm{H}$ is a central product of the members in $\mathrm{H}$.
Proof.
Suppose that $\mathscr{H}$ is a a finite central decomposition of $G$. Then define $\pi :{\prod}_{H\in \mathscr{H}}H\to G$ by $({x}_{H}:H\in \mathscr{H})\mapsto {\prod}_{H\in \mathscr{H}}{x}_{H}$. Then $G=\left({\prod}_{H\in \mathscr{H}}H\right)/\mathrm{ker}\pi $. Furthermore, $H\cap \mathrm{ker}\pi =1$ for each direct factor^{} $H$ of ${\prod}_{K\in \mathscr{H}}K$. Thus, $G$ is a central product of $\mathscr{H}$. ∎
2 Examples

1.
Every direct product^{} is also a central product and so also every direct decomposition is a central decomposition. The converse is generally false.

2.
Let $E=\{\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \alpha \hfill & \hfill \gamma \hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \beta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]:\alpha ,\beta ,\gamma \in K\}$, for a field $K$. Then $G$ is a centrally indecomposable group.

3.
If
$$F=\{\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill {\alpha}_{1}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {\alpha}_{n}\hfill & \hfill \gamma \hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \hfill & \hfill {\beta}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \hfill & \hfill \mathrm{\ddots}\hfill & \hfill 0\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill & \hfill {\beta}_{n}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right]:{\alpha}_{i},{\beta}_{i},\gamma \in K,1\le i\le n\}$$ and
$${H}_{i}=\{A\in F:{\alpha}_{j}=0={\beta}_{j}\forall j\ne i,1\le j\le n\}$$ then $\{{H}_{1},\mathrm{\dots},{H}_{n}\}$ is a central decomposition of $G$. Furthermore, each ${H}_{i}$ is isomorphic^{} to $E$ and so $\mathscr{H}$ is a fully refined central decomposition.

4.
If ${D}_{8}=\u27e8a,b{a}^{4},{b}^{2},{(ab)}^{2}\u27e9$ – the dihedral group^{} of order 8, and ${Q}_{8}=\u27e8i,j{i}^{4},{i}^{2}={j}^{2},{i}^{j}={i}^{1}\u27e9$ – the quaternion group^{} of order $8$, then ${D}_{8}\circ {D}_{8}={D}_{8}\times {D}_{8}/\u27e8({a}^{2},{a}^{2})\u27e9$ is isomorphic to ${Q}_{8}\circ {Q}_{8}=({Q}_{8}\times {Q}_{8})/\u27e8({i}^{2},{i}^{2})\u27e9$; 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 wellknown KrullRemakSchmidt theorem for direct products of groups.
3 History
The name central product appears to have been coined by Philip Hall [1, Section^{} 3.2] though the principal concept of such a product^{} appears in earlier work (e.g. [2, Theorem II]). Hall describes central products as “…the group obtained from the direct product by identifying the centres of the direct factors…”. The modern definition clearly out grows this original version as now centers may be only partially identified.
References
 1 P. Hall, Finitebynilpotent groups, Proc. Camb. Phil. Soc., 52 (1956), 611616.
 2 B. H. Neumann, and H. Neumann, A remark on generalized free products^{}, J. London Math. Soc. 25 (1950), 202204.
Title  central product of groups 

Canonical name  CentralProductOfGroups 
Date of creation  20130322 18:49:45 
Last modified on  20130322 18:49:45 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  6 
Author  Algeboy (12884) 
Entry type  Definition 
Classification  msc 20E34 
Synonym  central decomposition 
Defines  central decomposition 
Defines  central product 