central product of groups

1 Definitions

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

1.

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

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 $E=\left\{\begin{bmatrix}1&\alpha&\gamma\\ 0&1&\beta\\ 0&0&1\end{bmatrix}:\alpha,\beta,\gamma\in K\right\}$ , for a field $K$ . Then $G$ is a centrally indecomposable group.
3.

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.

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.

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, Finite-by-nilpotent groups, Proc. Camb. Phil. Soc., 52 (1956), 611-616.
2 B. H. Neumann, and H. Neumann, A remark on generalized free products, J. London Math. Soc. 25 (1950), 202-204.

Title	central product of groups
Canonical name	CentralProductOfGroups
Date of creation	2013-03-22 18:49:45
Last modified on	2013-03-22 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