central product of groups

1 Definitions

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

1. for $𝒥\subseteq \mathscr{H}$, $G=⟨𝒥⟩$if, and only if, $𝒥=\mathscr{H}$, and

2. $[H,⟨\mathscr{H}-\{H\}⟩]=1$ 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 $G\mathrm{=}\mathrm{⟨}\mathrm{H}\mathrm{⟩}$ but thishas the negative affect of allowing, for example, ${\mathrm{R}}^{\mathrm{2}}$ to have the central decompositionsuch sets as $\mathrm{\{}\mathrm{⟨}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{⟩}\mathrm{,}\mathrm{⟨}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{⟩}\mathrm{,}\mathrm{⟨}\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{)}\mathrm{⟩}\mathrm{,}\mathrm{0}\mathrm{,}{\mathrm{R}}^{\mathrm{2}}\mathrm{\}}$ andin general a decomposition of any possible size. By impossing 1, we then restrictthe 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 $G$. Thendefine $\pi :{\prod }_{H\in \mathscr{H}}H\to G$ by $(x_H:H\in \mathscr{H})\mapsto {\prod }_{H\in \mathscr{H}}{x}_H$.Then $G=\left({\prod }_{H\in \mathscr{H}}H\right)/\ker \pi$. Furthermore, $H\cap \ker \pi =1$ foreach 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 acentral decomposition. The converse is generally false.

2. Let , for a field $K$. Then $G$ is a centrally indecomposable group.

3. If

   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 $D_8=⟨a,b|a^4,b^2,{(ab)}^2⟩$ – the dihedral group of order 8,and $Q_8=⟨i,j|i^4,i^2=j^2,i^j=i^{-1}⟩$ – the quaternion group of order $8$, then$D_8\circ D_8=D_8\times D_8/⟨(a^2,a^2)⟩$ is isomorphic to$Q_8\circ Q_8=(Q_8\times Q_8)/⟨(i^2,i^2)⟩$; yet, $D_8$ and $Q_8$ are nonisomorphic andcentrally 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 identifyingthe centres of the direct factors…”. The modern definition clearly out grows this original versionas now centers may be only partially identified.