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\begin{document}

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\section*{central product of groups}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_riemannian_geometry_and_spin_geometry}{In Riemannian geometry and spin geometry}\dotfill \pageref*{in_riemannian_geometry_and_spin_geometry} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{CentralProduct}\hypertarget{CentralProduct}{}
\textbf{([[central product]])}

Given two [[groups]] $G_1$ and $G_2$ and a joint [[subgroup]]

\begin{equation}
C \xhookrightarrow{\iota_i} Z(G_i)
\label{SubgroupInclusion}\end{equation}
in each of their [[centers]], then the corresponding (``external'') \emph{central product} is the [[quotient group]]

\begin{displaymath}
G_1 \circ G_2
  \;\coloneqq\;
  \big( 
    G_1 \times G_2
  \big)/_{diag} C
\end{displaymath}
of the [[direct product group]] $G_1 \times G_2$ by the [[diagonal]] subgroup $C \xhookrightarrow{(\iota_1, \iota_2)} G_1 \times G_2$.

\end{defn}
(\hyperlink{Gorenstein80}{Gorenstein 80, p. 29})

\begin{remark}
\label{MaterialDefinition}\hypertarget{MaterialDefinition}{}
\textbf{([[structural set theory|structural]] over [[material set theory|material]] definition)}

Beware that most texts insists on stating the choices in Def. \ref{CentralProduct} as that of

\begin{enumerate}%
\item two separate subgroups $C_i \xhookrightarrow{\iota_i} Z(G_i)$


\item an [[isomorphism]] $C_1 \xrightarrow[\simeq]{\phi} C_2$ between them



\end{enumerate}
and insists that the second groups as via $(-)^{-1}\circ \phi$

These clauses matter if one thinks of the subgroup inclusions as in [[material set theory]]. But we speak [[structural set theory]], which means that a [[subgroup]] inclusion as in \eqref{SubgroupInclusion} is really a choice of \emph{[[monomorphism|monic]] [[homomorphism]]}, and this choice already absorbs the choice of $\phi$ and or of $(-)^{-1}\circ \phi$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{(notation)}

Beware that there is no widely accepted convention for the notation of central products, and that most notational conventions suppress the choices of central subgroups involved. The ``$\circ$''-notation is popular in [[finite group]]-theory, while in [[Riemannian geometry]] people tend to use ``$\cdot$'' (see [[Sp(n).Sp(1)]]) or just plain juxtaposition, with no symbol for the central product at all.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_riemannian_geometry_and_spin_geometry}{}\subsubsection*{{In Riemannian geometry and spin geometry}}\label{in_riemannian_geometry_and_spin_geometry}

In [[Riemannian geometry]] and [[spin geometry]] one considers the central products [[Sp(n).Sp(1)]] and [[Spin(n).Spin(m)]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item D. Gorenstein, p. 29 of \emph{Finite Groups}, New York (1980)

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Central_product}{Central product}}


\item GroupProps, \emph{\href{https://groupprops.subwiki.org/wiki/External_central_product}{External central product}}



\end{itemize}
[[!redirects central products of groups]]

[[!redirects external central product of groups]] [[!redirects external central products of groups]]

[[!redirects internal central product of groups]] [[!redirects internal central products of groups]]

[[!redirects central product group]] [[!redirects central product groups]]

[[!redirects external central product group]] [[!redirects external central product groups]]

[[!redirects internal central product group]] [[!redirects internal central product groups]]

[[!redirects central product]] [[!redirects central products]]

[[!redirects external central product]] [[!redirects external central products]]

[[!redirects internal central product]] [[!redirects internal central products]]



\end{document}