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\begin{document}

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\section*{semidirect product group}

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

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

[[!include group theory - contents]]

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{interior_semidirect_products}{Interior semidirect products}\dotfill \pageref*{interior_semidirect_products} \linebreak
\noindent\hyperlink{right_semidirect_products}{Right semidirect products}\dotfill \pageref*{right_semidirect_products} \linebreak
\noindent\hyperlink{as_a_grothendieck_construction}{As a Grothendieck construction}\dotfill \pageref*{as_a_grothendieck_construction} \linebreak
\noindent\hyperlink{semidirect_products_of_groupoids}{Semidirect products of groupoids}\dotfill \pageref*{semidirect_products_of_groupoids} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_split_group_extensions}{As split group extensions}\dotfill \pageref*{as_split_group_extensions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_automorphisms_on_the_circle_group}{The automorphisms on the circle group}\dotfill \pageref*{the_automorphisms_on_the_circle_group} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

If a [[group]] $G$ [[action|acts]] on a group $\Gamma$ (on the left, say) by group [[automorphism]]

\begin{displaymath}
\rho : G \to Aut(\Gamma)
  \,,
\end{displaymath}
then there is a \textbf{semidirect product} group $\Gamma \rtimes \, G$ whose underlying set is the [[Cartesian product]] $\Gamma \times  G$ but whose multiplication is twisted by $\rho$:

\begin{displaymath}
(\delta,h)(\gamma,g)= (\delta \rho(h)(\gamma) , h g)
\end{displaymath}
for $\delta, \gamma \in \Gamma,\; h,g \in G$, where $^h \gamma$ denotes the result of acting with $h$ on the left on $\gamma$.

If the twist is trivial, then this reduces to just the [[direct product group]] construction, whence the name.

There is a [[projection]] morphism $p:\Gamma \rtimes \, G \to G$ , $(\gamma, g) \to g$. A [[section]] $s$ of this can be identified with a [[derivation]] $d$, i.e. $d$ satisfies $d(h g) = (d h) \,^h (d g)$.

\hypertarget{interior_semidirect_products}{}\subsubsection*{{Interior semidirect products}}\label{interior_semidirect_products}

Let $H$ be any group. A decomposition of $H$ as an \textbf{internal} semidirect product consists of a [[subgroup]] $\Gamma$ and a [[normal subgroup]] $G$, such that every element of $H$ can be written uniquely in the form $\gamma g$, for $\gamma \in \Gamma$ and $g \in G$.

The internal and external concepts are equivalent. In particular, any (external) semidirect product $\Gamma \rtimes G$ is an internal semidirect product of the [[images]] of $\Gamma$ and $G$ in it.

\hypertarget{right_semidirect_products}{}\subsubsection*{{Right semidirect products}}\label{right_semidirect_products}

The definitions above are not symmetric in left and right; since the first definition begins with a left action, we may call it a \emph{left} semidirect product. Then a \textbf{right} semidirect product is given by an action on the right, or internally by the requirement that every element can be written in the form $g \gamma$.

However, right and left semidirect products are equivalent. Essentially, this is because any left action $(h,g) \mapsto {}^h{g}$ defines a right action $(g,h) \mapsto g^h \coloneqq {}^{h^{-1}}g$ and vice versa.

\hypertarget{as_a_grothendieck_construction}{}\subsubsection*{{As a Grothendieck construction}}\label{as_a_grothendieck_construction}

Writing $\mathbb{B} G$ for the [[category]] with a single [[object]] $\ast$ and the [[group]] $G$ as its [[hom set]] (i.e. the [[delooping]] [[groupoid]] of $G$), define a [[functor]] $F \colon \mathbb{B}G \to$ [[Cat]] to send that single object to the delooping groupoid of $\Gamma$, i.e. $* \mapsto \mathbb{B}\Gamma$ and to send the morphisms $G \to Aut(\Gamma)$ according to the given [[action]] of $G$ on $\Gamma$.

Then the delooping of the semidirect product group $\Gamma \rtimes G$ arises as the [[Grothendieck construction]] of this functor:

\begin{displaymath}
\mathbb{B}( \Gamma \rtimes G)
  \;\simeq\;
  \int_{\mathbb{B}G}F
\end{displaymath}
\hypertarget{semidirect_products_of_groupoids}{}\subsubsection*{{Semidirect products of groupoids}}\label{semidirect_products_of_groupoids}

It is useful to generalise this to the case $\Gamma$ is a [[groupoid]]. This occurs if for example $\Gamma = \pi_1 X$ where $X$ is a (left) $G$-space.

So if $X=Ob(\Gamma)$, then $\Gamma \rtimes \, G$ has object set $X$ and a morphism $y \to x$ is a pair $(\gamma,g)$ such that $\gamma: y \to
g x$ in $\Gamma$. The composition law is then given again by

\begin{displaymath}
(\delta,h)(\gamma,g)= (\delta \, ^h \gamma, h g)
\end{displaymath}
if $(\delta, h): z \to y$, so that $\delta: z \to h y$ in $\Gamma$.

If $\Gamma$ is a discrete groupoid, and so identified with $X$, then we get $X \rtimes \, G$ which is the [[action groupoid]] of the action. In this case the projection $p: X \rtimes \, G \to G$ is a covering morphism of groupoids, i.e. any $g \in G$ has a unique lifting with given initial point. Note that if $Y \to X$ is a covering map of spaces, then the induced morphism of fundamental groupoids is a covering morphism of groupoids. If $q: H \to \pi_1 X$ is a covering morphism of groupoids, and $X$ admits a universal covering map, then there is a topology on $Y=Ob(H)$ such that $H \cong \pi_1 Y$. In this way, the category of covering maps of $X$ is equivalent to the category of covering morphisms of $\pi_1 X$.

The utility of the more general construction is that there is notion of orbit groupoid $\Gamma //G$ (identify any $\gamma$ and $^g
\gamma$) and it is a theorem that the orbit groupoid is isomorphic to the quotient groupoid

\begin{displaymath}
(\Gamma \rtimes \, G)/N
\end{displaymath}
where $N$ is the [[normal closure]] in $\Gamma \rtimes \, G$ of all elements $(1_x,g)$. Details are in the book reference below (but the conventions are not quite the same).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_split_group_extensions}{}\subsubsection*{{As split group extensions}}\label{as_split_group_extensions}

Semidirect product groups $A \rtimes_\rho G$ are precisely the split [[group extensions]] of $G$ by $A$. See at \emph{\href{group+extension#SplitExtensionsAndSemidirectProductGroups}{group extension -- split extensions and semidirect product groups}}.

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

\hypertarget{the_automorphisms_on_the_circle_group}{}\subsubsection*{{The automorphisms on the circle group}}\label{the_automorphisms_on_the_circle_group}

For $U(1) = \mathbb{R}/\mathbb{Z}$ the [[circle group]], the [[automorphism group]] is

\begin{displaymath}
Aut(U(1)) \simeq \mathbb{Z}_2
  \,,
\end{displaymath}
where the nontrivial element in $\mathbb{Z}_2$ acts on $\mathbb{R}$ by multiplication with $-1$. Write $\rho_{aut} : U(1) \times \mathbb{Z}_2 \to U(1)$ for the automorphism [[action]]. The corresponding semidirect product group is the [[group extension]]

\begin{displaymath}
U(1) \stackrel{}{\hookrightarrow} U(1) \rtimes_{\rho_{aut}} \mathbb{Z}_2
  \to \mathbb{Z}_2
\end{displaymath}
where the group operation is given by

\begin{displaymath}
(c_1 \; mod \; \mathbb{Z}, \sigma_1)
  \cdot
  (c_2\; mod \; \mathbb{Z}, \sigma_2)
  = 
  (c_1 + \sigma_1(c_2) \; mod \; \mathbb{Z}, \sigma_1 + \sigma_2)
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[semidirect product Lie algebra]]

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

A general survey is in

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Semi-direct_product}{Semidirect product}}

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Patrick Morandi, \emph{Semidirect products} (\href{http://sierra.nmsu.edu/morandi/notes/semidirect.pdf}{pdf})

\end{itemize}
Relevant textbooks include

\begin{itemize}%
\item [[R. Brown]], \emph{Topology and groupoids}, Booksurge 2006.


\item [[P. J. Higgins]] and J. Taylor, \emph{The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space}, in K.H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, Springer LNM 962 (1982) 115--122.



\end{itemize}
[[!redirects semidirect product groups]]



\end{document}