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\begin{document}

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\section*{dihedral group}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{DihedralGroups}{Dihedral groups}\dotfill \pageref*{DihedralGroups} \linebreak
\noindent\hyperlink{BinaryDihedralGroup}{Binary dihedral/dicyclic groups}\dotfill \pageref*{BinaryDihedralGroup} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{group_cohomology}{Group cohomology}\dotfill \pageref*{group_cohomology} \linebreak
\noindent\hyperlink{as_part_of_the_ade_pattern}{As part of the ADE pattern}\dotfill \pageref*{as_part_of_the_ade_pattern} \linebreak
\noindent\hyperlink{group_presentation}{Group presentation}\dotfill \pageref*{group_presentation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{quaternion_group__and_triality}{Quaternion group $Q_8$ and triality}\dotfill \pageref*{quaternion_group__and_triality} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{DihedralGroups}{}\subsubsection*{{Dihedral groups}}\label{DihedralGroups}

The dihedral group, $D_{2n}$, is a [[finite group]] of [[order of a group|order]] $2n$. It may be defined as the [[symmetry group]] of a regular $n$-gon.

For instance $D_6$ is the symmetry group of the equilateral triangle and is [[isomorphism|isomorphic]] to the [[symmetric group]], $S_3$.

For $n \in \mathbb{N}$, $n \geq 1$, the dihedral group $D_{2n}$ is thus the [[subgroup]] of the [[orthogonal group]] $O(2)$ which is generated from the finite [[cyclic group|cyclic]] [[subgroup]] $C_n$ of $SO(2)$ and the [[reflection]] at the $x$-axis (say). It is a semi-direct product of $C_n$ and a $C_2$ corresponding to that reflection.

Under the further embedding $O(2)\hookrightarrow SO(3)$ the (cyclic and) dihedral groups are precisely those [[finite subgroups of SO(3)]] that, among their [[ADE classification]], are not in the exceptional series.

(see e.g. \hyperlink{Greenless01}{Greenless 01, section 2})

\begin{remark}
\label{NotationConvention}\hypertarget{NotationConvention}{}
\textbf{Warning on notation}

There are two different conventions for numbering the dihedral groups.

\begin{enumerate}%
\item The above is the \emph{algebraic convention} in which the suffix gives the [[order]] of the group: ${\vert D_{2 n}\vert} = 2 n$.


\item In the \emph{geometric convention} one writes ``$D_n$'' instead of ``$D_{2n}$'', recording rather the geometric nature of the object of which it is the symmetry group.

Also beware that there is yet another group denoted $D_n$ mentioned at \emph{[[Coxeter group]]}.



\end{enumerate}
\end{remark}
\hypertarget{BinaryDihedralGroup}{}\subsubsection*{{Binary dihedral/dicyclic groups}}\label{BinaryDihedralGroup}

Under the further lift through the [[spin group]]-[[double cover]] map $SU(2) \simeq Spin(3) \to SO(3)$ of the [[special orthogonal group]], the dihedral group $D_{2n}$ is covered by the \emph{binary dihedral group}, also known as the \emph{dicyclic group} and denoted

\begin{displaymath}
2 D_{2n} = Dic_n
\end{displaymath}
Equivalently, this is the lift of the dihedral group $D_{2n}$ (\hyperlink{DihedralGroups}{above}) through the [[pin group]] [[double cover]] of the [[orthogonal group]] [[O(2)]] to [[Pin(2)]]

\begin{displaymath}
\itexarray{
    2 D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Pin_-(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    Spin(3)
    \\
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    &{}^{(pb)}&
    \big\downarrow
    \\
    D_{2n}
    &\overset{\phantom{AA}}{\hookrightarrow}&
    O(2)
    &\overset{\phantom{AA}}{\hookrightarrow}&
    SO(3)    
  }
\end{displaymath}
Explicity, let $\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C}$ be the [[quaternions]] realized as the [[Cayley-Dickson double]] of the [[complex numbers]], and identify the [[circle group]]

\begin{displaymath}
SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}
\end{displaymath}
with the [[unit circle]] in $\mathbb{C} \hookrightarrow \mathbb{H}$ this way, with group structure given by multiplication of [[quaternions]]. Then the [[Pin group]] [[Pin(2)]] is [[isomorphism|isomorphic]] to the [[subgroup]] of the [[group of units]] $\mathbb{H}^\times$ of the [[quaternions]] which consists of this copy of [[SO(2)]] together with the multiplies of the imaginary quaternion $j$ with this copy of $SO(2)$:

\begin{displaymath}
Pin_-(2)
  \;\simeq\;
  S\big( \mathbb{C}\big) 
   \;\cup\; 
  j \cdot S\big( \mathbb{C}\big) 
  \;\subset\;
  S(\mathbb{H}) 
  \;\simeq\;
  Spin(3)
  \,.
\end{displaymath}
The binary dihedral group $2 D_{2n}$ is the [[subgroup]] of that generated from

\begin{enumerate}%
\item $a \coloneqq \exp\left( \pi i \tfrac{1}{n} \right) \in S(\mathbb{C}) \subset Pin_-(2) \subset Spin(3)$


\item $x \coloneqq j \in Pin_-(2) \subset Spin(3)$.



\end{enumerate}
It is manifest that these two generators satisfy the relations

\begin{displaymath}
a^{2n} = 1
  \,,
  \phantom{AA}
  x^2 = a^n \; (= -1)
  \,,
  \phantom{AA}
  x^{-1} a x = a^{-1}
\end{displaymath}
and in fact these [[generators and relations]] fully determine $2 D_{2n}$, up to [[isomorphism]].

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

\hypertarget{group_cohomology}{}\subsubsection*{{Group cohomology}}\label{group_cohomology}

The [[group cohomology]] of the dihedral group is discussed for instance at \hyperlink{Groupprops}{Groupprops}.

\hypertarget{as_part_of_the_ade_pattern}{}\subsubsection*{{As part of the ADE pattern}}\label{as_part_of_the_ade_pattern}

[[!include ADE -- table]]

\hypertarget{group_presentation}{}\subsubsection*{{Group presentation}}\label{group_presentation}

The dihedral group $D_{2n}$ has a group presentation

\begin{displaymath}
\langle x,y : x^n=y^2=(xy)^2=1\rangle.
\end{displaymath}
From this it is easy to see that it is a [[semi-direct product]] of the $C_n$ generated by $x$ and the $C_2$ generated by $y$. The action of $y$ on $x$ is given by $\,^y x= x^{-1}$.

It is a standard example considered in elementary [[combinatorial group theory]].

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

\hypertarget{quaternion_group__and_triality}{}\subsubsection*{{Quaternion group $Q_8$ and triality}}\label{quaternion_group__and_triality}

The first binary dihedral group $2 D_4$ is [[isomorphism|isomorphic]] to the [[quaternion group]] of order 8:

\begin{displaymath}
2 D_4 (= Dic_2) \simeq Q_8
  \,.
\end{displaymath}
In the [[ADE-classification]] this is the entry [[D4]].

[[!include character table of 2D4=Dic2=Q8]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[anti-cyclotomic extension]]


\item [[dihedral homology]]


\item [[group presentation]]


\item [[Coxeter group]]



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

Discussion in the context of the [[classification of finite rotation groups]] goes back to

\begin{itemize}%
\item [[Felix Klein]], chapter I.4 of \emph{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade}, 1884, translated as \emph{Lectures on the Icosahedron and the Resolution of Equations of Degree Five} by George Morrice 1888, \href{https://archive.org/details/cu31924059413439}{online version}

\end{itemize}
Discussion in the context of [[equivariant cohomology theory]]:

\begin{itemize}%
\item [[John Greenlees]], \emph{Rational SO(3)-Equivariant Cohomology Theories}, in \emph{Homotopy methods in algebraic topology} (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc. (2001) 99 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5444}{web})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Dihedral_group}{Dihedral group}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Dicyclic_group#Binary_dihedral_group}{Binary dihedral group}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Dicyclic_group}{Dicyclic group}}


\item Groupprops, \emph{\href{https://groupprops.subwiki.org/wiki/Dicyclic_group}{Dicyclic group}}


\item Groupprops, \emph{\href{https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups}{Linear representation theory of dicyclic groups}}


\item Groupprops, \emph{\href{http://groupprops.subwiki.org/wiki/Group_cohomology_of_dihedral_group:D8}{Group cohomology of dihedral group:D8}}


\item GroupNames, \emph{\href{https://people.maths.bris.ac.uk/~matyd/GroupNames/dicyclic.html}{Dicyclic groups $Dic_n$}}



\end{itemize}
[[!redirects dihedral groups]]

[[!redirects binary dihedral group]] [[!redirects binary dihedral groups]]

[[!redirects dicyclic group]] [[!redirects dicyclic groups]]



\end{document}