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\begin{document}

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\section*{quaternion 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{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{character_table}{Character table}\dotfill \pageref*{character_table} \linebreak
\noindent\hyperlink{matrix_representation}{Matrix representation}\dotfill \pageref*{matrix_representation} \linebreak
\noindent\hyperlink{SubgroupLattice}{Subgroup lattice}\dotfill \pageref*{SubgroupLattice} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{quaternion group} of [[order of a group|order]] 8, $Q_8$, is the [[finite subgroup of SU(2)]] $Q_8 \subset SU(2) \simeq S^3 \subset \mathbb{H}$ of unit [[quaternions]] which consists of the canonical four [[linear basis|basis]]-quaternions and their negatives:

\begin{displaymath}
Q_8
  \;=\;
  \big\{
     \pm 1,
     \, 
     \pm i,
     \, 
     \pm j,
     \, 
     \pm k
  \big\}
  \,.
\end{displaymath}
This is [[isomorphic|isomorphic]] to the [[binary dihedral group]] of the same [[order of a group|order]] $Q_8 \simeq 2 D_4$. As such, the [[Dynkin diagram]] that corresponds to $Q_8$ under the [[ADE-classification]] of [[finite subgroups of SU(2)]] is [[D4]], the [[triality]]-invariant one.

\begin{quote}%
graphics grabbed from Wikipedia \href{https://upload.wikimedia.org/wikipedia/commons/5/59/Dynkin_diagram_D4.png}{here}


\end{quote}
This order-8 quaternion group $Q_8$ is the first in a row of generalized quaternion groups, $Q_{2^n}$, which are also examples of dicyclic groups, which class forms part of an even larger family. We will treat both general dicyclic groups and the specific example of the quaternion group together.

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

The \emph{dicyclic} of order $4n$, $n\geq 2$, is the [[group]] $Dic_n$ defined by the presentation $\langle x,y | x^{2n}= x^{n} y^{-2}=y^{-1}x y x=1\rangle$.

The \emph{quaternion group} (of order 8) is then $Dic_n$ for $n=2$.

The \emph{generalised quaternion group} of order $2^{k+1}$ is $Dic_n$ with $n= 2^{k-1}$.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item $Dic_n$ is (isomorphic to) a finite subgroup of $\mathbb{H}^\ast$ as can be seen by taking generators $x=j$ and $y=\cos(\pi/n) + i\sin(\pi/n)$. For $n=2$ this simply yields the subgroup generated by $i$ and $j$.


\item $Dic_n$ has another presentation as $\langle R, S, T | R^2=S^2=T^n=R S T\rangle$. $R S T$ as a power of each of the generators is central and $Dic_n/\langle R S T\rangle= D_{2n}$, where $D_{2n}=\langle R, S, T | R^2=S^2=T^n=R S T=1\rangle$ is the [[dihedral group]] of order $2n$.


\item $Q_8$ is a \textbf{Hamiltonian group} i.e. a [[non-abelian group]] such that every [[subgroup]] is [[normal subgroup|normal]]. Moreover, a general structure theorem for Hamiltonian groups by Baer (1933) says that every Hamiltonian group has a [[direct product group]]-decomposition containing $Q_8$ as a factor hence, in particular, every Hamiltonian group contains $Q_8$ as a [[subgroup]]! (cf. \hyperlink{Scott87}{Scott (1987, p.253}))


\item $Q_8$ is the multiplicative part of the quaternionic near-field $J_9$. (cf. \hyperlink{Weibel07}{Weibel (2007)})



\end{itemize}
\begin{prop}
\label{InclusionInLargerFininteSubgroupsOfSU2}\hypertarget{InclusionInLargerFininteSubgroupsOfSU2}{}
\textbf{(inclusion of $Q_8$ into [[finite subgroups of SU(2)]])}

Among the [[finite subgroups of SU(2)]] (hence among all ``finite quaternion groups'') the quaternion group of [[order of a group|order]] 8, $Q_8$ is a proper [[subgroup]] precisely of the three [[ADE-classification|exceptional cases]]:

\begin{itemize}%
\item $Q_8 \subset 2 T$ the [[binary tetrahedral group]] ([[normal subgroup|normal]]),


\item $Q_8 \subset 2 O$ the [[binary octahedral group]] ([[normal subgroup|normal]])


\item $Q_8 \subset 2 I$ the [[binary icosahedral group]] (not [[normal subgroup|normal]])



\end{itemize}
\end{prop}
(e.g. \hyperlink{KocaMocKoca16}{Koca-Moc-Koca 16, p. 8}, pointing to \hyperlink{CoxeterMoser65}{Coxeter-Moser 65} and \hyperlink{Coxeter73}{Coxeter 73})

\hypertarget{character_table}{}\subsubsection*{{Character table}}\label{character_table}

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

\hypertarget{matrix_representation}{}\subsubsection*{{Matrix representation}}\label{matrix_representation}

There are lots of different ways of defining $Q:=Q_8$. One is that it is the subgroup of $Gl(2,\mathbb{C})$ generated by the matrices

\begin{displaymath}
\xi = \left(\itexarray{i&0\\0&-i}\right)
\end{displaymath}
and

\begin{displaymath}
\eta =\left(\itexarray{0&-1\\1&0}\right).
\end{displaymath}
In this form it is a nice exercise to derive a presentation of $Q_8$. Clearly $\xi^4=1$ and $\eta$ is not in $\langle \xi\rangle$ as is easiy checked, so the order of this group must be at least 8.

We note that $\eta^2 = \xi^2$ and that $\eta \xi \eta^{-1}= \xi^{-1}$, so a guess for a presentation would be

\begin{displaymath}
\langle x,y : x^4=1, y^2=x^2, y x y^{-1}=x^{-1}\rangle.
\end{displaymath}
Let us call $G$ the group presented by this presentation, then there is an obvious epimorphism from $G$ to $Q$ sending $x$ to $\xi$ and $y$ to $\eta$. This is an isomorphism as will be clear if we show that the order of $G$ is less than of equal to 8. Now every element of $G$ can be written in the form $x^i y^j$ with $0\leq i\leq 3$ and $0\leq j\leq 1$, since $y x=x^{-1}y$ so powers of $y$ can be shifted to the right in any expression and then if the resulting power of $y$ is greater than 2 we can use $y^2=x^2$ to replace even powers of $y$ by powers of $x$. We must therefore have that the group $G$ must contain at most 8 elements so the above presentation is a presentation of $Q_8$.

\hypertarget{SubgroupLattice}{}\subsubsection*{{Subgroup lattice}}\label{SubgroupLattice}

The following shows the [[subgroup lattices]] of the first few [[generalized quaternion groups]]:

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[quaternion]]


\item [[dihedral group]]


\item [[canonical formula of myth]]



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

\begin{itemize}%
\item [[Kenneth Brown|Kenneth S. Brown]], \emph{Cohomology of Groups} , GTM \textbf{87} Springer Heidelberg 1982. (pp.98-101)


\item H. S. M. Coxeter, \emph{The binary polyhedral groups, and other generalizations of the quaternion group} , Duke Math. J. \textbf{7} no.1 (1940) pp.367--379.


\item T. Y. Lam, \emph{Hamilton's Quaternions} , pp.429-454 in \emph{Handbook of Algebra III} , Elsevier Amsterdam 2004. (\href{http://math.berkeley.edu/~lam/quat.ps}{preprint})


\item W. R. Scott, \emph{Group Theory} , Dover New York 1987. (pp.189-194, 252-254)


\item [[Charles Weibel]], \emph{Survey of Non-Desarguesian Planes} , Notices of the AMS \textbf{54} no.10 (2007) pp.1294--1303. (\href{http://www.ams.org/notices/200710/tx071001294p.pdf}{pdf})


\item Mehmet Koca, Ramazan Koç, Nazife Ozdes Koca, \emph{Two groups $2^3.PSL_2(7)$ and $2^3:PSL_2(7)$ of order 1344} (\href{https://arxiv.org/abs/1612.06107}{arXiv:1612.06107})


\item H.S.M. Coxeter, W. O. J. Moser, \emph{Generators and Relations for Discrete Groups}, (Springer Verlag, 1965);


\item H.S.M. Coxeter, \emph{Regular Complex Polytopes} (Cambridge; Cambridge University Press, 1973).



\end{itemize}
See also

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


\item Groupprops, \emph{\href{https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_quaternion_group}{Linear representation theory of the quaternion group}}


\item GroupNames, \emph{\href{https://people.maths.bris.ac.uk/~matyd/GroupNames/quaternion.html}{Quaternion groups $Q_{2^n}$}}



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

[[!redirects quaternionic group]] [[!redirects quaternionic groups]]

[[!redirects generalized quaternionic group]] [[!redirects generalized quaternionic groups]]

[[!redirects generalized quaternion group]] [[!redirects generalized quaternion groups]]

[[!redirects Q8]]



\end{document}