octic group
OcticGroup
2004-11-05 17:08:38
2004-11-05 17:17:55
Example
dihedral group
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The \emph{octic group} also known as the $4^{th}$ dihedral group, is a non-Abelian group with eight elements. It is traditionally denoted by $D_4$. This group is defined by the presentation
$$< s,t \mid s^4 = t^2 = e, st = ts^{-1}>$$
or, equivalently, defined by the multiplication table
\begin{center}
\begin{tabular}{r|rrrrrrrr}
$\cdot$ & $e$ & $s$ & $s^2$ & $s^3$ & $t$ & $ts$ & $ts^2$ & $ts^3$\\
\hline
$e$ & $e$ & $s$ & $s^2$ & $s^3$ & $t$ & $ts$ & $ts^2$ & $ts^3$\\
$s$ & $s$ & $s^2$ & $s^3$ & $e$ & $ts^3$&$t$ & $ts$ & $ts^2$\\
$s^2$ & $s^2$&$s^3$ & $e$ & $s$ & $ts^2$&$ts^3$&$t$ & $ts$\\
$s^3$ & $s^3$&$e$ & $s$ & $s^2$ & $ts$ &$ts^2$&$ts^3$& $t$\\
$t$ & $t$ &$ts$ & $ts^2$& $ts^3$& $e$ & $s$ & $s^2$ & $s^3$\\
$ts$ & $ts$ &$ts^2$& $ts^3$& $t$ & $s^3$& $e$ & $s$ & $s^2$\\
$ts^2$ & $ts^2$&$ts^3$&$t$ & $ts$ & $s^2$& $s^3$&$e$ & $s$\\
$ts^3$ & $ts^3$&$t$ & $ts$ & $ts^2$& $s$ & $s^2$&$s^3$ & $e$\\
\end{tabular}
\end{center}
\noindent
where we have put each product $xy$ into row $x$ and column $y$. The lattice of the subgroups is given below:
\begin{center}
\begin{figure}
\centerline{
\xymatrix{
& & D_4 \\
& \{ e, s^2, t, ts^2 \} \edge{ur}\edge{dr} & < s > \edge{u}\edge{d} & \{ e, s^2, st, ts\} \edge{ul}\edge{dl} \\
<ts^2> \edge{ur}\edge{drr} & <t>\edge{u}\edge{dr} & <s^2>\edge{ul}\edge{u}\edge{ur}\edge{d} & <st> \edge{u}\edge{dl} & <ts> \edge{ul}\edge{dll}\\
& & <1>\edge{ull}\edge{ul}\edge{u}\edge{ur}\edge{urr}\\
}
}
\end{figure}
\end{center}
where $<a_1,\ldots,a_n>$ denotes the subgroup generated by $a_1,\ldots ,a_n$ and $\{b_1,\ldots ,b_n\}$ denotes the subgroup. Of those subgroups, the following are its proper normal subgroup: $\{e,s^2,t,ts^2\}$, $<s>$, $\{e,s^2,st,ts\}$, and $<s^2>$. In addition the center and commutator subgroup of the octic group is $<s^2>$. It can also be shown that the automorphism of the octic group \textit{($\operatorname{Aut}(D_4)$)} is isomorphic to itself\textit{($D_4$)}.\cite{PJ} An additional property is that the subgroup of the general linear group of dimension 2 over the real numbers generated by:
$$\left[ \begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right],\left[ \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right]$$
is isomorphic to the octic group.
\begin{thebibliography}{1}
\bibitem[PJ]{PJ} Pedersen, John: Groups of small order. \PMlinkexternal{http://www.math.usf.edu/~eclark/algctlg/small_groups.html}{http://www.math.usf.edu/~eclark/algctlg/small_groups.html}
\end{thebibliography}