7 The diagram below shows an equilateral triangle $A B C$. The three lines of reflection symmetry of $A B C$ (the lines $a , b$ and $c$ ) are shown as broken lines. The point of intersection of these three lines, $O$, is the centre of rotational symmetry of the triangle.\\
\includegraphics[max width=\textwidth, alt={}, center]{06496165-0b83-4050-ae26-fa5a0614bd46-4_533_538_884_246}

The group $D _ { 3 }$ is defined as the set of symmetries of $A B C$ under the composition of the following transformations.\\
$i$ : the identity transformation\\
$a$ : reflection in line $a$\\
$b$ : reflection in line $b$\\
$c$ : reflection in line $c$\\
$p$ : an anticlockwise rotation about $O$ through $120 ^ { \circ }$\\
$q$ : a clockwise rotation about $O$ through $120 ^ { \circ }$\\
Note that the lines $a , b$ and $c$ are unaffected by the transformations and remain fixed.
\begin{enumerate}[label=(\alph*)]
\item On the diagrams provided in the Printed Answer Booklet, show each of the six elements of $D _ { 3 }$ obtained when the above transformations are applied to triangle $A B C$.
\item Complete the Cayley table given in the Printed Answer Booklet.
\item List all the proper subgroups of $D _ { 3 }$.
\item State, with justification, whether $D _ { 3 }$ is
\begin{enumerate}[label=(\roman*)]
\item cyclic,
\item abelian.
\end{enumerate}\item The group $H$, also of order 6, is the set of rotational symmetries of the regular hexagon. Describe two structural differences between $D _ { 3 }$ and $H$.

\section*{END OF QUESTION PAPER}
\end{enumerate}

\hfill \mbox{\textit{OCR Further Additional Pure AS 2022 Q7 [13]}}