5 The set $S$ consists of all $2 \times 2$ matrices having determinant 1 or - 1 . For instance, the matrices $\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$ and $\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)$ are elements of $S$. It is given that $\times _ { \mathbf { M } }$ is the operation of matrix multiplication.
\begin{enumerate}[label=(\alph*)]
\item State the identity element of $S$ under $\times _ { \mathbf { M } }$.

The group $G$ is generated by $\mathbf { P }$, under $\times _ { \mathbf { M } }$.
\item Determine the order of $G$.

The group $H$ is generated by $\mathbf { Q }$ and $\mathbf { R }$, also under $\times _ { \mathbf { M } }$.
\item \begin{enumerate}[label=(\roman*)]
\item By finding each element of $H$, determine the order of $H$.
\item List all the proper subgroups of $H$.
\end{enumerate}\item State whether each of the following statements is true or false. Give a reason for each of your answers.

\begin{itemize}
  \item $G$ is abelian
  \item $G$ is cyclic
  \item $H$ is abelian
  \item $H$ is cyclic
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR Further Additional Pure AS 2024 Q5 [14]}}