8 The group $G$ is cyclic and of order 12.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item State the possible orders of all the proper subgroups of $G$. You must justify your answers.
\item List all the elements of each of these subgroups.
\item Explain why $G$ must be abelian.

The group $\mathbb { Z } _ { k }$ is the cyclic group of order $k$, consisting of the elements $\{ 0,1,2 , \ldots , k - 1 \}$ under the operation $+ _ { k }$ of addition modulo $k$.

The coordinate group $\mathrm { C } _ { \mathrm { mn } }$ is the group which consists of elements of the form $( x , y )$, where $\mathrm { x } \in \mathbb { Z } _ { \mathrm { m } }$ and $\mathrm { y } \in \mathbb { Z } _ { \mathrm { n } }$, under the operation $\oplus$ given by $\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) \oplus \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right) = \left( \mathrm { x } _ { 1 } + { } _ { \mathrm { m } } \mathrm { x } _ { 2 } , \mathrm { y } _ { 1 } + { } _ { \mathrm { n } } \mathrm { y } _ { 2 } \right)$. For example, for $m = 5$ and $n = 2 , ( 3,0 ) \oplus ( 4,1 ) = ( 2,1 )$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item List all the elements of $\mathrm { J } = \mathrm { C } _ { 34 }$.
\item Show that $G$ and $J$ are isomorphic.

There is a second coordinate group of order 12; that is, $\mathrm { K } = \mathrm { C } _ { \mathrm { mn } }$, where $1 < \mathrm { m } < \mathrm { n } < 12$ but neither $m$ nor $n$ is equal to 3 or 4 .
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item State the values of $m$ and $n$ which give $K$.
\item Hence list all of the elements of $K$.
\item Explain why $K$ must be abelian.
\end{enumerate}\item Show that $G$ and $K$ are not isomorphic.

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

\hfill \mbox{\textit{OCR Further Additional Pure 2024 Q8 [15]}}