4
\begin{enumerate}[label=(\alph*)]
\item The elements of the set $P = \{ 1,3,9,11 \}$ are combined under the binary operation, *, defined as multiplication modulo 16.
\begin{enumerate}[label=(\roman*)]
\item Demonstrate associativity for the elements $3,9,11$ in that order.

Assuming associativity holds in general, show that $P$ forms a group under the binary operation *.
\item Write down the order of each element.
\item Write down all subgroups of $P$.
\item Show that the group in part (i) is cyclic.
\end{enumerate}\item Now consider a group of order 4 containing the identity element $e$ and the two distinct elements, $a$ and $b$, where $a ^ { 2 } = b ^ { 2 } = e$. Construct the composition table. Show that the group is non-cyclic.
\item Now consider the four matrices $\mathbf { I } , \mathbf { X } , \mathbf { Y }$ and $\mathbf { Z }$ where

$$\mathbf { I } = \left( \begin{array} { l l } 
1 & 0 \\
0 & 1
\end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 
1 & 0 \\
0 & - 1
\end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } 
- 1 & 0 \\
0 & 1
\end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } 
- 1 & 0 \\
0 & - 1
\end{array} \right) .$$

The group G consists of the set $\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}$ with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
\item The distinct elements $\{ p , q , r , s \}$ are combined under the binary operation ${ } ^ { \circ }$. You are given that $p ^ { \circ } q = r$ and $q ^ { \circ } p = s$.

By reference to the group axioms, prove that $\{ p , q , r , s \}$ is not a group under ${ } ^ { \circ }$.

Option 5: Markov chains

\section*{This question requires the use of a calculator with the ability to handle matrices.}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI FP3 2016 Q4 [24]}}