AQA Further Paper 3 Discrete 2020 June — Question 6 8 marks

Exam BoardAQA
ModuleFurther Paper 3 Discrete (Further Paper 3 Discrete)
Year2020
SessionJune
Marks8
PaperDownload PDF ↗
TopicGroups
TypeComplete or analyse Cayley table
DifficultyChallenging +1.2 This is a structured Further Maths group theory question requiring systematic application of given relations to complete a Cayley table. While it involves abstract algebra (inherently harder), the question provides all necessary relations and most table entries, making it primarily methodical calculation rather than requiring deep insight. The isomorphism identification (D₃/S₃) is standard recognition for students at this level.
Spec8.03a Binary operations: and their properties on given sets8.03b Cayley tables: construct for finite sets under binary operation
6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e \\ & q \boldsymbol { \Delta } q = e \\ & r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6
    1. State the order of \(G\). 6
      1. (ii) Prove that the inverse of \(q r\) is \(q r\).
        6
    2. Complete the Cayley table for elements of \(G\). 6
    3. Complete the Cayley table for elements of \(G\).
      A\(e\)\(r\)\(r ^ { 2 }\)\(q\)\(q r\)\(q r ^ { 2 }\)
      \(e\)\(e\)\(r\)\(r ^ { 2 }\)\(q\)\(q r\)\(q r ^ { 2 }\)
      \(r\)\(r\)\(r ^ { 2 }\)\(e\)
      \(r ^ { 2 }\)\(r ^ { 2 }\)\(e\)\(r\)
      \(q\)\(q\)\(q r\)\(q r ^ { 2 }\)\(e\)
      \(q r\)\(q r\)\(q r ^ { 2 }\)\(q\)\(r ^ { 2 }\)
      \(q r ^ { 2 }\)\(q r ^ { 2 }\)\(q\)\(q r\)\(r\)\(r ^ { 2 }\)\(e\)
      6
    4. State the name of a group which is isomorphic to \(G\).
6 The group $( G , \boldsymbol { A } )$ has the elements $e , r , r ^ { 2 } , q , q r$ and $q r ^ { 2 }$, where $r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }$ and $e$ is the identity element of $G$.

The elements $q$ and $r$ have the following properties:

$$\begin{aligned}
& r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e \\
& q \boldsymbol { \Delta } q = e \\
& r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r
\end{aligned}$$

6
\begin{enumerate}[label=(\alph*)]
\item (i) State the order of $G$.

6 (a) (ii) Prove that the inverse of $q r$ is $q r$.\\

6
\item Complete the Cayley table for elements of $G$.

6 (b) Complete the Cayley table for elements of $G$.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
A & $e$ & $r$ & $r ^ { 2 }$ & $q$ & $q r$ & $q r ^ { 2 }$ \\
\hline
$e$ & $e$ & $r$ & $r ^ { 2 }$ & $q$ & $q r$ & $q r ^ { 2 }$ \\
\hline
$r$ & $r$ & $r ^ { 2 }$ & $e$ &  &  &  \\
\hline
$r ^ { 2 }$ & $r ^ { 2 }$ & $e$ & $r$ &  &  &  \\
\hline
$q$ & $q$ & $q r$ & $q r ^ { 2 }$ & $e$ &  &  \\
\hline
$q r$ & $q r$ & $q r ^ { 2 }$ & $q$ & $r ^ { 2 }$ &  &  \\
\hline
$q r ^ { 2 }$ & $q r ^ { 2 }$ & $q$ & $q r$ & $r$ & $r ^ { 2 }$ & $e$ \\
\hline
\end{tabular}
\end{center}

6
\item State the name of a group which is isomorphic to $G$.
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2020 Q6 [8]}}
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