Challenging +1.8 This is a Further Maths group theory question requiring verification of group axioms with a non-standard operation on complex numbers. Parts (a)-(c) involve systematic but algebraically intensive verification of identity, inverse, and associativity. Part (d) requires identifying why closure fails, and part (e) demands finding a specific finite subgroup. The algebraic manipulation is substantial and the conceptual demand (especially finding a order-3 subgroup) exceeds typical A-level questions, though it follows a standard group verification template.
9 The set \(C\) consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of \(C\) by \(\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }\) where \(\mathrm { a } , \mathrm { b } \in \mathrm { C }\).
Determine the identity element of \(C\) under ⊕.
For each element \(x\) in \(C\) show that it has an inverse element in \(C\).
Show that \(\oplus\) is associative on \(C\).
Explain why \(( C , \oplus )\) is not a group.
Find a subset, \(D\), of \(C\) such that \(( D , \oplus )\) is a group of order 3 .
\section*{END OF QUESTION PAPER}
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9 The set $C$ consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of $C$ by $\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }$ where $\mathrm { a } , \mathrm { b } \in \mathrm { C }$.
\begin{enumerate}[label=(\alph*)]
\item Determine the identity element of $C$ under ⊕.
\item For each element $x$ in $C$ show that it has an inverse element in $C$.
\item Show that $\oplus$ is associative on $C$.
\item Explain why $( C , \oplus )$ is not a group.
\item Find a subset, $D$, of $C$ such that $( D , \oplus )$ is a group of order 3 .
\section*{END OF QUESTION PAPER}
}{www.ocr.org.uk}) after the live examination series.\\
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.\\
For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.\\
OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2023 Q9 [11]}}