| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Standard +0.3 This is a straightforward group theory question testing basic definitions and table reading. Part (a) requires simple observation of the Cayley table for commutativity and identifying subgroups by inspection. Part (b) involves routine calculation of element orders in a cyclic group using the formula for powers. All parts are direct application of definitions with no problem-solving or proof required, making it easier than average even for FP3 students. |
| Spec | 8.03e Order of elements: and order of groups8.03f Subgroups: definition and tests for proper subgroups8.03i Properties of groups: structure of finite groups up to order 7 |
| \(e\) | \(a\) | \(b\) | \(p\) | \(q\) | \(r\) | |
| \(e\) | \(e\) | \(a\) | \(b\) | \(p\) | \(q\) | \(r\) |
| \(a\) | \(a\) | \(b\) | \(e\) | \(r\) | \(p\) | \(q\) |
| \(b\) | \(b\) | \(e\) | \(a\) | \(q\) | \(r\) | \(p\) |
| \(p\) | \(p\) | \(q\) | \(r\) | \(e\) | \(a\) | \(b\) |
| \(q\) | \(q\) | \(r\) | \(p\) | \(b\) | \(e\) | \(a\) |
| \(r\) | \(r\) | \(p\) | \(q\) | \(a\) | \(b\) | \(e\) |
\begin{enumerate}[label=(\alph*)]
\item A group $G$ of order 6 has the combination table shown below.
\begin{tabular}{c|cccccc}
& $e$ & $a$ & $b$ & $p$ & $q$ & $r$ \\
\hline
$e$ & $e$ & $a$ & $b$ & $p$ & $q$ & $r$ \\
$a$ & $a$ & $b$ & $e$ & $r$ & $p$ & $q$ \\
$b$ & $b$ & $e$ & $a$ & $q$ & $r$ & $p$ \\
$p$ & $p$ & $q$ & $r$ & $e$ & $a$ & $b$ \\
$q$ & $q$ & $r$ & $p$ & $b$ & $e$ & $a$ \\
$r$ & $r$ & $p$ & $q$ & $a$ & $b$ & $e$ \\
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item State, with a reason, whether or not $G$ is commutative. [1]
\item State the number of subgroups of $G$ which are of order 2. [1]
\item List the elements of the subgroup of $G$ which is of order 3. [1]
\end{enumerate}
\item A multiplicative group $H$ of order 6 has elements $e, c, c^2, c^3, c^4, c^5$, where $e$ is the identity. Write down the order of each of the elements $c^3, c^4$ and $c^5$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q1 [6]}}