| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Standard +0.3 This is a straightforward FP3 group theory question testing basic definitions and table reading. Part (a) requires identifying commutativity from a given Cayley table (trivial inspection), counting order-2 subgroups (direct observation), and listing an order-3 subgroup (simple identification). Part (b) asks for orders of elements in a cyclic group of order 6, which follows immediately from standard results about cyclic groups. While group theory is a Further Maths topic, these are foundational exercises requiring only recall and basic pattern recognition, making it easier than an average A-level question overall. |
| 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{center}
\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}
\end{center}
\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 2008 Q1 [6]}}