| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Commutativity and group classification |
| Difficulty | Standard +0.8 This question requires understanding of group theory concepts (cyclic groups, subgroups, element orders, isomorphism) and applying Lagrange's theorem. While the calculations are straightforward once you know the theory, it demands conceptual knowledge beyond standard A-level and requires systematic consideration of small group structures—typical of Further Maths pure topics but not requiring extended proof or novel insight. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03f Subgroups: definition and tests for proper subgroups8.03g Cyclic groups: meaning of the term8.03l Isomorphism: determine using informal methods |
1 In this question $G$ is a group of order $n$, where $3 \leqslant n < 8$.\\
(i) In each case, write down the smallest possible value of $n$ :
\begin{enumerate}[label=(\alph*)]
\item if $G$ is cyclic,
\item if $G$ has a proper subgroup of order 3,
\item if $G$ has at least two elements of order 2 .\\
(ii) Another group has the same order as $G$, but is not isomorphic to $G$. Write down the possible value(s) of $n$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2009 Q1 [5]}}