| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Standard +0.8 This FP3 group theory question requires understanding of cyclic groups, subgroup orders, and Lagrange's theorem. Part (a) involves identifying subgroup elements using generator powers (moderate), while part (b) requires conceptual understanding of proper subgroups and cyclic group structure. The final part tests knowledge that subgroup orders must divide group order. These are standard group theory results but require abstract reasoning beyond typical A-level, placing it moderately above average difficulty. |
| Spec | 8.03f Subgroups: definition and tests for proper subgroups8.03k Lagrange's theorem: order of subgroup divides order of group |
\begin{enumerate}[label=(\alph*)]
\item A cyclic multiplicative group $G$ has order 12. The identity element of $G$ is $e$ and another element is $r$, with order 12.
\begin{enumerate}[label=(\roman*)]
\item Write down, in terms of $e$ and $r$, the elements of the subgroup of $G$ which is of order 4. [2]
\item Explain briefly why there is no proper subgroup of $G$ in which two of the elements are $e$ and $r$. [1]
\end{enumerate}
\item A group $H$ has order $mnp$, where $m, n$ and $p$ are prime. State the possible orders of proper subgroups of $H$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q1 [5]}}