| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Challenging +1.2 This is a Further Maths FP3 group theory question requiring proof of group axioms and identification of subgroups. While it involves abstract algebra (inherently harder than standard A-level), the question is relatively straightforward: the group axioms follow directly from integer properties, and the subgroup checks are standard applications of closure/inverse criteria. The structure is guided and requires no novel insight, making it moderately above average difficulty. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03f Subgroups: definition and tests for proper subgroups |
The set $S$ consists of the numbers $3^n$, where $n \in \mathbb{Z}$. ($\mathbb{Z}$ denotes the set of integers $\{0, \pm 1, \pm 2, \ldots\}$.)
\begin{enumerate}[label=(\roman*)]
\item Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.) [6]
\item Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
\begin{enumerate}[label=(\alph*)]
\item The numbers $3^{2n}$, where $n \in \mathbb{Z}$. [2]
\item The numbers $3^n$, where $n \in \mathbb{Z}$ and $n \geqslant 0$. [2]
\item The numbers $3^{(±n^2)}$, where $n \in \mathbb{Z}$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q9 [12]}}