| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.8 This is a standard FP3 group theory question requiring table reading, verification of group axioms (closure, associativity, identity, inverses), and understanding of isomorphism to cyclic groups. While systematic, it requires careful checking and understanding of abstract algebra concepts beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 8.03b Cayley tables: construct for finite sets under binary operation8.03c Group definition: recall and use, show structure is/isn't a group8.03l Isomorphism: determine using informal methods |
| \(p\) | \(q\) | \(r\) | \(s\) | \(t\) | |
| \(p\) | \(t\) | \(s\) | \(p\) | \(r\) | \(q\) |
| \(q\) | \(s\) | \(p\) | \(q\) | \(t\) | \(r\) |
| \(r\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(s\) | \(r\) | \(t\) | \(s\) | \(q\) | \(p\) |
| \(t\) | \(q\) | \(r\) | \(t\) | \(p\) | \(s\) |
Elements of the set $\{p, q, r, s, t\}$ are combined according to the operation table shown below.
\begin{tabular}{c|ccccc}
& $p$ & $q$ & $r$ & $s$ & $t$ \\
\hline
$p$ & $t$ & $s$ & $p$ & $r$ & $q$ \\
$q$ & $s$ & $p$ & $q$ & $t$ & $r$ \\
$r$ & $p$ & $q$ & $r$ & $s$ & $t$ \\
$s$ & $r$ & $t$ & $s$ & $q$ & $p$ \\
$t$ & $q$ & $r$ & $t$ & $p$ & $s$ \\
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Verify that $q(st) = (qs)t$. [2]
\item Assuming that the associative property holds for all elements, prove that the set $\{p, q, r, s, t\}$, with the operation table shown, forms a group $G$. [4]
\item A multiplicative group $H$ is isomorphic to the group $G$. The identity element of $H$ is $e$ and another element is $d$. Write down the elements of $H$ in terms of $e$ and $d$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q4 [8]}}