| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Complete or analyse Cayley table |
| Difficulty | Standard +0.3 This is a straightforward group theory question requiring basic verification of associativity (table lookup), checking group axioms (closure is given by table, identity and inverses are routine to identify), and listing elements of an isomorphic group (simple pattern recognition). All steps are mechanical applications of definitions with no novel insight required, making it slightly easier than average for Further Maths. |
| Spec | 8.03b Cayley tables: construct for finite sets under binary operation8.03c Group definition: recall and use, show structure is/isn't a group8.03d Latin square property: for group tables8.03l Isomorphism: determine using informal methods |
| \(p\) | \(q\) | \(r\) | \(s\) | \(t\) | |
| \(p\) | \(t\) | \(s\) | \(p\) | \(r\) | \(q\) |
| \(q\) | \(s\) | \(p\) | \(q\) | \(t\) | \(r\) |
| \(r\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(s\) | \(r\) | \(t\) | \(s\) | \(q\) | \(p\) |
| \(t\) | \(q\) | \(r\) | \(t\) | \(p\) | \(s\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((qs)t = tt = s\) | B1 | For obtaining \(s\) |
| B1 2 | For obtaining \(s\) |
| Answer | Marks | Guidance |
|---|---|---|
| Identity \(= r\) | B1 | For stating closure with reason |
| B1 | For stating identity \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(s^{-1} = p\), \(t^{-1} = q\) | M1 | For checking for inverses |
| A1 4 | For stating inverses OR For giving sufficient explanation to justify each element has an inverse eg \(r\) occurs once in each row and/or column |
| Answer | Marks | Guidance |
|---|---|---|
| Identity \(= r\) | B1 | For stating identity \(r\) |
| M1 | For attempting to establish a generator \(\neq r\) | |
| eg \(p^2 = t\), \(p^3 = q\), \(p^4 = s\) | A1 | For showing powers of \(p\) (OR \(q\), \(s\) or \(t\)) are different elements of the set |
| \(\Rightarrow p^5 = r\), so \(p\) is a generator | A1 | For concluding \(p^5\) (OR \(q^5\), \(s^5\) or \(t^5\)) \(= r\) |
| (iii) \(e, d, d^2, d^3, d^4\) | B2 2 | For stating all elements AEF eg \(d^{-1}, d^{-2}, dd\) |
**(i)** $q(st) = qp = s$
$(qs)t = tt = s$ | B1 | For obtaining $s$
| B1 2 | For obtaining $s$
**(ii) METHOD 1**
Closed: see table
Identity $= r$ | B1 | For stating closure with reason
| B1 | For stating identity $r$
Inverses: $p^{-1} = s$, $q^{-1} = t$, $(r^{-1} = r)$,
$s^{-1} = p$, $t^{-1} = q$ | M1 | For checking for inverses
| A1 4 | For stating inverses OR For giving sufficient explanation to justify each element has an inverse eg $r$ occurs once in each row and/or column
**METHOD 2**
Identity $= r$ | B1 | For stating identity $r$
| M1 | For attempting to establish a generator $\neq r$
eg $p^2 = t$, $p^3 = q$, $p^4 = s$ | A1 | For showing powers of $p$ (OR $q$, $s$ or $t$) are different elements of the set
$\Rightarrow p^5 = r$, so $p$ is a generator | A1 | For concluding $p^5$ (OR $q^5$, $s^5$ or $t^5$) $= r$
**(iii)** $e, d, d^2, d^3, d^4$ | B2 2 | For stating all elements AEF eg $d^{-1}, d^{-2}, dd$4 Elements of the set $\{ p , q , r , s , t \}$ are combined according to the operation table shown below.
\begin{center}
\begin{tabular}{ c | c c c c c }
& $p$ & $q$ & $r$ & $s$ & $t$ \\
\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}
\end{center}
(i) Verify that $q ( s t ) = ( q s ) t$.\\
(ii) 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 $G$.\\
(iii) 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 $d$.
\hfill \mbox{\textit{OCR FP3 2007 Q4 [8]}}