| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Isomorphism between groups |
| Difficulty | Challenging +1.2 This is a structured group theory question requiring students to construct operation tables for cyclic and Klein-4 groups, identify the identity element from a given table, and demonstrate isomorphism by finding element correspondences. While it involves abstract algebra (a Further Maths topic), the question is highly scaffolded with explicit group definitions and clear steps. The main challenge is systematic verification of group properties and pattern matching between tables, which is methodical rather than requiring deep insight. This is moderately above average difficulty due to the abstract nature of groups, but remains accessible to well-prepared FM students. |
| 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 |
| \(a\) | \(b\) | \(c\) | \(d\) | |
| \(a\) | \(b\) | \(a\) | \(d\) | \(c\) |
| \(b\) | \(a\) | \(b\) | \(c\) | \(d\) |
| \(c\) | \(d\) | \(c\) | \(a\) | \(b\) |
| \(d\) | \(c\) | \(d\) | \(b\) | \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct table for \(H\) | B2 | SR: allow B1 for 1 or 2 errors |
| Correct table for \(K\) | B2 | SR: allow B1 for 1 or 2 errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Identity \(= b\) | B1 | For correct identity |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(G\) is isomorphic to \(H\) | B1 | For \(H\) identified as isomorphic to \(G\) (may be implied by table) |
| \(a \leftrightarrow r^2\) at least once | B1 | |
| \(c,d \leftrightarrow r, r^3\) either way | B1 | |
| \(c,d \leftrightarrow r,r^3\) both ways and \(b\) corresponds to \(e\) explicit. Award fourth B1 only for completely correct answer. If none of last 3 marks gained, then SC1 for order of all elements of \(G\) and \(H\) | B1 |
# Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct table for $H$ | B2 | SR: allow B1 for 1 or 2 errors |
| Correct table for $K$ | B2 | SR: allow B1 for 1 or 2 errors |
---
# Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identity $= b$ | B1 | For correct identity |
---
# Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $G$ is isomorphic to $H$ | B1 | For $H$ identified as isomorphic to $G$ (may be implied by table) |
| $a \leftrightarrow r^2$ at least once | B1 | |
| $c,d \leftrightarrow r, r^3$ either way | B1 | |
| $c,d \leftrightarrow r,r^3$ both ways **and** $b$ corresponds to $e$ explicit. Award fourth B1 only for completely correct answer. If none of last 3 marks gained, then SC1 for order of all elements of $G$ and $H$ | B1 | |
---4 The elements $a , b , c , d$ are combined according to the operation table below, to form a group $G$ of order 4.
\begin{center}
\begin{tabular}{ l | l l l l }
& $a$ & $b$ & $c$ & $d$ \\
\hline
$a$ & $b$ & $a$ & $d$ & $c$ \\
$b$ & $a$ & $b$ & $c$ & $d$ \\
$c$ & $d$ & $c$ & $a$ & $b$ \\
$d$ & $c$ & $d$ & $b$ & $a$ \\
\end{tabular}
\end{center}
Group $G$ is isomorphic either to the multiplicative group $H = \left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}$ or to the multiplicative group $K = \{ e , p , q , p q \}$. It is given that $r ^ { 4 } = e$ in group $H$ and that $p ^ { 2 } = q ^ { 2 } = e$ in group $K$, where $e$ denotes the identity in each group.\\
(i) Write down the operation tables for $H$ and $K$.\\
(ii) State the identity element of $G$.\\
(iii) Demonstrate the isomorphism between $G$ and either $H$ or $K$ by listing how the elements of $G$ correspond to the elements of the other group. If the correspondence can be shown in more than one way, list the alternative correspondence(s).
\hfill \mbox{\textit{OCR FP3 2012 Q4 [9]}}