| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Matrix groups |
| Difficulty | Challenging +1.2 This is a structured Further Maths group theory question with clear scaffolding through five parts. While it involves complex cube roots of unity and matrix multiplication (FP3 topics), each part is relatively routine: (i) is immediate inspection, (ii) requires checking a few cases, (iii-iv) are direct matrix calculations, and (v) is a standard isomorphism check comparing group structures. The question tests understanding of abstract algebra concepts but doesn't require deep insight or novel problem-solving—it's a typical exam question testing whether students can apply learned techniques to a moderately sophisticated context. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar8.03c Group definition: recall and use, show structure is/isn't a group8.03f Subgroups: definition and tests for proper subgroups8.03l Isomorphism: determine using informal methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{A, D\}\) OR \(\{A, E\}\) OR \(\{A, F\}\) | B1 1 | For stating any one subgroup |
| Answer | Marks | Guidance |
|---|---|---|
| \(A\) is the identity | B1 | For identifying \(A\) as the identity |
| 5 is not a factor of 6 | B1 2 | For reference to factors of 6 |
| OR elements can be only of order 1, 2, 3, 6 |
| Answer | Marks | Guidance |
|---|---|---|
| \(BE = \begin{pmatrix}0&1\\1&0\end{pmatrix} = D\), \(EB = \begin{pmatrix}0&\omega\\\omega^2&0\end{pmatrix} = F\) | M1 | For finding \(BE\) and \(EB\) AND using \(\omega^3 = 1\) |
| A1 | For correct \(BE\) (\(D\) or matrix) | |
| A1 | For correct \(EB\) (\(F\) or matrix) | |
| \(D\) or \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\), \(F\) or \(\begin{pmatrix}0&\omega\\\omega^2&0\end{pmatrix} \in M\) | A1 4 | For justifying closure |
| \(\Rightarrow\) closure property satisfied |
| Answer | Marks | Guidance |
|---|---|---|
| \(B^{-1} = \frac{1}{\omega}\begin{pmatrix}\omega&0\\0&\omega\end{pmatrix} = C\) | M1 | For correct method of finding either inverse |
| A1 | For correct \(B^{-1} = C\). Allow \(\begin{pmatrix}\omega^2&0\\0&\omega\end{pmatrix}\) | |
| \(E^{-1} = \frac{1}{-1}\begin{pmatrix}0&-\omega^2\\-\omega&0\end{pmatrix} = E\) | A1 3 | For correct \(E^{-1} = E\). Allow \(\begin{pmatrix}0&\omega^2\\\omega&0\end{pmatrix}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(M\) is not commutative e.g. from \(BE \neq EB\) in part (iii) | B1 | For justification of \(M\) being not commutative |
| \(N\) is commutative (as \(\times \bmod 9\) is commutative) | B1 | For statement that \(N\) is commutative |
| \(\Rightarrow M\) and \(N\) not isomorphic | B1 # 3 | For correct conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Elements of \(M\) have orders 1, 3, 3, 2, 2, 2 | B1* | For all orders of one group correct |
| Elements of \(N\) have orders 1, 6, 3, 2, 3, 6 | B1 (*dep) | For sufficient orders of the other group correct |
| Different orders OR self-inverse elements | B1 # | For correct conclusion |
| \(\Rightarrow M\) and \(N\) not isomorphic | SR Award up to B1 B1 B1 if the self-inverse elements are sufficiently well identified for the groups to be non-isomorphic |
| Answer | Marks | Guidance |
|---|---|---|
| \(M\) has no generator since there is no element of order 6 | B1 | For all orders of \(M\) shown correctly |
| \(N\) has 2 OR 5 as a generator | B1 | For stating that \(N\) has generator 2 OR 5 |
| \(\Rightarrow M\) and \(N\) not isomorphic | B1 # | For correct conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| B1* | For stating correctly all 6 squared elements of one group | |
| B1 (*dep) | For stating correctly sufficient squared elements of the other group | |
| \(\Rightarrow M\) and \(N\) not isomorphic | B1 # | For correct conclusion |
| # In all Methods, the last B1 is dependent on at least one preceding B1 |
## (i)
$\{A, D\}$ OR $\{A, E\}$ OR $\{A, F\}$ | B1 1 | For stating any one subgroup
## (ii)
$A$ is the identity | B1 | For identifying $A$ as the identity
5 is not a factor of 6 | B1 2 | For reference to factors of 6
OR elements can be only of order 1, 2, 3, 6 | |
## (iii)
$BE = \begin{pmatrix}0&1\\1&0\end{pmatrix} = D$, $EB = \begin{pmatrix}0&\omega\\\omega^2&0\end{pmatrix} = F$ | M1 | For finding $BE$ and $EB$ AND using $\omega^3 = 1$
| A1 | For correct $BE$ ($D$ or matrix)
| A1 | For correct $EB$ ($F$ or matrix)
$D$ or $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $F$ or $\begin{pmatrix}0&\omega\\\omega^2&0\end{pmatrix} \in M$ | A1 4 | For justifying closure
$\Rightarrow$ closure property satisfied | |
## (iv)
$B^{-1} = \frac{1}{\omega}\begin{pmatrix}\omega&0\\0&\omega\end{pmatrix} = C$ | M1 | For correct method of finding either inverse
| A1 | For correct $B^{-1} = C$. Allow $\begin{pmatrix}\omega^2&0\\0&\omega\end{pmatrix}$
$E^{-1} = \frac{1}{-1}\begin{pmatrix}0&-\omega^2\\-\omega&0\end{pmatrix} = E$ | A1 3 | For correct $E^{-1} = E$. Allow $\begin{pmatrix}0&\omega^2\\\omega&0\end{pmatrix}$
## (v) METHOD 1
$M$ is not commutative e.g. from $BE \neq EB$ in part (iii) | B1 | For justification of $M$ being not commutative
$N$ is commutative (as $\times \bmod 9$ is commutative) | B1 | For statement that $N$ is commutative
$\Rightarrow M$ and $N$ not isomorphic | B1 # 3 | For correct conclusion
## (v) METHOD 2
Elements of $M$ have orders 1, 3, 3, 2, 2, 2 | B1* | For all orders of one group correct
Elements of $N$ have orders 1, 6, 3, 2, 3, 6 | B1 (*dep) | For sufficient orders of the other group correct
Different orders OR self-inverse elements | B1 # | For correct conclusion
$\Rightarrow M$ and $N$ not isomorphic | | SR Award up to B1 B1 B1 if the self-inverse elements are sufficiently well identified for the groups to be non-isomorphic
## (v) METHOD 3
$M$ has no generator since there is no element of order 6 | B1 | For all orders of $M$ shown correctly
$N$ has 2 OR 5 as a generator | B1 | For stating that $N$ has generator 2 OR 5
$\Rightarrow M$ and $N$ not isomorphic | B1 # | For correct conclusion
## (v) METHOD 4
| B1* | For stating correctly all 6 squared elements of one group
| B1 (*dep) | For stating correctly sufficient squared elements of the other group
$\Rightarrow M$ and $N$ not isomorphic | B1 # | For correct conclusion
| | # In all Methods, the last B1 is dependent on at least one preceding B1
---A set of matrices $M$ is defined by
$$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$
where $\omega$ and $\omega^2$ are the complex cube roots of 1. It is given that $M$ is a group under matrix multiplication.
\begin{enumerate}[label=(\roman*)]
\item Write down the elements of a subgroup of order 2. [1]
\item Explain why there is no element $X$ of the group, other than $A$, which satisfies the equation $X^2 = A$. [2]
\item By finding $BE$ and $EB$, verify the closure property for the pair of elements $B$ and $E$. [4]
\item Find the inverses of $B$ and $E$. [3]
\item Determine whether the group $M$ is isomorphic to the group $N$ which is defined as the set of numbers $\{1, 2, 4, 8, 7, 5\}$ under multiplication modulo 9. Justify your answer clearly. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2010 Q8 [13]}}