| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Challenging +1.3 This is a straightforward group theory question requiring identification of subgroups from a given Cayley table and recognition of the dihedral group D₃. Part (a) involves checking closure for subsets (standard technique), and part (b) requires knowing that non-abelian groups of order 6 are isomorphic to D₃ or S₃. While this is Further Maths content (inherently harder), it's a routine application of definitions without requiring proof or novel insight. |
| Spec | 8.03f Subgroups: definition and tests for proper subgroups8.03l Isomorphism: determine using informal methods |
| G | \(e\) | \(a\) | \(a ^ { 2 }\) | \(b\) | \(a b\) | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) |
| \(e\) | \(e\) | \(a\) | \(a ^ { 2 }\) | \(b\) | \(a b\) | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) |
| \(a\) | \(a\) | \(a ^ { 2 }\) | \(e\) | \(a b\) | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(b\) |
| \(a ^ { 2 }\) | \(a ^ { 2 }\) | \(e\) | \(a\) | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(b\) | \(a b\) |
| \(b\) | b | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(a b\) | \(e\) | \(a ^ { 2 }\) | \(a\) |
| \(a b\) | \(a b\) | b | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(a\) | \(e\) | \(a ^ { 2 }\) |
| \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(\mathrm { a } ^ { 2 } \mathrm {~b}\) | \(a b\) | b | \(a ^ { 2 }\) | \(a\) | \(e\) |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | {e, a, a2} |
| {e, b} {e, ab} {e, a2b} | B1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Ignore inclusion of {e}, G |
| (b) | G is isomorphic to the symmetric group of three |
| Answer | Marks | Guidance |
|---|---|---|
| permutations of 3 symbols) | B1 | |
| [1] | 2.5 | Accept “the non-cyclic group of order 6” |
Question 2:
2 | (a) | {e, a, a2}
{e, b} {e, ab} {e, a2b} | B1
B1
B1
B1
[4] | 1.1
1.1
1.1
1.1 | Ignore inclusion of {e}, G
(b) | G is isomorphic to the symmetric group of three
elements
i.e. D (the symmetries of the ) or S (the
3 3
permutations of 3 symbols) | B1
[1] | 2.5 | Accept “the non-cyclic group of order 6”2 The following Cayley table is for $G$, a group of order 6. The identity element is $e$ and the group is generated by the elements $a$ and $b$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
G & $e$ & $a$ & $a ^ { 2 }$ & $b$ & $a b$ & $\mathrm { a } ^ { 2 } \mathrm {~b}$ \\
\hline
$e$ & $e$ & $a$ & $a ^ { 2 }$ & $b$ & $a b$ & $\mathrm { a } ^ { 2 } \mathrm {~b}$ \\
\hline
$a$ & $a$ & $a ^ { 2 }$ & $e$ & $a b$ & $\mathrm { a } ^ { 2 } \mathrm {~b}$ & $b$ \\
\hline
$a ^ { 2 }$ & $a ^ { 2 }$ & $e$ & $a$ & $\mathrm { a } ^ { 2 } \mathrm {~b}$ & $b$ & $a b$ \\
\hline
$b$ & b & $\mathrm { a } ^ { 2 } \mathrm {~b}$ & $a b$ & $e$ & $a ^ { 2 }$ & $a$ \\
\hline
$a b$ & $a b$ & b & $\mathrm { a } ^ { 2 } \mathrm {~b}$ & $a$ & $e$ & $a ^ { 2 }$ \\
\hline
$\mathrm { a } ^ { 2 } \mathrm {~b}$ & $\mathrm { a } ^ { 2 } \mathrm {~b}$ & $a b$ & b & $a ^ { 2 }$ & $a$ & $e$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item List all the proper subgroups of $G$.
\item State another group of order 6 to which $G$ is isomorphic.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2021 Q2 [5]}}