| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2018 |
| Session | March |
| Marks | 11 |
| Topic | Groups |
| Type | Groups of symmetries |
| Difficulty | Standard +0.8 This is a standard introduction to the dihedral group D_3 covering basic group theory concepts (order, subgroups, commutativity). While it requires understanding symmetries and composition, all parts follow textbook patterns for this topic. The multi-part structure and need to visualize transformations elevate it slightly above average, but it requires no novel insights—just systematic application of definitions. |
| Spec | 8.03a Binary operations: and their properties on given sets8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03f Subgroups: definition and tests for proper subgroups8.03j Properties of groups: higher finite order or infinite order8.03k Lagrange's theorem: order of subgroup divides order of group |
| Answer | Marks |
|---|---|
| E1 | 2.4 |
| [1] |
| Answer | Marks |
|---|---|
| B1 | 2.2a |
| B1 | 2.2a |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | 1.1 | Might be called \(k^{-1}\) |
| M1, A1 | 2.5, 1.1 | Method for either \(jk\) or \(kj\); (Note that the intermediate triangles may not be seen.) (Note that \(k^2 j\) may be seen instead of \(jk\) and \(jk^2\) instead of \(kj\) these have different intermediate triangles) |
| A1 | 1.1 | Give one A1 if \(jk\) and \(kj\) (reversed) are correct |
| [4] |
| Answer | Marks |
|---|---|
| E1 | 2.4 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | 1.1 | Any one subgroup of order 2 |
| B1 | 1.1 | All 3 and no extras |
| B1 | 1.1 | |
| Condone the appearance of \(\{i\}\) and/or \(G\); Note that instead of \(\{i,jk\}, \{i,k^2j\}\) may be given and instead of \(\{i,kj\}, \{i,jk^2\}\) may be given | ||
| [3] |
## Part (i)
**Answer:** There are $3!$ arrangements of the three vertices (A, B, C) and hence $3! = 6$ symmetries of the triangle
| **E1** | **2.4** | |
| **[1]** | | |
## Part (ii)
**Answer:** $j^2 = i$ so the order of $j$ is 2
$k^3 = i$ so the order of $k$ is 3
| **B1** | **2.2a** | |
| **B1** | **2.2a** | |
| **[2]** | | |
## Part (iii)
**Answer:** $k^2: \triangle \to \triangle$
$B \quad A \quad C \quad C \quad A$
$jk: \triangle \to \triangle \to \triangle$
$B \quad A \quad C \quad A \quad B \quad B \quad A$
$kj: \triangle \to \triangle \to \triangle$
$B \quad C \quad C \quad B \quad A \quad B \quad A$
| **B1** | **1.1** | Might be called $k^{-1}$ |
| **M1, A1** | **2.5, 1.1** | Method for either $jk$ or $kj$; (Note that the intermediate triangles may not be seen.) (Note that $k^2 j$ may be seen instead of $jk$ and $jk^2$ instead of $kj$ these have different intermediate triangles) |
| **A1** | **1.1** | Give one A1 if $jk$ and $kj$ (reversed) are correct |
| **[4]** | | |
## Part (iv)
**Answer:** No …since (e.g.) $jk \neq kj$
| **E1** | **2.4** | |
| **[1]** | | |
## Part (v)
**Answer:** $\{i,j\}, \{i,jk\}, \{i,kj\}$
$\{i, k, k^2\}$
| **B1** | **1.1** | Any one subgroup of order 2 |
| **B1** | **1.1** | All 3 and no extras |
| **B1** | **1.1** | |
| | | Condone the appearance of $\{i\}$ and/or $G$; Note that instead of $\{i,jk\}, \{i,k^2j\}$ may be given and instead of $\{i,kj\}, \{i,jk^2\}$ may be given |
| **[3]** | | |
---4 The group $G$ consists of the symmetries of the equilateral triangle $A B C$ under the operation of composition of transformations (which may be assumed to be associative).
Three elements of $G$ are
\begin{itemize}
\item $\boldsymbol { i }$, the identity
\item $\boldsymbol { j }$, the reflection in the vertical line of symmetry of the triangle
\item $\boldsymbol { k }$, the anticlockwise rotation of $120 ^ { \circ }$ about the centre of the triangle.
\end{itemize}
These are shown in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772}\\
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762}\\
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}\\
(i) Explain why the order of $G$ is 6 .\\
(ii) Determine
\begin{itemize}
\item the order of $\boldsymbol { j }$,
\item the order of $\boldsymbol { k }$.\\
(iii) - Express, in terms of $\boldsymbol { j }$ and/or $\boldsymbol { k }$, each of the remaining three elements of $G$.
\item Draw a diagram for each of these elements.\\
(iv) Is the operation of composition of transformations on $G$ commutative? Justify your answer.\\
(v) List all the proper subgroups of $G$.
\end{itemize}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2018 Q4 [11]}}