| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Topic | Groups |
| Type | Complete or analyse Cayley table |
| Difficulty | Challenging +1.2 This is a structured group theory question requiring Cayley table construction, verification of group axioms, and isomorphism analysis. While it involves multiple parts and abstract algebra concepts (modular arithmetic, group properties, subgroups, isomorphism), each step is methodical and follows standard procedures. The modular arithmetic is straightforward, the group axioms check is routine given associativity, and determining isomorphism between two order-6 groups (cyclic vs non-cyclic) is a standard classification problem. This is moderately above average difficulty due to the abstract nature and multiple components, but remains accessible to well-prepared Further Maths students. |
| Spec | 8.02e Finite (modular) arithmetic: integers modulo n8.03c Group definition: recall and use, show structure is/isn't a group8.03g Cyclic groups: meaning of the term |
| Answer | Marks | Guidance |
|---|---|---|
| \(G\) | 1 | 2 |
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 4 | 4 | 8 |
| 8 | 8 | 16 |
| 16 | 16 | 32 |
| 32 | 32 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H\) | \(e\) | \(x\) |
| \(e\) | \(e\) | \(x\) |
| \(x\) | \(x\) | \(e\) |
| \(y\) | \(y\) | \(yx\) |
| \(y^2\) | \(y^2\) | \(xy\) |
| \(xy\) | \(xy\) | \(y^2\) |
| \(yx\) | \(yx\) | \(y\) |
**Question 8** [2+3+4+2+1 marks]
**(i)(a)**
| $G$ | **1** | **2** | **4** | **8** | **16** | **32** |
|-----|-------|-------|-------|-------|--------|--------|
| **1** | 1 | 2 | 4 | 8 | 16 | 32 |
| **2** | 2 | 4 | 8 | 16 | 32 | 1 |
| **4** | 4 | 8 | 16 | 32 | 1 | 2 |
| **8** | 8 | 16 | 32 | 1 | 2 | 4 |
| **16** | 16 | 32 | 1 | 2 | 4 | 8 |
| **32** | 32 | 1 | 2 | 4 | 8 | 16 |
M1 (for mostly correct), A1 (for all correct) **[2]**
**(i)(b)**
$(S, \times_{63})$ closed, since no new elements in table B1
$\times_{63}$ is associative (given) B1
1 is the identity element B1
Each (non-identity) element has a unique inverse: $2 \leftrightarrow 32$, $4 \leftrightarrow 16$ and $8$ is self-inverse B1 (All must be identified)
**[3]**
**(ii)(a)**
| $H$ | $e$ | $x$ | $y$ | $y^2$ | $xy$ | $yx$ |
|-----|-----|-----|-----|--------|------|------|
| $e$ | $e$ | $x$ | $y$ | $y^2$ | $xy$ | $yx$ |
| $x$ | $x$ | $e$ | $xy$ | $yx$ | $y$ | $y^2$ |
| $y$ | $y$ | $yx$ | $y^2$ | $e$ | $x$ | $xy$ |
| $y^2$ | $y^2$ | $xy$ | $e$ | $y$ | $yx$ | $x$ |
| $xy$ | $xy$ | $y^2$ | $yx$ | $x$ | $e$ | $y$ |
| $yx$ | $yx$ | $y$ | $x$ | $xy$ | $y^2$ | $e$ |
B1 (for last 3 elements (any forms)), B1 (for identity row/column), B1 (for easy elements (gold) or $\geqslant 14$ others), B1 (for all) **[4]**
**(ii)(b)**
Proper subgroups of $H$ are (condone inclusion of $\{e\}$ and $H$):
$\{e,x\}$, $\{e,xy\}$, $\{e,yx\}$ and $\{e,y,y^2\}$ B1B1 (B1 Any 2; +B1 all 4 and no extras)
**[2]**
**(ii)(c)**
$G$ and $H$ are NOT isomorphic B1 (Correct conclusion WITH a valid reason)
e.g. Different numbers of self-inverse elements / elements of order 3, or $G$ cyclic, $H$ non-cyclic, or $G$ abelian, $H$ non-abelian
**[1]**8 (i) $S$ is the set $\{ 1,2,4,8,16,32 \}$ and $\times _ { 63 }$ is the operation of multiplication modulo 63 .
\begin{enumerate}[label=(\alph*)]
\item Construct the multiplication table for $\left( S , \times _ { 63 } \right)$.
\item Show that $\left( S , \times _ { 63 } \right)$ forms a group, $G$. (You may assume that $\times _ { 63 }$ is associative.)\\
(ii) The group $H$, also of order 6, has identity element $e$ and contains two further elements $x$ and $y$ with the properties
$$x ^ { 2 } = y ^ { 3 } = e \quad \text { and } \quad x y x = y ^ { 2 } .$$
(a) Construct the group table of $H$.\\
(b) List all the proper subgroups of $H$.
\item State, with justification, whether $G$ and $H$ are isomorphic.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q8 [12]}}