| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Topic | Groups |
| Type | Isomorphism between groups |
| Difficulty | Challenging +1.2 This is a standard group theory question requiring construction of Cayley tables and verification of group axioms. While it involves abstract algebra (a Further Maths topic), the execution is mechanical: compute modular products, check closure/identity/inverses systematically, and identify isomorphism type (both are Klein four-group). The question guides students through each step explicitly, requiring careful bookkeeping rather than creative insight. |
| Spec | 8.02e Finite (modular) arithmetic: integers modulo n8.03c Group definition: recall and use, show structure is/isn't a group8.03l Isomorphism: determine using informal methods |
$G$ is the set $\{2, 4, 6, 8\}$, $H$ is the set $\{1, 5, 7, 11\}$ and $\times_n$ denotes the operation of multiplication modulo $n$.
\begin{enumerate}[label=(\roman*)]
\item Construct the multiplication tables for $(G, \times_{10})$ and $(H, \times_{12})$. [2]
\item By verifying the four group axioms, show that $G$ and $H$ are groups under their respective binary operations, and determine whether $G$ and $H$ are isomorphic. [6]
\end{enumerate}
[You may assume that $\times_n$ is associative.]
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q6 [8]}}