| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Topic | Groups |
| Type | Isomorphism between groups |
| Difficulty | Challenging +1.8 This is a substantial group theory question requiring understanding of subgroups, Lagrange's theorem, element orders, and isomorphism. While the concepts are advanced (Further Maths), the actual execution is methodical: listing subgroups systematically, applying Lagrange's theorem, computing orders in Z_8, and comparing group structures. The isomorphism determination requires recognizing that different element order distributions prevent isomorphism, which is a key insight but follows from systematic analysis. |
| Spec | 8.03f Subgroups: definition and tests for proper subgroups8.03g Cyclic groups: meaning of the term8.03i Properties of groups: structure of finite groups up to order 78.03l Isomorphism: determine using informal methods |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | 1 | 2 |
| 1 | 8 | 4 |
**(i)** Any four of: $\{e, a, bc, abc\}$, $\{e, b, ac, abc\}$, $\{e, c, ab, abc\}$, $\{e, a, b, ab\}$, $\{e, a, c, ac\}$, $\{e, b, c, bc\}$, $\{e, ab, bc, ac\}$ B1B1B1B1 **[4]**
**(ii)** 8 is not divisible by 3, or order of a group is divisible by order of any subgroup (accept 'Lagrange's Theorem') B1 **[1]**
**(iii)**
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| 1 | 8 | 4 | 8 | 2 | 8 | 4 | 8 |
($-1$ per error, but no negative marks) B3 **[3]**
**(iv)** e.g. $H$ has only one element of order 2, $G$ has at least 2 elements of order 2; hence $G$ and $H$ are not isomorphic (or e.g. $H$ is cyclic, $G$ is not cyclic; or any other valid method) B1 **[1]**7 The multiplicative group $G$ has eight elements $e , a , b , c , a b , a c , b c , a b c$, where $e$ is the identity. The group is commutative, and the order of each of the elements $a , b , c$ is 2 .\\
(i) Find four subgroups of $G$ of order 4.\\
(ii) Give a reason why no group of order 8 can have a subgroup of order 3 .
The group $H$ has elements $0,1,2 , \ldots , 7$ with group operation addition modulo 8 .\\
(iii) Find the order of each element of $H$.\\
(iv) Determine whether $G$ and $H$ are isomorphic and justify your conclusion.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q7 [9]}}