| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Challenging +1.8 This is a comprehensive group theory question requiring multiple proofs and applications of Lagrange's theorem, finding generators, analyzing composition tables, and identifying subgroups. While the individual parts use standard techniques (proving subgroups of order 10 are cyclic, showing a group is cyclic, identifying symmetries), the question requires sustained abstract reasoning across multiple parts, knowledge of isomorphisms, and careful analysis of a 10×10 composition table—significantly harder than typical A-level questions but standard for Further Maths group theory. |
| Spec | 8.03f Subgroups: definition and tests for proper subgroups8.03i Properties of groups: structure of finite groups up to order 78.03j Properties of groups: higher finite order or infinite order |
| A | B | C | D | E | F | G | H | I | J | |
| A | C | J | G | H | A | B | I | F | E | D |
| B | F | E | H | G | B | A | D | C | J | I |
| C | G | D | I | F | C | J | E | B | A | H |
| D | J | C | B | E | D | G | F | I | H | A |
| E | A | B | C | D | E | F | G | H | I | J |
| F | H | I | D | C | F | E | J | A | B | G |
| G | I | H | E | B | G | D | A | J | C | F |
| H | D | G | J | A | H | I | B | E | F | C |
| I | E | F | A | J | I | H | C | D | G | B |
| J | B | A | F | I | J | C | H | G | D | E |
| Answer | Marks |
|---|---|
| (i) Prove every proper subgroup of order-10 group is cyclic | [4] |
| (ii) Show M is cyclic | [4] |
| (iii) List all proper subgroups of M | [3] |
| (iv) Identify identity, rotations, reflections in P | [4] |
| (v) State whether P is isomorphic to M, with reason | [2] |
| (vi) Find order of each element of P | [3] |
| (vii) List all proper subgroups of P | [4] |
## Question 4 (Option 4: Groups)
**(i)** Prove every proper subgroup of order-10 group is cyclic | [4] |
**(ii)** Show M is cyclic | [4] |
**(iii)** List all proper subgroups of M | [3] |
**(iv)** Identify identity, rotations, reflections in P | [4] |
**(v)** State whether P is isomorphic to M, with reason | [2] |
**(vi)** Find order of each element of P | [3] |
**(vii)** List all proper subgroups of P | [4] |
---4 (i) Prove that, for a group of order 10, every proper subgroup must be cyclic.
The set $M = \{ 1,2,3,4,5,6,7,8,9,10 \}$ is a group under the binary operation of multiplication modulo 11.\\
(ii) Show that $M$ is cyclic.\\
(iii) List all the proper subgroups of $M$.
The group $P$ of symmetries of a regular pentagon consists of 10 transformations
$$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$
and the binary operation is composition of transformations. The composition table for $P$ is given below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& A & B & C & D & E & F & G & H & I & J \\
\hline
A & C & J & G & H & A & B & I & F & E & D \\
\hline
B & F & E & H & G & B & A & D & C & J & I \\
\hline
C & G & D & I & F & C & J & E & B & A & H \\
\hline
D & J & C & B & E & D & G & F & I & H & A \\
\hline
E & A & B & C & D & E & F & G & H & I & J \\
\hline
F & H & I & D & C & F & E & J & A & B & G \\
\hline
G & I & H & E & B & G & D & A & J & C & F \\
\hline
H & D & G & J & A & H & I & B & E & F & C \\
\hline
I & E & F & A & J & I & H & C & D & G & B \\
\hline
J & B & A & F & I & J & C & H & G & D & E \\
\hline
\end{tabular}
\end{center}
One of these transformations is the identity transformation, some are rotations and the rest are reflections.\\
(iv) Identify which transformation is the identity, which are rotations and which are reflections.\\
(v) State, giving a reason, whether $P$ is isomorphic to $M$.\\
(vi) Find the order of each element of $P$.\\
(vii) List all the proper subgroups of $P$.
\hfill \mbox{\textit{OCR MEI FP3 2007 Q4 [24]}}