| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Proper subgroups identification |
| Difficulty | Standard +0.8 This is a multi-part group theory question requiring systematic computation of inverses and orders in a multiplicative group mod 20, identification of a subgroup, and application of Lagrange's theorem. While the calculations are methodical rather than conceptually deep, group theory is an advanced Further Maths topic, and the question requires understanding of multiple abstract concepts (inverse, order, subgroup, Lagrange's theorem). The computational work is substantial but routine for students at this level. |
| Spec | 8.03a Binary operations: and their properties on given sets8.03b Cayley tables: construct for finite sets under binary operation8.03e Order of elements: and order of groups8.03f Subgroups: definition and tests for proper subgroups8.03k Lagrange's theorem: order of subgroup divides order of group |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 1, 9, 11 and 19 are self-inverse | M1 | For any 2 of the self-inverse elements correctly identified |
| 1, 9, 11 and 19 are self-inverse | A1 | All 4 self-inverse elements correctly identified |
| \(3 \mapsto 7\), \(7 \mapsto 3\), \(13 \mapsto 17\), \(17 \mapsto 13\) | B1 | Correct inverses for the other elements |
| (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Table with at least 3 correct orders | M1 | At least 3 correct orders |
| \(1,3,7,9,11,13,17,19\) with orders \(1,4,4,2,2,4,4,2\) | A1 | 6 correct orders |
| All orders correct | A1 | All correct |
| (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\{1,3,7,9\}\) or \(\{1,9,13,17\}\) or \(\{1,9,11,19\}\) | B1 | Describes a correct subgroup of order 4 |
| (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Because 4 is a factor of 8 | B1 | Correct explanation |
| (1 mark) |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 1, 9, 11 and 19 are self-inverse | M1 | For any 2 of the self-inverse elements correctly identified |
| 1, 9, 11 and 19 are self-inverse | A1 | All 4 self-inverse elements correctly identified |
| $3 \mapsto 7$, $7 \mapsto 3$, $13 \mapsto 17$, $17 \mapsto 13$ | B1 | Correct inverses for the other elements |
| **(3 marks)** | | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Table with at least 3 correct orders | M1 | At least 3 correct orders |
| $1,3,7,9,11,13,17,19$ with orders $1,4,4,2,2,4,4,2$ | A1 | 6 correct orders |
| All orders correct | A1 | All correct |
| **(3 marks)** | | |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{1,3,7,9\}$ or $\{1,9,13,17\}$ or $\{1,9,11,19\}$ | B1 | Describes a correct subgroup of order 4 |
| **(1 mark)** | | |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Because 4 is a factor of 8 | B1 | Correct explanation |
| **(1 mark)** | | |\begin{enumerate}
\item The set $G = \{ 1,3,7,9,11,13,17,19 \}$ under the binary operation of multiplication modulo 20 forms a group.\\
(a) Find the inverse of each element of $G$.\\
(b) Find the order of each element of $G$.\\
(c) Find a subgroup of $G$ of order 4\\
(d) Explain how the subgroup you found in part (c) satisfies Lagrange's theorem.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 AS 2020 Q1 [8]}}