| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.8 This is a Further Maths group theory question requiring systematic verification of group axioms, completion of a Cayley table, determination of element orders, and proof that the group is cyclic. While the individual steps are methodical rather than requiring deep insight, the topic itself (abstract algebra) is advanced, the question has multiple parts requiring careful reasoning, and part (e) requires knowledge of non-isomorphic groups of the same order. This is moderately challenging for Further Maths students but follows standard group theory verification procedures. |
| Spec | 8.03a Binary operations: and their properties on given sets8.03b Cayley tables: construct for finite sets under binary operation8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups8.03g Cyclic groups: meaning of the term |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | 1 5 7 11 13 17 |
| Answer | Marks |
|---|---|
| 17 17 13 11 7 5 1 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | General form/layout |
| Answer | Marks |
|---|---|
| (b) | Closure since there are no new elements in the table |
| Answer | Marks |
|---|---|
| ( Group) | B1 |
| Answer | Marks |
|---|---|
| [3] | 2.2a |
| Answer | Marks |
|---|---|
| 2.4 | Statements like “Closure from the table” are insufficient |
| Answer | Marks |
|---|---|
| (c) | Elements: 1 5 7 11 13 17 |
| Orders: 1 6 3 6 3 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1a |
| 1.1 | At least 2 (non-identity) elements correct |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | It has (at least) one element of order 6 | |
| or noting that there is a generator (5 or 11) | B1 | |
| [1] | 2.4 | |
| (e) | (i) | e.g. S , the group of six permutations of 3 symbols |
| Answer | Marks | Guidance |
|---|---|---|
| 3 2 | B1 | |
| [1] | 1.2 | or the corresponding transformations |
| Answer | Marks |
|---|---|
| (ii) | The non-cyclic group has elements of orders (1), 2, 2, 2, 3, 3 |
| Answer | Marks | Guidance |
|---|---|---|
| or this group is not abelian (or non-commutative) ( it is non-cyclic) | B1 | |
| [1] | 2.4 | |
| 1 | 5 | 7 |
| 5 | 7 | 17 |
| 7 | 17 | 13 |
| 11 | 1 | 5 |
| 13 | 11 | 1 |
Question 7:
7 | (a) | 1 5 7 11 13 17
1 1 5 7 11 13 17
5 5 7 17 1 11 13
7 7 17 13 5 1 11
11 11 1 5 13 17 7
13 13 11 1 17 7 5
17 17 13 11 7 5 1 | M1
A1
A1
A1
[4] | 1.1a
1.1
1.1
1.1 | General form/layout
At least R and C correct (shaded)
1 1
LSP observed
All correct
(b) | Closure since there are no new elements in the table
[Associativity assumed]
Identity is 1
Inverses: (5, 11) and (7, 13) are inverse-pairs; 17 is self-inverse
( Group) | B1
B1
B1
[3] | 2.2a
1.1
2.4 | Statements like “Closure from the table” are insufficient
Statements such as “1 appears in every row and column”
are insufficient since one should justify that each element’s
left-inverse and right-inverse are the same.
(c) | Elements: 1 5 7 11 13 17
Orders: 1 6 3 6 3 2 | M1
A1
[2] | 1.1a
1.1 | At least 2 (non-identity) elements correct
All 5 (non-identity) elements correct
Ignore missing (or incorrect) order for 1
(d) | It has (at least) one element of order 6
or noting that there is a generator (5 or 11) | B1
[1] | 2.4
(e) | (i) | e.g. S , the group of six permutations of 3 symbols
3
or D , the (dihedral) group of symmetries of the triangle
3
or the product group ℤ ⊗ ℤ
3 2 | B1
[1] | 1.2 | or the corresponding transformations
Allow wordy descriptions if complete
(ii) | The non-cyclic group has elements of orders (1), 2, 2, 2, 3, 3
or noting that all (non-identity) elements have order 2 or 3
or this group is not abelian (or non-commutative) ( it is non-cyclic) | B1
[1] | 2.4
1 | 5 | 7 | 11 | 13 | 17
5 | 7 | 17 | 1 | 11
7 | 17 | 13 | 5 | 1
11 | 1 | 5 | 13 | 17
13 | 11 | 1 | 17 | 7
177 You are given the set $S = \{ 1,5,7,11,13,17 \}$ together with $\times _ { 18 }$, the operation of multiplication modulo 18.
\begin{enumerate}[label=(\alph*)]
\item Complete the Cayley table for $\left( S , \times _ { 18 } \right)$ given in the Printed Answer Booklet.
\item Prove that ( $S , \times _ { 18 }$ ) is a group. (You may assume that $\times _ { 18 }$ is associative.)
\item Write down the order of each element of the group.
\item Show that $\left( S , \times _ { 18 } \right)$ is a cyclic group.
\item \begin{enumerate}[label=(\roman*)]
\item Give an example of a non-cyclic group of order 6 .
\item Give one reason why your example is structurally different to $\left( S , { } _ { 18 } \right)$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2019 Q7 [12]}}