| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Non-group structures |
| Difficulty | Standard +0.3 This is a straightforward group theory question testing basic axiom verification. Part (i) requires checking closure/identity (routine), part (ii) needs recognizing that a=1 (the identity element), and part (iii) asks about isomorphism to C₄ by checking if the group is cyclic. All parts follow standard procedures with no novel insight required, making it slightly easier than average for Further Maths content. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03d Latin square property: for group tables8.03l Isomorphism: determine using informal methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3\times3=1\), \(5\times5=1\), \(7\times7=1\) | M1, A1 | For showing operation table or otherwise; for a convincing reason |
| OR: Show \(a\times e=a\) has no solution | M1 | For attempt to find identity OR for showing operation table |
| Identity is not 3, not 5, and not 7 | A1 2 | By reference to operation table or otherwise |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((a=)\ 1\) | B1 1 | For value of \(a\) stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: \(\{e,r,r^2,r^3\}\) is cyclic; (ii) group is not cyclic | B1* | For a pair of correct statements |
| OR: \(\{e,r,r^2,r^3\}\) has 2 self-inverse elements; (ii) group has 4 self-inverse elements | B1* | For a pair of correct statements |
| OR: \(\{e,r,r^2,r^3\}\) has 1 element of order 2; (ii) group has 3 elements of order 2 | B1* | For a pair of correct statements |
| OR: \(\{e,r,r^2,r^3\}\) has element(s) of order 4; (ii) group has no element of order 4 | B1* | For a pair of correct statements |
| Not isomorphic | B1(dep*) 2 | For correct conclusion |
## Question 1:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\times3=1$, $5\times5=1$, $7\times7=1$ | M1, A1 | For showing operation table or otherwise; for a convincing reason |
| OR: Show $a\times e=a$ has no solution | M1 | For attempt to find identity OR for showing operation table |
| Identity is not 3, not 5, and not 7 | A1 **2** | By reference to operation table or otherwise |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(a=)\ 1$ | B1 **1** | For value of $a$ stated |
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: $\{e,r,r^2,r^3\}$ is cyclic; (ii) group is not cyclic | B1* | For a pair of correct statements |
| OR: $\{e,r,r^2,r^3\}$ has 2 self-inverse elements; (ii) group has 4 self-inverse elements | B1* | For a pair of correct statements |
| OR: $\{e,r,r^2,r^3\}$ has 1 element of order 2; (ii) group has 3 elements of order 2 | B1* | For a pair of correct statements |
| OR: $\{e,r,r^2,r^3\}$ has element(s) of order 4; (ii) group has no element of order 4 | B1* | For a pair of correct statements |
| Not isomorphic | B1(dep*) **2** | For correct conclusion |
---1 (i) Show that the set of numbers $\{ 3,5,7 \}$, under multiplication modulo 8, does not form a group.\\
(ii) The set of numbers $\{ 3,5,7 , a \}$, under multiplication modulo 8 , forms a group. Write down the value of $a$.\\
(iii) State, justifying your answer, whether or not the group in part (ii) is isomorphic to the multiplicative group $\left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}$, where $e$ is the identity and $r ^ { 4 } = e$.
\hfill \mbox{\textit{OCR FP3 2007 Q1 [5]}}