| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Subgroups and cosets |
| Difficulty | Standard +0.3 This is a straightforward group theory question testing basic definitions and subgroup criteria. Part (i) requires verifying the four group axioms (closure, associativity, identity, inverses) which is routine given the hint about integer addition. Part (ii) involves checking standard subgroup conditions for three sets, with (b) failing due to lack of inverses being a common textbook example. The question requires careful verification but no novel insight or complex reasoning beyond applying definitions. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03f Subgroups: definition and tests for proper subgroups |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3^n \times 3^m = 3^{n+m}\), \(n + m \in \mathbb{Z}\) | B1 | For showing closure |
| \((3^p \times 3^q) \times 3^r = (3^{p+q}) \times 3^r = 3^{p+q+r}\) | M1 | For considering 3 distinct elements, seen bracketed 2+1 or 1+2 |
| \(= 3^p \times (3^{q+r}) = 3^p \times (3^q \times 3^r) \Rightarrow\) associativity | A1 | For correct justification of associativity |
| Identity is \(3^0\) | B1 | For stating identity. Allow 1 |
| Inverse is \(3^{-n}\) | B1 | For stating inverse |
| \(3^n \times 3^m = 3^{n+m} = 3^{m+n} = 3^m \times 3^n \Rightarrow\) commutativity | B1 6 | For showing commutativity |
| (ii) (a) \(3^{2n} \times 3^{2m} = 3^{2n+2m} (= 3^{2(n+m)})\) | B1* | For showing closure |
| Identity, inverse OK | B1 | For stating other two properties satisfied and hence a subgroup |
| 2 | (*dep) |
| Answer | Marks | Guidance |
|---|---|---|
| \(-n \notin\) subset | A1 2 | For justification of not being a subgroup. \(3^{-n}\) must be seen here or in (i) |
| (c) EITHER: eg \(3^1 \times 3^2 = 3^5\) | M1 | For attempting to find a specific counter-example of closure |
| \(\neq 3^{3^r} \Rightarrow\) not a subgroup | A1 2 | For a correct counter-example and statement that it is not a subgroup |
| OR: \(3^{n^2} \times 3^{m^2} = 3^{n^2+m^2}\) | M1 | For considering closure in general |
| \(\neq 3^{r^2}\) eg \(1^2 + 2^2 = 5\) \(\Rightarrow\) not a subgroup | A1 | For explaining why \(n^2 + m^2 \neq r^2\) in general and statement that it is not a subgroup |
**(i)** $3^n \times 3^m = 3^{n+m}$, $n + m \in \mathbb{Z}$ | B1 | For showing closure
$(3^p \times 3^q) \times 3^r = (3^{p+q}) \times 3^r = 3^{p+q+r}$ | M1 | For considering 3 distinct elements, seen bracketed 2+1 or 1+2
$= 3^p \times (3^{q+r}) = 3^p \times (3^q \times 3^r) \Rightarrow$ associativity | A1 | For correct justification of associativity
Identity is $3^0$ | B1 | For stating identity. Allow 1
Inverse is $3^{-n}$ | B1 | For stating inverse
$3^n \times 3^m = 3^{n+m} = 3^{m+n} = 3^m \times 3^n \Rightarrow$ commutativity | B1 6 | For showing commutativity
**(ii) (a)** $3^{2n} \times 3^{2m} = 3^{2n+2m} (= 3^{2(n+m)})$ | B1* | For showing closure
Identity, inverse OK | B1 | For stating other two properties satisfied and hence a subgroup
| 2 | (*dep)
**(b)** For $3^{-n}$
$-n \notin$ subset | A1 2 | For justification of not being a subgroup. $3^{-n}$ must be seen here or in (i)
**(c)** **EITHER:** eg $3^1 \times 3^2 = 3^5$ | M1 | For attempting to find a specific counter-example of closure
$\neq 3^{3^r} \Rightarrow$ not a subgroup | A1 2 | For a correct counter-example and statement that it is not a subgroup
**OR:** $3^{n^2} \times 3^{m^2} = 3^{n^2+m^2}$ | M1 | For considering closure in general
$\neq 3^{r^2}$ eg $1^2 + 2^2 = 5$ $\Rightarrow$ not a subgroup | A1 | For explaining why $n^2 + m^2 \neq r^2$ in general and statement that it is not a subgroup9 The set $S$ consists of the numbers $3 ^ { n }$, where $n \in \mathbb { Z }$. ( $\mathbb { Z }$ denotes the set of integers $\{ 0 , \pm 1 , \pm 2 , \ldots \}$.)\\
(i) Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.)\\
(ii) Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
\begin{enumerate}[label=(\alph*)]
\item The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
\item The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
\item The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$.
4
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2007 Q9 [12]}}