| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Groups with generators and relations |
| Difficulty | Challenging +1.2 This is a standard FP3 group theory question covering basic properties of the dihedral group D₅. Parts (i)-(iii) require simple recall and application of definitions (commutativity, Lagrange's theorem, subgroup listing). Part (iv) involves routine order calculations using the given relations. Part (v) requires systematic use of the relations r⁴a = ar to compute products, which is mechanical but requires care. While this is Further Maths content (inherently harder), it's a textbook exercise testing standard techniques rather than requiring novel insight or proof. |
| Spec | 8.03b 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 groups8.03f Subgroups: definition and tests for proper subgroups |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(r^4, a \neq a, r^4\) | B1 1 | For stating the non-commutative product in the given table, or justifying another correct one |
| (ii) Possible subgroups order 2, 5 | B1 B1 2 | For either order stated; For both orders stated, and no more (Ignore 1) |
| (iii) (a) \(\{e, a\}\) | B1 | For correct subgroup |
| (b) \(\{e, r, r^2, r^3, r^4\}\) | B1 2 | For correct subgroup |
| (iv) order of \(r^3 = 5\); \((ar)^2 = ar \cdot ar = r^4 a \cdot ar = e \Rightarrow\) order of \(ar = 2\); \((ar^2)^2 = ar^2 a r \cdot r = ar^2 a r = ara \cdot r = e \Rightarrow\) order of \(ar^2 = 2\) | B1; M1; A1; A1 4 | For correct order; For attempt to find \((ar)^m = e\) OR \((ar^2)^m = e\); For correct order; For correct order |
| (v) Cayley table with 16 entries, all 4 elements of form \(e\) or \(r^m\); All 4 elements in leading diagonal = \(e\); No repeated elements in any completed row or column; Any two rows or columns correct; All elements correct | B1; B1; B1; B1 B1 5 [14] | For all 16 elements of the form \(e\) or \(r^m\); For all 4 elements in leading diagonal = \(e\); For no repeated elements in any completed row or column; For any two rows or columns correct; For all elements correct |
**(i)** $r^4, a \neq a, r^4$ | B1 1 | For stating the non-commutative product in the given table, or justifying another correct one
**(ii)** Possible subgroups order 2, 5 | B1 B1 2 | For either order stated; For both orders stated, and no more (Ignore 1)
**(iii)** **(a)** $\{e, a\}$ | B1 | For correct subgroup
**(b)** $\{e, r, r^2, r^3, r^4\}$ | B1 2 | For correct subgroup
**(iv)** order of $r^3 = 5$; $(ar)^2 = ar \cdot ar = r^4 a \cdot ar = e \Rightarrow$ order of $ar = 2$; $(ar^2)^2 = ar^2 a r \cdot r = ar^2 a r = ara \cdot r = e \Rightarrow$ order of $ar^2 = 2$ | B1; M1; A1; A1 4 | For correct order; For attempt to find $(ar)^m = e$ **OR** $(ar^2)^m = e$; For correct order; For correct order
**(v)** Cayley table with 16 entries, all 4 elements of form $e$ or $r^m$; All 4 elements in leading diagonal = $e$; No repeated elements in any completed row or column; Any two rows or columns correct; All elements correct | B1; B1; B1; B1 B1 5 [14] | For all 16 elements of the form $e$ or $r^m$; For all 4 elements in leading diagonal = $e$; For no repeated elements in any completed row or column; For any two rows or columns correct; For all elements correctA group $D$ of order 10 is generated by the elements $a$ and $r$, with the properties $a^2 = e$, $r^5 = e$ and $r^4a = ar$, where $e$ is the identity. Part of the operation table is shown below.
\includegraphics{figure_1}
\begin{enumerate}[label=(\roman*)]
\item Give a reason why $D$ is not commutative. [1]
\item Write down the orders of any possible proper subgroups of $D$. [2]
\item List the elements of a proper subgroup which contains
\begin{enumerate}[label=(\alph*)]
\item the element $a$, [1]
\item the element $r$. [1]
\end{enumerate}
\item Determine the order of each of the elements $r^3$, $ar$ and $ar^2$. [4]
\item Copy and complete the section of the table marked E, showing the products of the elements $ar$, $ar^2$, $ar^3$ and $ar^4$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q8 [14]}}