| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Session | Specimen |
| Marks | 10 |
| 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 verification of four group axioms (closure, associativity, identity, inverses) for a non-standard operation, finding subgroups, and testing isomorphism. While systematic, it demands understanding of abstract algebra concepts beyond standard A-level and careful verification of multiple properties, placing it moderately above average difficulty. |
| 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 group |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| As all answers to \(a * b\) are reduced modulo 6, they are in the given set and thus the set is closed under \(*\) | R1 | Must show that under modulo 6, \(a * b\) can only result in a member of the given set |
| Identity element \(= 2\) | B1 | Clearly identifies the identity element |
| 0 and 4 are inverses of each other; 1 and 3 are inverses of each other; 2 and 5 are self-inverse elements | B1 | Finds and states the inverse of each element (PI), FT from 'their' identity |
| \(a*(b*c) = a+(b+c+4)+4\) is shown to equal \((a*b)*c = (a+b+4)+c+4\) | R1 | Shows associativity between elements of the set under the operation \(*\) |
| As \(G\) satisfies each of the four group axioms under the binary operation \(*\), \(G\) is a group | R1 | States correct conclusion; only award if completely correct solution, clear and no slips |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\{2\}, \{0,2,4\}, \{2,5\}\) — identifies two correct subgroups | B1 | Condone inclusion of \(\{0,1,2,3,4,5\}\) |
| Identifies all three proper subgroups and no others included | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(G = (\langle 1 \rangle, *)\) OR \(K = (\{3,9,13,11,5,1\}, \times_{14})\) | B1 | Identifies the generator of \(G\) OR generates every element of the group \(K\) (PI) |
| \(1 \mapsto 3,\ 0 \mapsto 9,\ 5 \mapsto 13,\ 4 \mapsto 11,\ 3 \mapsto 5,\ 2 \mapsto 1\) | B1 | Finds correctly a one-to-one mapping between each element of \(G\) and \(K\) (condone use of equal sign) |
| As there is a one-to-one mapping between the elements of \(G\) and the elements of \(K\), \(G \cong K\) | E1 | Deduces that \(G\) is isomorphic to \(K\) with concluding statement using correct mathematical language and notation throughout |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| As all answers to $a * b$ are reduced modulo 6, they are in the given set and thus the set is closed under $*$ | R1 | Must show that under modulo 6, $a * b$ can only result in a member of the given set |
| Identity element $= 2$ | B1 | Clearly identifies the identity element |
| 0 and 4 are inverses of each other; 1 and 3 are inverses of each other; 2 and 5 are self-inverse elements | B1 | Finds and states the inverse of each element (PI), FT from 'their' identity |
| $a*(b*c) = a+(b+c+4)+4$ is shown to equal $(a*b)*c = (a+b+4)+c+4$ | R1 | Shows associativity between elements of the set under the operation $*$ |
| As $G$ satisfies each of the four group axioms under the binary operation $*$, $G$ is a group | R1 | States correct conclusion; only award if completely correct solution, clear and no slips |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\{2\}, \{0,2,4\}, \{2,5\}$ — identifies two correct subgroups | B1 | Condone inclusion of $\{0,1,2,3,4,5\}$ |
| Identifies all three proper subgroups and no others included | B1 | |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $G = (\langle 1 \rangle, *)$ OR $K = (\{3,9,13,11,5,1\}, \times_{14})$ | B1 | Identifies the generator of $G$ OR generates every element of the group $K$ (PI) |
| $1 \mapsto 3,\ 0 \mapsto 9,\ 5 \mapsto 13,\ 4 \mapsto 11,\ 3 \mapsto 5,\ 2 \mapsto 1$ | B1 | Finds correctly a one-to-one mapping between each element of $G$ and $K$ (condone use of equal sign) |
| As there is a one-to-one mapping between the elements of $G$ and the elements of $K$, $G \cong K$ | E1 | Deduces that $G$ is isomorphic to $K$ with concluding statement using correct mathematical language and notation throughout |
---5 The binary operation * is defined as
$$a * b = a + b + 4 ( \bmod 6 )$$
where $a , b \in \mathbb { Z }$.
5
\begin{enumerate}[label=(\alph*)]
\item Show that the set $\{ 0,1,2,3,4,5 \}$ forms a group $G$ under *.\\
5
\item Find the proper subgroups of the group $G$ in part (a).\\
5
\item Determine whether or not the group $G$ in part (a) is isomorphic to the group $K = \left( \langle 3 \rangle , \times _ { 14 } \right)$\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete Q5 [10]}}