Challenging +1.2 This is a multi-part group theory question requiring proof of associativity (algebraic manipulation), demonstrating a group is cyclic (finding a generator), listing subgroups (systematic enumeration), and describing an isomorphism (pattern recognition). While covering several concepts, each part follows standard procedures taught in Further Maths without requiring deep insight or novel problem-solving approaches. The associativity proof is routine algebra, and the finite group analysis involves straightforward computation with small sets.
A binary operation * is defined on positive real numbers by
$$a * b = a + b + a b$$
Prove that the operation * is associative.
(ii) The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7
Show that \(G\) is cyclic.
The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
- M1: Attempts to identify an isomorphism — at least 2 correct non-identity pairings, or rearranges group tables, or maps powers of a generator e.g. \((\text{their } 3)^k \to (\text{their } 5)^k\), or matches non-trivial proper subgroups
- A1: Identifies 4 correct pairings, or sets up mapping with one correct generator
- A1: All pairings correct, or sets up mapping with generators of each group correct, e.g. \(3^k \to 5^k\)
# Question 6(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $(a*b)*c = (a+b+ab)*c = a+b+ab+c+(a+b+ab)c$ | M1 | 2.1 |
| $a*(b*c) = a*(b+c+bc) = a+b+c+bc+a(b+c+bc)$ | M1 | 2.1 |
| $a+b+ab+c+(a+b+ab)c = a+b+c+bc+ab+ac+abc$ $= a+b+c+bc+a(b+c+bc)$ | A1 | 2.2a |
| So $(a*b)*c = a*(b*c)$, which means $*$ is associative | A1 | 2.4 |
**Notes:**
- M1: Begins proof by correctly expanding $(a*b)*c$ **or** $a*(b*c)$ to an expression in $a$, $b$, $c$
- M1: Makes progress by attempting to expand both $(a*b)*c$ **and** $a*(b*c)$
- A1: Both underlined expressions correct with correct expansion seen for each independently
- A1: Explains $(a*b)*c = a*(b*c)$ means $*$ is associative; depends on both M marks
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# Question 6(ii)(a):
| Working | Mark | Guidance |
|---------|------|----------|
| $3^2=2,\ 3^3=6,\ 3^4=4,\ 3^5=5,\ 3^6=1$ **or** $5^2=4,\ 5^3=6,\ 5^4=2,\ 5^5=3,\ 5^6=1$ | M1 | 2.1 |
| 3 (or 5) has order 6 and so generates the group so $G$ is cyclic | A1 | 2.4 |
**Notes:**
- M1: Demonstrates understanding of cyclic by attempting all powers of 3 or 5; accept $\langle 3\rangle = \{3,2,6,4,5,1\}$
- A1: Must evaluate all powers correctly and explain why $G$ is cyclic; must refer to cyclic in conclusion
- Special case: Allow M1A0 if order of 3 (or 5) stated as 6 without justification but must reference generator or same order as $G$
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# Question 6(b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\{1\}$, $H$ | B1 | 1.1b |
| $\{1,17\}$ **or** $\{1,7,13\}$ | M1 | 1.1b |
| $\{1,17\}$ **and** $\{1,7,13\}$ (and no others) | A1 | 1.1b |
**Notes:**
- B1: Identifies $\{1\}$ and $H$ as subgroups
- M1: Identifies $\{1,17\}$ **or** $\{1,7,13\}$ as a subgroup
- A1: Identifies $\{1,17\}$ **and** $\{1,7,13\}$ as subgroups and no others
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# Question 6(c):
| Working | Mark | Guidance |
|---------|------|----------|
| $\begin{array}{c\|cccccc} G & 1 & 2 & 3 & 4 & 5 & 6 \\ H & 1 & 7 & 5 & 13 & 11 & 17 \end{array}$ **or** $\begin{array}{c\|cccccc} G & 1 & 2 & 3 & 4 & 5 & 6 \\ H & 1 & 13 & 11 & 7 & 5 & 17 \end{array}$ | M1, A1, A1 | 3.1a, 1.1b, 1.1b |
**Notes:**
- M1: Attempts to identify an isomorphism — at least 2 correct non-identity pairings, or rearranges group tables, or maps powers of a generator e.g. $(\text{their } 3)^k \to (\text{their } 5)^k$, or matches non-trivial proper subgroups
- A1: Identifies 4 correct pairings, or sets up mapping with one correct generator
- A1: All pairings correct, or sets up mapping with generators of each group correct, e.g. $3^k \to 5^k$
---
\begin{enumerate}
\item (i) A binary operation * is defined on positive real numbers by
\end{enumerate}
$$a * b = a + b + a b$$
Prove that the operation * is associative.\\
(ii) The set $G = \{ 1,2,3,4,5,6 \}$ forms a group under the operation of multiplication modulo 7\\
(a) Show that $G$ is cyclic.
The set $H = \{ 1,5,7,11,13,17 \}$ forms a group under the operation of multiplication modulo 18\\
(b) List all the subgroups of $H$.\\
(c) Describe an isomorphism between $G$ and $H$.
\hfill \mbox{\textit{Edexcel FP2 2019 Q6 [12]}}