## Question 4(a)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Identity is $e$ | B1 | |
| Inverse table: Element $a,b,c,d,e,f,g,h$ → Inverse $b,a,c,g,e,h,d,f$ | B2 | Give B1 for four correct |
| **[3]** | | |

## Question 4(a)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $d^2 = a,\ d^4 = c$ | M1, A1, A1 | Finding powers of an element; Identifying $d$ (or $f$, $g$, or $h$) as generator; Or $f^2=b,\ f^4=c$; Or $g^2=b,\ g^4=c$; Or $h^2=a,\ h^4=c$ | At least fourth power; Implies previous M1 |
| Hence $d$ has order 8, and $G$ is cyclic | E1 | Correctly shown |
| **[4]** | | |

## Question 4(a)(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Isomorphism table mapping $H: 0,2,4,6,8,10,12,14$ to $G$ elements | B1 | For $e\leftrightarrow 0$ and $c\leftrightarrow 8$ | |
| $\{d,f,g,h\} \leftrightarrow \{2,6,10,14\}$ | B1 | In any order |
| Fully correct isomorphism | B1 | |
| **[3]** | | |

## Question 4(a)(iv):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Rotations have order 2 or 4; Reflections have order 2 | B1 | Correct statement about rotations and/or reflections which implies non-IM | Or (4) reflections (and 180° rotation) have order 2; Or composition of reflections (or 90° rotation and reflection) is not commutative |
| There is no element of order 8; Hence not isomorphic | E1 | Fully correct explanation | Dependent on previous B1 |
| **[2]** | | |

## Question 4(b)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $f_m f_n(x) = \dfrac{\dfrac{x}{1+nx}}{1+m\left(\dfrac{x}{1+nx}\right)}$ | M1 | Composition of functions | In either order |
| $= \dfrac{x}{1+nx+mx} = \dfrac{x}{1+(m+n)x} = f_{m+n}(x)$ | E1 | Correctly shown | E0 if in wrong order |
| **[2]** | | |

## Question 4(b)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(f_m f_n)f_p = f_{m+n}f_p = f_{m+n+p}$ | M1 | Combining three functions | |
| $f_m(f_n f_p) = f_m f_{n+p} = f_{m+n+p}$ | | | M1E1 bod for $(f_m f_n)f_p = f_{m+n+p} = f_m(f_n f_p)$ |
| Hence $S$ is associative | E1 | Correctly shown |
| **[2]** | | |

## Question 4(b)(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| For any $f_m$, $f_n$ in $S$, $f_m f_n = f_{m+n}$ | M1 | Referring to this in context | |
| $f_m f_n$ is in $S$ (so $S$ is closed) | A1 | | |
| Identity is $f_0$ | B1 | B0 for $x$; B1 for $n=0$ | |
| Inverse of $f_n$ is $f_{-n}$ since $f_n f_{-n} = f_{n-n} = f_0$ | B1, B1 | | |
| $S$ is also associative, and hence is a group | E1 | Closure, associativity, identity and inverses must all be mentioned | Dependent on previous 5 marks |
| **[6]** | | |

## Question 4(b)(iv):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{f_{2n}\}$ for all integers $n$ | B2 | Or $\{f_{3n}\}$ etc; Give B1 for multiples of 2 (or 3 etc) but not completely correctly described | e.g. $\{f_0, f_2, f_4, f_6, \ldots\}$ |
| **[2]** | | |