## (i)

Group A: $e = 6$

Group B: $e = 1$

Group C: $e = 2^0$ OR $1$

Group D: $e = 1$

| B1 | For any two correct identities |
| B1 | For two other correct identities |
| 2 | AEF for D, but not "$m = n$" |

## (ii)

| B1* | For showing group table OR sufficient details of orders of elements OR stating cyclic / non-cyclic / Klein group (as appropriate) |
| B1* | for one of groups A, B, C |
| B1* | for another of groups A, B, C |

$A \not\cong B$

$B \not\cong C$

$A \cong C$

| B1 (dep*) | For stating non-isomorphic with sufficient detail relating to the first 2 marks |
| B1 (dep*) | For stating non-isomorphic |
| B1 (dep*) | For stating isomorphic |
| 5 | |

## (iii)

$\frac{1+2m}{1+2n} \cdot \frac{1+2p}{1+2q} \cdot \frac{1+2m+2p+4mp}{1+2q+4nq}$

| M1* | For considering product of 2 distinct elements of this form |
| M1 (dep*) | For multiplying out |
| A1 | For simplifying to form shown |

$= \frac{1+2(m+p+2mp)}{1+2(n+q+2nq)} = \frac{1+2r}{1+2x}$

| A1 | For identifying as correct form, so closed (4 marks) |

| SR | $\frac{\text{odd}}{\text{odd}} \times \frac{\text{odd}}{\text{odd}} \times \frac{\text{odd}}{\text{odd}}$ earns full credit |
| SR | If clearly attempting to prove commutativity, allow at most M1 |

## (iv)

Closure not satisfied

Identity and inverse not satisfied

| B1 | For stating closure |
| B1 | For stating identity and inverse (2 marks) |

| SR | If associativity is stated as not satisfied then award at most B1 B0 OR B0 B1 |