The group $\{ G,{\text{ }} * \}$ has a subgroup $\{ H,{\text{ }} * \}$. The relation $R$ is defined, for $x,{\text{ }}y \in G$, by $xRy$ if and only if ${x^{ - 1}} * y \in H$.

(a)     Show that $R$ is an equivalence relation.

(b)     Given that $G = \{ 0,{\text{ }} \pm 1,{\text{ }} \pm 2,{\text{ }} \ldots \}$, $H = \{ 0,{\text{ }} \pm 4,{\text{ }} \pm 8,{\text{ }} \ldots \}$ and $*$ denotes addition, find the equivalence class containing the number $3$.

(a)     $\underline {{\text{reflexive}}}$

${x^{ - 1}}x = e \in H$     A1

therefore $xRx$ and $R$ is reflexive     R1

$\underline {{\text{symmetric}}}$

Note: Accept the word commutative.

let $xRy$ so that ${x^{ - 1}}y \in H$     M1

the inverse of ${x^{ - 1}}y$ is ${y^{ - 1}}x \in H$     A1

therefore $yRx$ and $R$ is symmetric     R1

$\underline {{\text{transitive}}}$

let $xRy$ and $yRz$ so ${x^{ - 1}}y \in H$ and ${y^{ - 1}}z \in H$     M1

therefore ${x^{ - 1}}y\,{y^{ - 1}}z = {x^{ - 1}}z \in H$     A1

therefore $xRz$ and $R$ is transitive     R1

hence $R$ is an equivalence relation     AG

[8 marks]

(b)     the identity is $0$ so the inverse of $3$ is $-3$     (R1)

the equivalence class of 3 contains $x$ where $- 3 + x \in H$     (M1)

$- 3 + x = 4n{\text{ }}(n \in \mathbb{Z})$     (M1)

$x = 3 + 4n{\text{ (n}} \in \mathbb{Z})$     A1

Note: Accept $\{  \ldots  - 5,{\text{ }} - 1,{\text{ }}3,{\text{ }}7,{\text{ }} \ldots \}$ or $x \equiv 3(\bmod 4)$.

Note: If no other relevant working seen award A3 for $\{ 3 + 4n\}$ or $\{  \ldots  - 5,{\text{ }} - 1,{\text{ }}3,{\text{ }}7,{\text{ }} \ldots \}$ seen anywhere.

[4 marks]