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]