State the possible orders of an element of $\{ G,{\text{ }} * \}$ and for each order give an example of an element of that order.

(i)     State a generator for $\{ H,{\text{ }} * \}$.

(ii)     Write down the elements of $\{ H,{\text{ }} * \}$.

(iii)     Write down the elements of the coset of $H$ containing $a$.

orders are 1 2 3 4 6 12     A2

Note: A1 for four or five correct orders.

Note: For the rest of this question condone absence of xxx and accept equivalent expressions.

$\begin{array}{*{20}{l}} {{\text{order:}}}&1&{{\text{element:}}}&2&{A1} \\ {}&2&{}&{{a^2}}&{A1} \\ {}&3&{}&{b{\text{ or }}{{\text{b}}^2}}&{A1} \\ {}&4&{}&{a{\text{ or }}{a^3}}&{A1} \\ {}&6&{}&{{a^2} * b{\text{ or }}{a^2} * {b^2}}&{A1} \\ {}&{12}&{}&{a * b{\text{ or }}a * {b^2}{\text{ or }}{a^3} * b{\text{ or }}{a^3} * {b^2}}&{A1} \end{array}$

[8 marks]

(i)     $H$ has order 6     (R1)

generator is ${a^2} * b$ or ${a^2} * {b^2}$     A1

(ii)     $H = \left\{ {e,{\text{ }}{a^2} * b,{\text{ }}{b^2},{\text{ }}{a^2},{\text{ }}b,{\text{ }}{a^2} * {b^2}} \right\}$     A3

Note: A2 for 4 or 5 correct. A1 for 2 or 3 correct.

(iii)     required coset is $Ha$ (or $aH$)     (R1)

$Ha = \left\{ {a,{\text{ }}{a^3} * b,{\text{ }}a * {b^2},{\text{ }}{a^3},{\text{ }}a * b,{\text{ }}{a^3} * {b^2}} \right\}$    A1

[7 marks]