The set \(S\) contains the eighth roots of unity given by \(\left\{ {{\text{cis}}\left( {\frac{{n\pi }}{4}} \right),{\text{ }}n \in \mathbb{N},{\text{ }}0 \leqslant n \leqslant 7} \right\}\).

(i)     Show that \(\{ S,{\text{ }} \times \} \) is a group where \( \times \) denotes multiplication of complex numbers.

(ii)     Giving a reason, state whether or not \(\{ S,{\text{ }} \times \} \) is cyclic.

(i)     closure: let \({a_1} = {\text{cis}}\left( {\frac{{{n_1}\pi }}{4}} \right)\) and \({a_2} = {\text{cis}}\left( {\frac{{{n_2}\pi }}{4}} \right) \in S\)     M1

then \({a_1} \times {a_2} = {\text{cis}}\left( {\frac{{({n_1} + {n_2})\pi }}{4}} \right)\) (which \( \in S\) because the addition is carried out modulo 8)     A1

identity: the identity is 1 (and corresponds to \(n = 0\))     A1

inverse: the inverse of \({\text{cis}}\left( {\frac{{n\pi }}{4}} \right)\) is \({\text{cis}}\left( {\frac{{(8 - n)\pi }}{4}} \right) \in S\)     A1

associatively: multiplication of complex numbers is associative     A1

the four group axioms are satisfied so \(S\) is a group     AG

(ii)     \(S\) is cyclic     A1

because \({\text{cis}}\left( {\frac{\pi }{4}} \right)\), for example, is a generator     R1

[7 marks]