Theorem 6.15: Let G; be a cyclic group, and let g be a generator of G. Then the following results hold (1)If the order of g is infinite, then G; Z;+ (2)If the order of g is n, then G; =lZn; eI Proof:(1)G={gk∈Z}, q:G→>Z,(gk)=k (2)G={e,g,g2,g} q:G→>Zm,q(g)=kTheorem 6.15: Let [G; *] be a cyclic group, and let g be a generator of G. Then the following results hold. (1)If the order of g is infinite, then [G;*] [Z;+] (2)If the order of g is n, then [G;*][ Zn ;] Proof:(1)G={gk |kZ}, :G→Z, (gk )=k (2)G={e,g,g2 ,g n-1 }, :G→Zn , (gk )=[k]