❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧
♣♦r●❧❛✉❜❡r ❉❛♥t❛s ▼♦r❛✐s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡s✲ tr❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣♦✐s s❡♠ ❡❧❡ ♥❛❞❛ s❡r✐❛ ♣♦ssí✈❡❧✳ ❆♦s ♠❡✉s ♣❛✐s ❏♦ã♦ ❞❡ ❉❡✉s ▼♦r❛✐s ❡ ▼❛r✐❛ ■s❛❜❡❧ ❉❛♥t❛s ▼♦r❛✐s q✉❡ ♠❡ ❡♥s✐♥❛r❛♠ q✉❡ ❛ ❡❞✉❝❛çã♦ é ❛ ♠❡❧❤♦r ❤❡r❛♥ç❛ ❞❡✐①❛❞❛ ♣♦r ❡❧❡s✳ ❆♦s ♠❡✉s ✐r♠ã♦s ❨✉r✐ ❉❛♥t❛s ▼♦r❛✐s ❡ ❆❧❞r✐♥ ❏♦sé ❉❛♥t❛s ▼♦r❛✐s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❡s♣♦s❛s ❙✉③② ❑❛r✐♥❡ ❡ ❉❛♥✐❡❧❧❡ ❇❛r❜♦s❛✱ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❡ ❛♣♦✐❛r❛♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣❡r❝✉rs♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦❧❡❣❛s q✉❡ t✐✈❡ ♥❛ ✈✐❞❛ q✉❡ ❝♦♠♣❛rt✐❧❤❛r❛♠ ❝♦♠✐❣♦ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ s✉❛s ❡①♣❡r✐ê♥❝✐❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛q✉❡❧❡s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞♦ P❘❖❋▼❆❚ ♥♦ ♣♦❧♦ ❞❡ ❏♦ã♦ P❡ss♦❛✳
❆♦ ♣r♦❢❡ss♦r ❏♦ã♦ ▼❛r❝♦s ❞♦ Ó ❡ ❛ ♣r♦❢❡ss♦r❛ ❋❧á✈✐❛ ❏❡rô♥✐♠♦ ♣❡❧♦s s❡✉s ❡s❢♦rç♦s ♣❛r❛ ❢❛③❡r ❞♦ P❘❖❋▼❆❚ ✉♠❛ r❡❛❧✐❞❛❞❡ ♥❛ ❯❋P❇✳
❆♦ ♣r♦❢❡ss♦r ❇r✉♥♦ ❘✐❜❡✐r♦ q✉❡ ♠❡ ❛❥✉❞♦✉ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ ❝✉rs♦✱ ♣r✐♥❝✐✲ ♣❛❧♠❡♥t❡ ♥❡ss❛ ❞✐ss❡rt❛çã♦✱ s❛♥❛♥❞♦ ❛s ❡✈❡♥t✉❛✐s ❞ú✈✐❞❛s q✉❡ s✉r❣✐r❛♠ ❡ s✉❣❡r✐♥❞♦ t❡①t♦s ♣❛r❛ ❡♥r✐q✉❡❝❡r ♦ tr❛❜❛❧❤♦✳
❖ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡s❝r✐t♦ ♣❡❧♦ ❡❣í♣❝✐♦ ❚❛❤❡r ❊❧●❛♠❛❧ ❝❛❧❝✉❧❛ ❧♦❣❛✲ r✐t♠♦s ❞✐s❝r❡t♦s ❝♦♠ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ●r✉♣♦ ❈í❝❧✐❝♦ ✜♥✐t♦ G✳ ❊ss❡s ❡❧❡♠❡♥t♦s
♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❡st✉❞❛r❡♠♦s ♥♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦ ✶✳ ❈♦♥❤❡❝❡♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡st✉❞❛❞❛s✱ ♣♦❞❡r❡♠♦s ❞❡✜♥✐r ❡ ❝❛❧❝✉❧❛r ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s✱ ✉t✐❧✐③❛♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❛ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❈♦♥❣r✉ê♥❝✐❛s✱ ❜❡♠ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s✳ ❱❛♠♦s ❡st✉❞❛r ❛❧❣♦r✐t♠♦s ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r ♦ ❛❧❣♦r✐t♠♦ ❡s❝r✐t♦ ♣♦r ❊❧●❛♠❛❧✱ ❜✉s❝❛♥❞♦ ❡♥t❡♥❞❡r ❛s ❞✐✜❝✉❧❞❛❞❡s ❛♣r❡s❡♥t❛❞❛s ♣♦r ❡❧❡ ❡ ♠♦str❛r s✉❛s ❛♣❧✐❝❛çõ❡s ♥♦ ❝❛♠♣♦ ❞❛ ❈r✐♣t♦❣r❛✜❛✳ ❆♣r❡s❡♥✲ t❛r❡♠♦s ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❛t✐✈✐❞❛❞❡s✱ ✈♦❧t❛❞❛s ♣❛r❛ ❡st✉❞❛♥t❡s ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✈✐s❛♥❞♦ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❡ ❛❧❣✉♥s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s ♥♦ tr❛❜❛❧❤♦✳
P❛❧❛✈r❛s ❝❤❛✈❡✿ ❊❧●❛♠❛❧✱ ❣r✉♣♦s ❝í❝❧✐❝♦s✱ r❛✐③ ♣r✐♠✐t✐✈❛✱ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✱ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❝❤❛✈❡ ♣ú❜❧✐❝❛✳
❚❤❡ ❡♥❝r②♣t✐♦♥ ❛❧❣♦r✐t❤♠ ✇r✐tt❡♥ ❜② ❊❣②♣t✐❛♥ ❚❛❤❡r ❊❧●❛♠❛❧ ❝♦♠♣✉t❡s ❞✐s✲ ❝r❡t❡ ❧♦❣❛r✐t❤♠s ✇✐t❤ ❡❧❡♠❡♥ts ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❈②❝❧✐❝❛❧✳ ❚❤❡s❡ ❡❧❡♠❡♥ts ❤❛✈❡
♣r♦♣❡rt✐❡s t❤❛t ❞✉r✐♥❣ t❤❡ st✉❞② ❈❤❛♣t❡r ✶✳ ❑♥♦✇✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ s♦♠❡ ♣r♦✲ ♣❡rt✐❡s st✉❞✐❡❞✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ❛♥❞ ❝♦♠♣✉t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s✱ ✉s✐♥❣ ❦♥♦✇❧❡❞❣❡ ♦❢ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❝♦♥❣r✉❡♥❝❡ ♦❢ ❘❡♠❛✐♥s ❛♥❞ ❚❤❡♦r❡♠ ❘❡♠❛✐♥❞❡r ♦❢ ❈❤✐♥❡s❡✳ ❲❡ ✇✐❧❧ st✉❞② ♣✉❜❧✐❝ ❦❡② ❛❧❣♦r✐t❤♠s✱ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ❛❧❣♦r✐t❤♠ ✇r✐tt❡♥ ❜② ❊❧●❛♠❛❧✱ s❡❡❦✐♥❣ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❞✐✣❝✉❧t✐❡s ♣r❡s❡♥t❡❞ ❜② ✐t ❛♥❞ s❤♦✇ ✐ts ❛♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❝r②♣t♦❣r❛♣❤②✳ ❲❡ ♣r❡s❡♥t ❛ s❡q✉❡♥❝❡ ♦❢ ❛❝t✐✈✐t✐❡s✱ ❛✐♠❡❞ ❛t st✉❞❡♥ts ♦❢ t❤❡ ✜rst ❣r❛❞❡ ♦❢ ❤✐❣❤ s❝❤♦♦❧✱ t❛r❣❡t✐♥❣ t❤❡ ❧❡❛r♥✐♥❣ ♦❢ s♦♠❡ s✉❜❥❡❝ts ❝♦✈❡r❡❞ ❛t ✇♦r❦✳
❑❡②✇♦r❞s✿ ❊❧●❛♠❛❧✱ ❝②❝❧✐❝ ❣r♦✉♣s✱ ♣r✐♠✐t✐✈❡ r♦♦t✱ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠✱ ❡♥❝r②♣✲ t✐♦♥ ❛❧❣♦r✐t❤♠✱ ♣✉❜❧✐❝ ❦❡②✳
✶ ●r✉♣♦s✱ ❙✉❜❣r✉♣♦s ❡ ♦ ●r✉♣♦ Zp ✶
✶✳✶ ●r✉♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ●r✉♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✷ ❙✉❜❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❖ ❣r✉♣♦ Zp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✸ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ P❡rs♦♥❛❧✐❞❛❞❡s ▼❛t❡♠át✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹✳✶ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹✳✷ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷ ▲♦❣❛r✐t♠♦s ❉✐s❝r❡t♦s ✶✾
✷✳✶ ▲♦❣❛r✐t♠♦s ❉✐s❝r❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❆❧❣♦r✐t♠♦ ❞❡ ❙✐❧✈❡r✱ P♦❤❧✐❣ ❡ ❍❡❧❧♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸ ❆❧❣♦r✐t♠♦ ❡ ❈r✐♣t♦❣r❛✜❛ ✷✻
✸✳✶ ❆❧❣♦r✐t♠♦ ❡ ❈r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶✳✶ ❆❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶✳✷ ❈r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ❙✐st❡♠❛s ❞❡ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷✳✶ ❖ ❙✐st❡♠❛ ❉✐✣❡✲❍❡❧❧♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷✳✷ ❈r✐♣t♦s✐st❡♠❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹ ❆❧❣♦r✐t♠♦ ❞❡ ❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧ ✹✵
✹✳✶ ❖ ❆❧❣♦r✐t♠♦ ❞❡ ❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✶✳✶ ❱❡r✐✜❝❛♥❞♦ ❛ ❛✉t❡♥t✐❝✐❞❛❞❡ ❞♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ P♦ssí✈❡✐s ❆t❛q✉❡s ❛♦ ❈r✐♣t♦ss✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷✳✶ ❘❡❝✉♣❡r❛♥❞♦ ❛ ❝❤❛✈❡ ♣❛rt✐❝✉❧❛r xA ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹✳✷✳✷ ❋♦r❥❛♥❞♦ ❛ss✐♥❛t✉r❛s s❡♠ r❡❝✉♣❡r❛r ❛ ❝❤❛✈❡ ♣❛rt✐❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✹✹
✺ ❆t✐✈✐❞❛❞❡s ♣❛r❛ s❛❧❛ ❞❡ ❛✉❧❛ ✹✺
✺✳✶ ❆❧❣♦r✐t♠♦s ♣❛r❛ ❛♣r❡♥❞❡r ❚❡♦r❡♠❛s ❡ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✺✳✶✳✶ ❆t✐✈✐❞❛❞❡ ✶ ✲ ❉❡t❡r♠✐♥❛♥❞♦ ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
p ✳ ✳ ✳ ✳ ✳ ✹✺
✶✳✶ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶ ❆❜✉ ❏❛❢❛r ▼♦❤❛♠❡❞ ✐❜♥ ▼✉s❛ ❛❧✲❑❤✇❛r✐③♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✸ ▲❡♦♥❛r❞♦ ❞❛ ❱✐♥❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ❉❛♥ ❇r♦✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ❊sq✉❡♠❛ ❞♦ ❆❧❣♦r✐t♠♦ ❆ss✐♠étr✐❝♦ ✭❝♦♥str✉çã♦ ❞♦ ♣ró♣r✐♦ ❛✉t♦r✮ ✳ ✳ ✸✻ ✸✳✻ ❲❤✐t✜❡❧❞ ❉✐✣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✼ ▼❛rt✐♥ ❍❡❧♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✽ ❘♦♥❛❧❞ ❘✐✈❡st✱ ❆❞✐ ❙❤❛♠✐r ❡ ▲❡♦♥❛r❞ ❆❞❧❡♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✶ ❚❛❤❡r ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✶ ❈❛❧❝✉❧❛❞♦r❛ ❈✐❡♥tí✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ✜♥❛❧✐❞❛❞❡ ❡①♣❧✐❝❛r ❛ ♠❛t❡♠át✐❝❛ ♣♦r trás ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧✳ ❆tr❛✈és ❞❡❧❡✱ ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ♥í✈❡❧ ❋✉♥❞❛✲ ♠❡♥t❛❧ ❡ ▼é❞✐♦ ♣♦❞❡♠ ❡①♣❧✐❝❛r ❝♦♠♦ ❛❝♦♥t❡❝❡ ❛ tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s s✐❣✐❧♦s❛s ♣❡❧❛ ✐♥t❡r♥❡t✱ ❛ss✉♥t♦ q✉❡ ❣❡r❛ ✐♥t❡r❡ss❡ ❡♠ ♣❡ss♦❛s q✉❡ ❣♦st❛♠ ❞❡ ❛♣r❡♥❞❡r ❛ ❧✐♥❣✉❛❣❡♠ ❝♦♠♣✉t❛❝✐♦♥❛❧✳ ❉❡ss❛ ❢♦r♠❛ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ✐♥❝❡♥t✐✈❛r s❡✉s ❛❧✉♥♦s ♥❛ ❛♣r❡♥❞✐③❛✲ ❣❡♠ ❞❛ ✧▼❛t❡♠át✐❝❛ ❆❜str❛t❛✧✱ ♣♦❞❡♥❞♦ ✐♥❝❧✉s✐✈❡ ❢❛③❡r ❝♦♠ q✉❡ ❡ss❡ ❛❧✉♥♦ ✐♥❣r❡ss❡ ♥✉♠ ❝✉rs♦ ❞❡ ♠❛t❡♠át✐❝❛ ♥♦ ♥í✈❡❧ ❙✉♣❡r✐♦r✳
❆ ♣r✐♥❝✐♣❛❧ ❞✐✜❝✉❧❞❛❞❡ ♥❡ss❡ ❛❧❣♦r✐t♠♦ é ♦ ❝á❧❝✉❧♦ ❞❡ ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s q✉❡✱ ❞✐❢❡r❡♥t❡ ❞♦ ❧♦❣❛r✐t♠♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ♣♦❞❡♠ ♥ã♦ ♣♦ss✉✐r s♦❧✉çã♦✳ P❛r❛ ❡♥✲ t❡♥❞❡r♠♦s ❡ss❛ ❞✐✜❝✉❧❞❛❞❡✱ ❝♦♠❡ç❛♠♦s ♠♦str❛♥❞♦ ❛ ❚❡♦r✐❛ ❞♦s ●r✉♣♦s ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❡♠ ♣❛rt✐❝✉❧❛r ❡①♣❧✐❝❛♠♦s ❝♦♠♦ ❡♥❝♦♥tr❛r ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❝í❝❧✐❝♦Z∗
p✱
❝♦♠p♣r✐♠♦✳ ❉❡st❛❝❛♠♦s ♦ ✧P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✧ ❡ ♦ ✧❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s
❘❡st♦s✧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞♦s ❝á❧❝✉❧♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳
◆♦ ❝❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s✱ ♠♦str❛♥❞♦ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ b ❞♦ ❣r✉♣♦ Z∗
p ♥❛
❜❛s❡ a✱ ❝♦♠ a, b ∈ Z∗
p✱ ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ s❡r✈❡ ♣❛r❛ ❝❛❧❝✉❧❛r ❡ss❡
❧♦❣❛r✐t♠♦ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ♣r✐♠♦p❢♦r ♠✉✐t♦ ❣r❛♥❞❡✱ ♦ ✧❆❧❣♦r✐t♠♦ ❞❡ ❙✐❧✈❡r✱ P♦❤❧✐❣
❡ ❍❡❧❧♠❛♥✧✳ P♦ré♠✱ ♠❡s♠♦ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ss❡ ❧♦❣❛r✐t♠♦✱ ♦ ❧❡✐t♦r ♣♦❞❡ ♣❡r❝❡❜❡r q✉❡ ♦s ❝á❧❝✉❧♦s sã♦ ❝♦♠♣❧❡①♦s ❡ ❞❡♠♦r❛❞♦s✱ ❡ss❡ ❢❛t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧Pr♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✧ ❡ é ❛ ❜❛s❡ ♣❛r❛ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ r❡❛❧✐③❛♠♦s ✉♠❛ ✐♥tr♦❞✉çã♦ ❤✐stór✐❝❛ s♦❜r❡ ❛❧❣♦r✐t♠♦ ❡ ❝r✐♣t♦❣r❛✜❛✱ ♠♦str❛♥❞♦ ❡①❡♠♣❧♦s ♥♦ ❝♦t✐❞✐❛♥♦ ❡s❝♦❧❛r ❞❡ ✉♠ ❛❧✉♥♦ ♥♦ ♥í✈❡❧ ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦✳ ❉❡♣♦✐s ❡①♣❧✐❝❛♠♦s ♦ q✉❡ é ❡ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ♦s s✐st❡♠❛s ❞❡ ❝r✐♣t♦❣r❛✜❛ ❞❡ ✧❝❤❛✈❡ ♣ú❜❧✐❝❛✧✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ✧❝r✐♣t♦ss✐st❡♠❛s✧✱ ❡ ♠♦str❛♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❞❡ss❡s s✐st❡♠❛s✱ ♦ ♠❛✐s ❛♥t✐❣♦ ❡ ♦ ♠❛✐s ✉t✐❧✐③❛❞♦ ♥❛ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s s✐❣✐❧♦s❛s✳
❖ ❝❛♣ít✉❧♦ ✹ é ❞❡❞✐❝❛❞♦ ❛♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❊❧●❛♠❛❧✳ ◆❡❧❡ ♠♦str❛✲ ♠♦s ❝♦♠♦ ❛❝♦♥t❡❝❡ ❛ tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s ❡ ❡①♣❧✐❝❛♠♦s ❛ ♠❛t❡♠át✐❝❛ q✉❡ é ✉t✐❧✐③❛❞❛ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✳ ▼♦str❛♠♦s t❛♠❜é♠ ❛s ❞✉❛s ❢♦r♠❛s q✉❡ ✉♠ ✐♥✈❛s♦r ♣♦❞❡r✐❛ t❡♥t❛r q✉❡❜r❛r ♦ ❛❧❣♦r✐t♠♦✱ r❡❝✉♣❡r❛♥❞♦ ❛ ❝❤❛✈❡ ♣❛rt✐❝✉❧❛r ♦✉ ❢♦r❥❛♥❞♦ ❛ss✐♥❛t✉r❛s s❡♠ r❡❝✉♣❡r❛r ❛ ❝❤❛✈❡ ♣❛rt✐❝✉❧❛r✳ ▼♦str❛♠♦s q✉❡ sã♦ ♣♦✉❝❛s
❞❡ ❛✉❧❛ ❞❛ ✶♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✈✐s❛♥❞♦ ♠♦str❛r ❝♦♠♦ ✉t✐❧✐③❛r ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧ ♣❛r❛ tr♦❝❛r ♠❡♥s❛❣❡♥s ❝r✐♣t♦❣r❛❢❛❞❛s ❡ ❡st✐♠✉❧❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦s ❛❧✉♥♦s s♦❜r❡ ♦s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s ♥♦ tr❛❜❛❧❤♦✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡ ✉t✐❧✐③❛r ❡ss❛s ❛t✐✈✐❞❛❞❡ ♣❛r❛ ❝r✐❛r ♦✉tr❛s ❝♦♠ ❛ ♠❡s♠❛ ✜♥❛❧✐❞❛❞❡✱ ❝♦♠♦ ✧r❡s♦❧✈❡r ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s✧ ♦✉ ✧❝❛❧❝✉❧❛r ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ❆❧❣♦r✐t♠♦ ❞❡ ❙✐❧✈❡r✱ P♦❤❧✐❣ ❡ ❍❡❧❧♠❛♥✧✳
●r✉♣♦s✱ ❙✉❜❣r✉♣♦s ❡ ♦ ●r✉♣♦
Z
p
❊st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❣r✉♣♦s✱ s✉❜❣r✉♣♦s ❡ ♦ ❣r✉♣♦ ❝í❝❧✐❝♦Zp✱
❝♦♠ p ♣r✐♠♦✳ ❊ss❡ ❡st✉❞♦ s❡rá ❡str✐t❛♠❡♥t❡ ♥❡❝❡ssár✐♦ ♣❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦s ❧♦❣❛✲
r✐t♠♦s ❞✐s❝r❡t♦s✱ ❜❡♠ ❝♦♠♦ ♦ ❡st✉❞♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧✱ ♣♦✐s ♦ ♠❡s♠♦ é ❞❡✜♥✐❞♦ ❝♦♠ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ✜♥✐t♦✱ ♣♦r ✐ss♦ ♣r❡❝✐s❛♠♦s s❛❜❡r ✧♦ q✉❡ s✐❣♥✐✜❝❛ ✉♠ ❣r✉♣♦ s❡r ❝í❝❧✐❝♦ ❡ ✜♥✐t♦❄✧ P❛r❛ r❡s♣♦♥❞❡r ❡ss❛ ♣❡r❣✉♥t❛ ✈❛♠♦s ♣r✐♠❡✐r♦ ❞❡✜♥✐r ✧❣r✉♣♦✧ ❡ ❡♠ s❡❣✉✐❞❛ ✈❛♠♦s ✈❡r ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳
✶✳✶ ●r✉♣♦
✶✳✶✳✶ ●r✉♣♦
❖ ❡st✉❞♦ ❞♦s ●r✉♣♦s é r❡❛❧✐③❛❞♦ ♥♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r ❡♠ ❞✐s❝✐♣❧✐♥❛s ✐♥✐❝✐❛✐s ❞❡ ➪❧❣❡❜r❛✱ ❡ss❡ ❡st✉❞♦ é ✐♠♣♦rt❛♥t❡✱ ♣♦✐s ♥❡❧❡ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❡ ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ❞❡✈❡ s❛t✐s❢❛③❡r ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ♣❛r❛ ♦❜t❡r♠♦s ✉♠ ❣r✉♣♦✳ ❊♠ ❬✸❪ ✧●r✉♣♦✧ é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❡✜♥✐çã♦ ✶ ❯♠ ❝♦♥❥✉♥t♦ G ❝♦♠ ✉♠❛ ♦♣❡r❛çã♦
G×G−→G
(a, b)7−→a·b
é ✉♠ ❣r✉♣♦ s❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✶✳ ❆ ♦♣❡r❛çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ✐st♦ é✱
a·(b·c) = (a·b)·c, ∀a, b, c∈G.
✷✳ ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ ✐st♦ é✱
∃ e∈G t❛❧ q✉❡ a·e=e·a=a.
✸✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ ♣♦ss✉✐ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦✱ ✐st♦ é✱
∀ a∈G, ∃ b ∈G t❛❧ q✉❡ a·b=·a=e.
❉❡♥♦t❛♠♦s b ♣♦r a−1
◆♦t❡ q✉❡ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❣r✉♣♦✱ é ♣r❡❝✐s♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❡ ❛ ♦♣❡r❛çã♦ t❡♥❤❛♠ ✧❡❧❡♠❡♥t♦ ♥❡✉tr♦✧ ❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡✈❡♠ ♣♦ss✉✐r ✧❡❧❡♠❡♥t♦ ✐♥✈❡rs♦✧✳ ➱ ❝♦♠✉♠ ❡♥tã♦ ❛♣❛r❡❝❡r ❛s s❡❣✉✐♥t❡s ♣❡r❣✉♥t❛s✿ ✧❙❡rá q✉❡ ❡①✐st❡ ♠❛✐s ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❡♠ ✉♠ ❣r✉♣♦❄✧ ♦✉ ✧❛❧❣✉♠ ❡❧❡♠❡♥t♦ ♣♦❞❡ ♣♦ss✉✐r ♠❛✐s ❞❡ ✉♠ ✐♥✈❡rs♦❄✧ ❖s t❡♦r❡♠❛s ❛ s❡❣✉✐r r❡s♣♦♥❞❡♠ ❡ss❛s ♣❡r❣✉♥t❛s✳
❚❡♦r❡♠❛ ✷ ❖ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ ✉♠ ❣r✉♣♦ é ú♥✐❝♦✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡♠ e, e′ ∈ Gt❛✐s q✉❡ ♦s ❞♦✐s s❡❥❛♠ ❡❧❡♠❡♥t♦s ♥❡✉tr♦s ❞❡ G✳ ❈♦♠♦e é ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ Gt❡♠♦s q✉❡✿
e=e·e′
❡ e′ é ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ G✱ t❡♠♦s q✉❡✿
e′ =e′·e
∴e′ =e
P♦rt❛♥t♦ ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ ✉♠ ❣r✉♣♦ é ú♥✐❝♦✳
❚❡♦r❡♠❛ ✸ ❖ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❣r✉♣♦ é ú♥✐❝♦✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡♠ b, c∈G ❞♦✐s ❡❧❡♠❡♥t♦s ✐♥✈❡rs♦s ❞❡ a∈G✳
❉❛ ❞❡✜♥✐çã♦ ❞❡ ❣r✉♣♦ t❡♠♦s✿
b =e·b = (c·a)·b=c·(a·b) =c·e=c
P♦rt❛♥t♦ ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❣r✉♣♦ é ú♥✐❝♦✳
❉✐③❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ é ❛❜❡❧✐❛♥♦ ♦✉ ❝♦♠✉t❛t✐✈♦ s❡ ❡❧❡ t❛♠❜é♠ ❛♣r❡s❡♥t❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛✱ ✐st♦ é✱
a·b=b·a, ∀a, b∈G
❊ss❡ t✐♣♦ ❞❡ ❣r✉♣♦ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♥♦ss♦s ❡st✉❞♦s✱ ♣♦✐s ✈❡r❡♠♦s ❛✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦ q✉❡ t♦❞♦ ❣r✉♣♦ ❝í❝❧✐❝♦ é ❛❜❡❧✐❛♥♦✳
✶✳✶✳✷ ❙✉❜❣r✉♣♦s
◆♦ ❡st✉❞♦ ❞♦s ❣r✉♣♦s é ❡ss❡♥❝✐❛❧ s❛❜❡r♠♦s ♦ q✉❡ é ✉♠ s✉❜❣r✉♣♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ é ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ ♣r✐♥❝✐♣❛❧ ❡st✉❞❛❞♦✱ ♦✉ s❡❥❛✱ ♦ ❣r✉♣♦Z∗
p é ✉♠ s✉❜❣r✉♣♦ ❞❡ Z✳ ❙✉❜❣r✉♣♦ é ❞❡✜♥✐❞♦ ❡♠ ❬✸❪ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❡✜♥✐çã♦ ✹ ❙❡❥❛ ✭G,·✮ ✉♠ ❣r✉♣♦✳ ❯♠ ❙✉❜❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ H ❞❡ G é ✉♠ s✉❜✲
❣r✉♣♦ ❞❡ G✭❞❡♥♦t❛♠♦s H < G✮ q✉❛♥❞♦✱ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ G✱ ♦ ❝♦♥❥✉♥t♦ H é ✉♠
❣r✉♣♦✱ ✐st♦ é✱ q✉❛♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✶✳ h1·h2 ∈H,∀ h1, h2 ∈H❀
✷✳ h1·(h2·h3) = (h1·h2)·h3,∀ h1, h2, h3 ∈H
✸✳ ∃ e ∈H t❛❧ q✉❡ e·h=h·e=h,∀ h∈H
✹✳ ♣❛r❛ ❝❛❞❛ h∈H ∃ h−1 ∈H t❛❧ q✉❡ h·h−1 =h−1·h=e
◆♦t❡ q✉❡ ❛s ❝♦♥❞✐çõ❡s ✷✱ ✸✱ ✹ ❞❡❝♦rr❡♠ ❞♦ ❢❛t♦ ❞❡ H s❡r ✉♠ ❣r✉♣♦✱ ❥á ❛ ♣r✐♠❡✐r❛
❝♦♥❞✐çã♦ ❞✐③ q✉❡ ❛ ♦♣❡r❛çã♦ ✭·✮ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡♠ H✳
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❡st❛❜❡❧❡❝❡ q✉❛♥❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ H é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ s❡♠ ♣r❡❝✐s❛r ✈❡r✐✜❝❛r s❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ ❉❡✜♥✐çã♦ ✹ sã♦ s❛t✐s❢❡✐t❛s✳
Pr♦♣♦s✐çã♦ ✶ ❙❡❥❛ H ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞♦ ❣r✉♣♦ G✳ H é ✉♠ s✉❜❣r✉♣♦ ❞❡ G s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✶✳ h1·h2 ∈H,∀ h1, h2 ∈H
✷✳ h−1 ∈H,∀ h∈H
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ H < G✱ t❡♠♦s q✉❡ ❛ ♦♣❡r❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡♠ H
✭❞❡✜♥✐çã♦ ❞❡ s✉❜❣r✉♣♦s✮✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ t❡♠ ✉♠ ✐♥✈❡rs♦ ♥♦ ♣ró♣r✐♦ s✉❜❣r✉♣♦✱ ♣♦✐s t♦❞♦ s✉❜❣r✉♣♦ é ✉♠ ❣r✉♣♦✳
❱❛♠♦s ♣r♦✈❛r ❛❣♦r❛ q✉❡ s❡ ❛♣❡♥❛s ❛s ❞✉❛s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✱ H é ✉♠
s✉❜❣r✉♣♦✱ ♦✉ s❡❥❛✱ ✈❛♠♦s ✈❡r✐✜❝❛r ❛s q✉❛tr♦ ❝♦♥❞✐çõ❡s ❞♦s s✉❜❣r✉♣♦s✳ ❆ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ é ❛ ♠❡s♠❛ ♥♦s ❞♦✐s ❝❛s♦s✱ ♣♦rt❛♥t♦ ❡❧❛ é s❛t✐s❢❡✐t❛✳ ❆ s❡❣✉♥❞❛ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛ ♣♦✐s ❛ ♦♣❡r❛çã♦ ❞♦ s✉❜❣r✉♣♦ é ❛ss♦❝✐❛t✐✈❛ ♣❛r❛ t♦❞♦s ❡❧❡♠❡♥t♦s ❞❡ H✳
❙❡❥❛ h ∈ H ♣♦r ✷ t❡♠♦s q✉❡ h−1 ∈ H✱ ♣♦r ✶ t❡♠♦s q✉❡ h· h−1 ∈ H✱ ❝♦♠♦
h·h−1 =e✱ t❡♠♦s q✉❡ e∈H✱ ♦✉ s❡❥❛✱ ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ♣❡rt❡♥❝❡ ❛ H✱ ♣♦rt❛♥t♦✱ ❛
t❡r❝❡✐r❛ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛ ❡ ❛ q✉❛rt❛ ❝♦♥❞✐çã♦ t❛♠❜é♠✳
❱❛♠♦s ❛❣♦r❛ ❝♦♠❡ç❛r ❛ ✧❝♦♥str✉✐r✧ ♦ ❣r✉♣♦ Z∗
p✱ ♦✉ s❡❥❛✱ ✈❛♠♦s ♠♦str❛r ❝♦♠♦
sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦ ❡ ❛ ♦♣❡r❛çã♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛✳ Pr✐♠❡✐r♦ ✈❛♠♦s ❞❡✜♥✐r ✉♠❛ ♦♣❡r❛çã♦ ♣❛r❛ ❡ss❡ ❝♦♥❥✉♥t♦ q✉❡ s❛t✐s❢❛ç❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ s✉❜❣r✉♣♦✳
❉❡✜♥✐çã♦ ✺ ❙❡❥❛♠ ✭G,·✮ ✉♠ ❣r✉♣♦ ❡ a ∈ G✳ ❉❡✜♥✐♠♦s ❛s ♣♦tê♥❝✐❛s ❞❡ a ❞❛
s❡❣✉✐♥t❡ ❢♦r♠❛✿
a0 = e;
an = an−1·a, ❝♦♠ n∈N;
a−n = (an)−1, ❝♦♠ n∈N.
❉❡♥♦t❛♠♦s ♣♦r hai ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣♦tê♥❝✐❛s ❞❡ a✱ ♦✉ s❡❥❛✿
hai={an : n
∈Z}
❱❛♠♦s ♠♦str❛r ❛ s❡❣✉✐r q✉❡ ❡ss❛ ❉❡✜♥✐çã♦ ❢❛③ ❝♦♠ q✉❡ hai s❡❥❛ s✉❜❣r✉♣♦ G✱
❞❡ss❛ ❢♦r♠❛ ♣♦❞❡r❡♠♦s ❝❛r❛❝t❡r✐③❛r ♦ ●r✉♣♦ Z∗
p✳
❆✜r♠❛çã♦ ✶ hai é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ✈❡r✐✜❝❛r s❡ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣♦s✐çã♦ ✶ sã♦ s❛t✐s✲ ❢❡✐t❛s✿
✶✳ ❙❡❥❛♠ n, m ∈ Z✱ t❡♠♦s q✉❡ an, am ∈ hai✱ ❝♦♠ ✐ss♦ an·am = an+m✳ ❈♦♠♦
n+m ∈Z⇒an+m
∈ hai❀
✷✳ P♦r ❞❡✜♥✐çã♦✱a−1 ∈ hai✳
P♦rt❛♥t♦✱ hai é s✉❜❣r✉♣♦ ❞❡G✳
◆♦t❡ q✉❡ ♥♦ ❝♦♥❥✉♥t♦ ❞❛s ♣♦tê♥❝✐❛s ♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠ s✉❜❣r✉♣♦ ✉t✐❧✐③❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✉♠ ❡❧❡♠❡♥t♦✱ ❝♦♠ ✐ss♦ ♣♦❞❡r❡♠♦s ❝❛r❛❝t❡r✐③❛r ❡ss❡ ❡❧❡♠❡♥t♦ ❝♦♠♦ ♦ ✧❣❡r❛❞♦r✧ ❞♦ s✉❜❣r✉♣♦✳
❉❡✜♥✐çã♦ ✻ hai é ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r a✳ ❈❤❛♠❛♠♦s a ❞❡ ❣❡r❛❞♦r ❞❡ hai
P♦❞❡♠♦s ❛❣♦r❛ ❞❡♥♦♠✐♥❛r ❣r✉♣♦s q✉❡ ♣♦❞❡♠ s❡r ❣❡r❛❞♦s ♣♦r ✉♠ ❡❧❡♠❡♥t♦✱ ❝♦♠ ✐ss♦ ✈♦❧t❛r❡♠♦s ♥♦ss♦s ❡st✉❞♦s ♣❛r❛ ❡ss❡ t✐♣♦ ❞❡ ❣r✉♣♦✳
❉❡✜♥✐çã♦ ✼ ❯♠ ❣r✉♣♦ Gé ❝í❝❧✐❝♦ q✉❛♥❞♦ ❡❧❡ ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❞❡ G✳
❆❣♦r❛ q✉❡ s❛❜❡♠♦s ♦ q✉❡ é ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦✱ ♣♦❞❡♠♦s ♣r♦❝✉r❛r s❛❜❡r ♦ ✧t❛♠❛✲ ♥❤♦✧ ❞❡ss❡ ❣r✉♣♦✱ ♦✉ s❡❥❛✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ ✉♠❛ ❝❛r❞✐♥❛❧✐✲ ❞❛❞❡✱ ✉♠ ❣r✉♣♦ ♣♦❞❡ ♣♦ss✉✐r ✉♠ ✧t❛♠❛♥❤♦✧✱ ❜❛st❛ ✈❡r✐✜❝❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ss❡ ❣r✉♣♦✳
❉❡✜♥✐çã♦ ✽ ❆ ♦r❞❡♠ ❞❡ ✉♠ ❣r✉♣♦ G é ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❡♠ G✳ ❊❧❛ s❡rá
❞❡♥♦t❛❞❛ ♣♦r |G|✳
◆♦t❡ q✉❡ ✉♠ ❣r✉♣♦ ♣♦❞❡ t❡r ✉♠❛ ♦r❞❡♠ ✧✜♥✐t❛✧✱ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ss❡ ❣r✉♣♦ é ✜♥✐t♦✱ ♦✉ ✉♠❛ ♦r❞❡♠ ✐♥✜♥✐t❛✱ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ss❡ ❣r✉♣♦ é ✐♥✜♥✐t♦✳
❈♦♠♦ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ é ✉♠ ❣r✉♣♦ q✉❡ é ❣❡r❛❞♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ●r✉♣♦ G ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❡✜♥✐çã♦ ✾ ❈❤❛♠❛✲s❡ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ a ∈ G ❛♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n
t❛❧ q✉❡ an=e✳ ❯s❛♠♦s ❛ ♥♦t❛çã♦ O(a) ♣❛r❛ ✐♥❞✐❝❛r♠♦s ❛ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦✳
❆ ❖r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✶✳ O(e) = 1
❉❡♠♦♥str❛çã♦✿ ❚r✐✈✐❛❧
✷✳ O(a) = O(a−1)
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s ♣❡❧❛ ❉❡✜♥✐çã♦ ✾ q✉❡ O(a) = n ⇒ an = e✱ ♣❡❧❛
❉❡✜♥✐çã♦ ✺ a−n = (an)−1 = e✳ ❙❡❥❛ k ∈ {1,2, . . . , n − 1} ♣❡❧❛ ❉❡✜♥✐çã♦
✾ s❛❜❡♠♦s q✉❡ ak 6= e✱ ❛❣♦r❛ s✉♣♦♥❤❛ q✉❡ (a−1)k = e✱ ❡♥tã♦ ❡①✐st❡ j ∈
{1,2, . . . , n−1} t❛❧ q✉❡ (aj)−1 = e✱ ❛ss✐♠✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ✐♥✈❡rs♦✱ ❡①✐st❡
j ∈ {1,· · · , n−1} t❛❧ q✉❡ aj =e ♦ q✉❡ é ❛❜s✉r❞♦✳
✸✳ ❙❡ a=cbc−1 ❡♥tã♦ O(a) = O(b)
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ O(a) = n✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ O(b) = n✳
❚❡♠♦s q✉❡
an= (cbc−1)n= (cbc−1)· · ·(cbc−1) = cbnc−1
❆ss✐♠✱ ❞❡
cbnc−1 =an =e
s❡❣✉❡ q✉❡
bn=e
❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ ❡①✐st❡ j ∈ {1, . . . , n−1} t❛❧ q✉❡ bj = e✿ ❉❡ a = cbc−1
r❡s✉❧t❛ b =c−1ac❞♦♥❞❡ bj =c−1ajc=e✳ ❆ss✐♠✱ ❡①✐st❡ j
∈ {1, . . . , n−1} t❛❧
q✉❡ aj =e ♦ q✉❡ é ❛❜s✉r❞♦ ❥á q✉❡ O(a) = n✳
✹✳ O(am)
≤O(a);∀m ∈Z.
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ O(a) = n ❡ q✉❡ O(am) = k✿ ❖r❛✱ ♣❡❧❛
❉❡✜♥✐çã♦ ✾✱ k é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ (am)k = e✿ ▼❛s (am)n = e❀
❡♥tã♦ n ≥k✳
✺✳ ❙❡❥❛ O(a) =n✳ ❙❡ ♠❞❝✭m;n✮ ❂ ✶ ❡♥tã♦ O(am) = O(a) =n✳
❉❡♠♦♥str❛çã♦✿ P♦r ✹✱ t❡♠♦s q✉❡ O(am)
≤ n✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ O(a) =
n ≤O(am)✳ ❈♦♠♦ ♠❞❝✭m;n✮ ❂ ✶✱ t❡♠♦s q✉❡ ❡①✐st❡♠ ✐♥t❡✐r♦s x, y t❛✐s q✉❡
1 =xm+yn
❊♥tã♦✱ a = axm+yn = (am)x(an)y = (am)x✳ ❆ss✐♠✱ ❞❡ a = (am)x✱ ❝♦♥❝❧✉í♠♦s
q✉❡ O(a)≤O(am)✳
Pr♦♣♦s✐çã♦ ✷ ❙❡ O(a) =n ❡ m ∈Z✱ ❡♥tã♦ am =e s❡✱ ❡ s♦♠❡♥t❡ s❡✱ m=kn;k
∈ Z✱ ♦✉ s❡❥❛ m é ♠ú❧t✐♣❧♦ ❞❡ n✿
❉❡♠♦♥str❛çã♦✿ ✭❈♦♥❞✐çã♦ ◆❡❝❡ssár✐❛✮ ❙✉♣♦♥❤❛ q✉❡ O(a) = n ❡ am = e❀ ♦♥❞❡
m ∈ Z✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ m é ♠ú❧t✐♣❧♦ ❞❡ n✳ ❙✉♣♦♥❤❛ q✉❡ m ∈ Z+✳ ❊♥tã♦✱ ♣❡❧❛
❉❡✜♥✐çã♦ ✾✱ m≥n✳ ❚❡♠♦s q✉❡
m=nq+r;r ∈ {0, . . . , n−1};q, r∈Z+.
❊♥tã♦
e=am =anq+r=anqar = (an)qar =ar.
❆ss✐♠✱ r= 0 ♣♦✐s r∈ {0, . . . , n−1} ❡ O(a) =n✳ P♦rt❛♥t♦ am =anqar =anqe =anq
✳
✭❈♦♥❞✐çã♦ ❙✉✜❝✐❡♥t❡✮ ❙✉♣♦♥❤❛ q✉❡m=nk;k∈Z✳ ❊♥tã♦am =ank = (an)k=e✳
❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ✈❛✐ r❡❧❛❝✐♦♥❛r ❛ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❝♦♠ ♦ t❛♠❛♥❤♦ ❞♦ ❣r✉♣♦ ❝í❝❧✐❝♦✳ ❈♦♠♦ ❝♦r♦❧ár✐♦ ❞❡ss❡ t❡♦r❡♠❛✱ ♣♦❞❡r❡♠♦s s❛❜❡r s❡ ✉♠ ❡❧❡♠❡♥t♦ é ❣❡r❛❞♦r ❛tr❛✈és ❞❛ ♦r❞❡♠ ❞❡ss❡ ❡❧❡♠❡♥t♦✳
❚❡♦r❡♠❛ ✶✵ ❙❡❥❛ a∈ G✳ ❙❡ O(a) = n✱ ❡♥tã♦ hai é ✉♠ s✉❜❣r✉♣♦ ❝í❝❧✐❝♦ G ❝♦♠ n
❡❧❡♠❡♥t♦s✳
❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ a∈G✳ P❡❧❛ ❉❡✜♥✐çã♦ ✾ t❡♠♦s O(a) =n ⇒an =e
❆❣♦r❛ ❝❛❧❝✉❧❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ a✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✺✱ ♦❜t❡♠♦s an+1 =an
·a=e·a=a an+2 =an+1·a =a·a =a2 an+3 =an+2·a=a2·a=a3
✳✳✳
an+n−1 =an
·an−1 =e
·an−1 =an−1
an+n=an−1·a=an=e
❈♦♠ ✐ss♦✱ t❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ hai sã♦ ♦❜t✐❞♦s ❞❡ ❢♦r♠❛ ❝í❝❧✐❝❛ ❡ hai =
{e, a, a2, . . . , an−1}✳ P♦rt❛♥t♦✱ hai é ✉♠ s✉❜❣r✉♣♦ ❝í❝❧✐❝♦ ❝♦♠ n ❡❧❡♠❡♥t♦s
❖ ❈♦r♦❧ár✐♦ ❛ s❡❣✉✐r é ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞♦ ❚❡♦r❡♠❛ ✶✵✳
❈♦r♦❧ár✐♦ ✷✳✶ |G|=n ❡ O(a) = n s❡✱ ❡ s♦♠❡♥t❡ s❡✱ hai=G✳
❖ ❝♦r♦❧ár✐♦ ❛❝✐♠❛ ♥♦s ❣❛r❛♥t❡ q✉❡ a é ✉♠ ❣❡r❛❞♦r ❞❡ G q✉❛♥❞♦ ♦ s✉❜❣r✉♣♦
❣❡r❛❞♦ ♣♦r a ♣♦ss✉✐ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G✳ ❊ss❡ ❢❛t♦ s❡rá
❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ♣ró①✐♠❛ s❡çã♦✱ ♣♦✐s é ❝♦♠ s❡✉ ✉s♦ q✉❡ ♣♦❞❡r❡♠♦s s❛❜❡r q✉❛♥❞♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Z∗
p ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ❛❧❣✉♠ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳
❈♦♠ ❡ss❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❞❛ t❡♦r✐❛ ❞♦s ❣r✉♣♦s✱ ♣♦❞❡♠♦s ❝♦♠❡ç❛r ❛ ❡st✉❞❛r ♦ ❣r✉♣♦ ❝í❝❧✐❝♦ Z∗
p✳ ❆s ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s ❛♣r❡s❡♥t❛❞♦s s❡r✈✐rã♦ ♣❛r❛ ✧❝♦♥str✉✐r✧
❡ss❡ ❣r✉♣♦✳
✶✳✷ ❖ ❣r✉♣♦
Z
p❊st✉❞❛r❡♠♦s ❛❣♦r❛ ❛ t❡♦r✐❛ ❞♦s ❣r✉♣♦s ❝í❝❧✐❝♦s ✜♥✐t♦s Zp ❝♦♠ p ♣r✐♠♦✳ ❊ss❡
❡st✉❞♦ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♣♦❞❡r♠♦s tr❛❜❛❧❤❛r ❝♦♠ ♦s ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s✱ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❞❡ss❡ t❡①t♦✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❞❡✜♥✐r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❡♥tr❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✿
❉❡✜♥✐çã♦ ✶✶ ❙❡❥❛ n ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❉✐r❡♠♦s q✉❡ ❞♦✐s ♥ú✲
♠❡r♦s ✐♥t❡✐r♦s a ❡ b sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ n q✉❛♥❞♦ ♦s r❡st♦s ❞❡ s✉❛ ❞✐✈✐sã♦
❡✉❝❧✐❞✐❛♥❛ ♣♦r n sã♦ ✐❣✉❛✐s✳ ◗✉❛♥❞♦ ♦s ♥ú♠❡r♦s a ❡ b sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ n✱
❡s❝r❡✈❡♠♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
a≡b(mod n)
❖❜s❡r✈❛çã♦ ✶ ◆♦t❡ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♣♦✐s✿
✶✳ a≡a(mod n)✭r❡✢❡①✐✈❛✮❀
✷✳ a≡b(mod n)⇒b≡a(mod n)✭s✐♠étr✐❝❛✮❀
✸✳ a≡b(mod n) e b≡c(mod n)⇒a≡c(mod n)✭tr❛♥s✐t✐✈❛✮✳
❈♦♠♦ ♦s ♣♦ssí✈❡✐s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ ✉♠ ♥ú♠❡r♦ a ♣♦r n é ✉♠
♥ú♠❡r♦ ♠❡♥♦r ❞♦ q✉❡ n✱ é ❢á❝✐❧ ✈❡r q✉❡ ❡ss❡ ♥ú♠❡r♦ é ❝♦♥❣r✉❡♥t❡ ❛ ✉♠✱ ❡ s♦♠❡♥t❡
✉♠✱ ❞♦s ♥ú♠❡r♦s 0,1, . . . , n−1✳
➱ ♣♦ssí✈❡❧ s❛❜❡r s❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s sã♦ ❝♦♥❣r✉❡♥t❡s s❡♠ ♣r❡❝✐s❛r r❡❛❧✐③❛r ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ss❡s ♥ú♠❡r♦s ♣♦r n✱ ✐ss♦ ❢❛❝✐❧✐t❛rá ♦s ❡st✉❞♦s ❡ ❛❧❣✉♠❛s ❞❡✲
♠♦♥str❛çõ❡s✱ ♣♦✐s ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ ❝r✐tér✐♦s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳
Pr♦♣♦s✐çã♦ ✸ ❙❡❥❛♠ a, b∈Z+ e b≥a✳ ❚❡♠♦s q✉❡✿
a≡b(mod n)⇔n |b−a
❉❡♠♦♥str❛çã♦✿ ❉❛❞♦s a, b∈Z+✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✶✶✱ t❡♠♦s q✉❡
a≡b(mod n)⇔a=nq+r e b=np+r.
❈♦♠ ✐ss♦✱
b−a=np+r−(nq+r)⇔b−a=np−nq+r−r⇔b−a=n(p−q).
❈♦♠♦p❡qsã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱p−qt❛♠❜é♠ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ♦✉ s❡❥❛n|b−a✱
♣♦rt❛♥t♦✱ a ≡b(mod n)⇔n|b−a
❱✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❱❡r❡♠♦s ❛ s❡❣✉✐r ♠❛✐s ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ss❛ r❡❧❛çã♦ q✉❡ s❡r✈✐rã♦ ♣❛r❛ ♦s ❡st✉❞♦s ❞♦s ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s✳
❚❡♦r❡♠❛ ✶✷ ❙❡❥❛♠a, b, c, d, x ∈Z❡n ∈N✳ ❆s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✶✳ ❙❡a≡b(mod n)❡c≡d(mod n),❡♥tã♦a+c≡b+d(mod n)❡ac≡bd(mod n);
✷✳ ❙❡ a ≡b(mod n), ❡♥tã♦ ax ≡bx(mod n);
✸✳ ❙❡ a ≡b(mod n), ❡♥tã♦ ak≡bk(mod n),∀k∈N.
❉❡♠♦♥str❛çã♦✿
✶ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✱ t❡♠♦s q✉❡ n | b−a ❡ n | c−d✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ k, q ∈ N
t❛✐s q✉❡ nk=b−a ❡nq =c−d✳ ❉❛í✱
(b+d)−(a+c) = (b−a) + (d−c) =nk+nq =n(k+q)⇔a+c≡b+d(mod n).
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ t❡♠♦s✱
ac−bd= (b+kn)(d+qn)−bd=bd+bqn+dkn+kqn2−bd=n(bq+dk+kqn)⇔
⇔ac≡bd(mod n);
✷ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✱ t❡♠♦s q✉❡ n|b−a✳ ▲♦❣♦✱
bx−ax=x(b−a) = xnq ⇔ax ≡bx(mod n);
✸ ❱❛♠♦s ♣r♦✈❛r ❡ss❡ ✐t❡♠ ✉s❛♥❞♦ ✐♥❞✉çã♦ s♦❜r❡ k✿ P❛r❛ k = 1✱ ♥❛❞❛ t❡♠♦s ❛
❞❡♠♦♥str❛r✳
❙✉♣♦♥❤❛ q✉❡ ❛ ♣r♦♣♦s✐çã♦ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠ k ∈ N✱ ✈❛♠♦s ✈❡r✐✜❝❛r ❛
✈❛❧✐❞❛❞❡ ❞❛ ♣r♦♣♦s✐çã♦ ♣❛r❛ k+ 1✳ ❈♦♠♦ a ≡ b(mod n) ❡ ak
≡ bk(mod n) t❡♠♦s✱
♣❡❧♦ ✐t❡♠ ✶✱ q✉❡ak+1 ≡bk+1(mod n)✳ P♦rt❛♥t♦✱ ❛ ♣r♦♣♦s✐çã♦ é ✈á❧✐❞❛ ♣❛r❛k+ 1✳
❙❛❜❡♥❞♦ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❞❡✜♥✐♠♦s ❡♥tã♦ ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ a∈Z ♠ó❞✉❧♦n ♣♦r✿
a={b∈Z:a≡b(mod n)}
P♦❞❡♠♦s ❛❣♦r❛ ♦❜t❡r ✉♠ s✐st❡♠❛ ❝♦♠ t♦❞♦s ♦s ♣♦ssí✈❡✐s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐✲ ❞✐❛♥❛ ❞❡a ❡b ♣♦rn✱ ❞❡ss❛ ❢♦r♠❛ ♣♦❞❡r❡♠♦s r❡❧❛❝✐♦♥❛r ✉♠ ❡❧❡♠❡♥t♦ ❝♦♠ s✉❛ ❝❧❛ss❡
❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❡✜♥✐çã♦ ✶✸ ❯♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦n é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s
♥❛t✉r❛✐s r1, . . . rs t❛✐s q✉❡✿
✶✳ ♠❞❝✭ri, n✮❂✶✱ ♣❛r❛ t♦❞♦ i= 1, . . . , s❀
✷✳ ri 6=rj ♠♦❞ ✭n✮✱ s❡ i6=j❀
✸✳ P❛r❛ ❝❛❞❛ m∈N t❛❧ q✉❡ ♠❞❝✭m, n✮❂✶✱ ❡①✐st❡ i t❛❧ q✉❡ m≡ri(mod n).
P❡❧❛ ❉❡✜♥✐çã♦ ❛❝✐♠❛✱ ✉♠ s✐st❡♠❛ ❞❡ r❡sí❞✉♦s é ❝♦♥st✐t✉í❞♦ ♣♦r ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❆ ❉❡✜♥✐çã♦ ❛ s❡❣✉✐r ❢❛rá ✉♠❛ r❡❧❛çã♦ ❡♥tr❡n❡ ♦ ♥ú♠❡r♦
❞❡ ❡❧❡♠❡♥t♦s ❞♦ s✐st❡♠❛ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ n✳
❉❡✜♥✐çã♦ ✶✹ ✭❋✉♥çã♦ ❞❡ ❊✉❧❡r✮ ❉❡s✐❣♥❛♠♦s ♣♦r φ(n) ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s
❞❡ ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ n✱
❆s ♦❜s❡r✈❛çõ❡s ❛ s❡❣✉✐r sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ✐♠❡❞✐❛t❛s ❞❛ ❉❡✜♥✐çã♦ ✶✹✳
❖❜s❡r✈❛çã♦ ✷ φ(n)≤n−1✳
❖❜s❡r✈❛çã♦ ✸ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❡♥tã♦ φ(p) =p−1✳
❙❡❥❛Z∗
p✱ ❝♦♠p ♣r✐♠♦✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s a∈ {1,2, . . . , p−1},
❝♦♠ a = {b ∈ Z : a ≡ b(mod p)}✱ ❝♦♠ 1 ≤ a ≤ p−1✳ ❚❡♠♦s q✉❡ t♦❞♦ c ∈ Z∗ é ❝♦♥❣r✉❡♥t❡ ❛ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s 1,2, . . . , p−1✳
❉❡✜♥✐çã♦ ✶✺ ❉❛❞♦s a, b∈Z∗
p✱ ❡♥tã♦
a⊕b =a+b e
a⊙b=a·b.
P❡❧♦ ✐t❡♠ ✭✶✮ ❞♦ ❚❡♦r❡♠❛ ✶✷ t❡♠♦s q✉❡ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ♥♦ ❣r✉♣♦ Z∗
p✳ ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ tr❛❜❛❧❤❛r❡♠♦s
❝♦♠ ♦ ❣r✉♣♦ Z∗
p ♠✉❧t✐♣❧✐❝❛t✐✈♦✳
❆✜r♠❛çã♦ ✷ 1 é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ❣r✉♣♦ Z∗
p ♠✉❧t✐♣❧✐❝❛t✐✈♦✳
❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ a∈Z∗
p✱ t❡♠♦sa⊙1 =a·1 = a
❊st❛♠♦s ✧❝♦♥str✉✐♥❞♦✧ ♦ ❣r✉♣♦ Z∗
p✱ ✉s❛♥❞♦ ❛s ❉❡✜♥✐çõ❡s ✺ ❡ ✶✶✳ ❆♣❛r❡❝❡ ❛ s❡✲
❣✉✐♥t❡ ♣❡r❣✉♥t❛ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♥♦ss♦s ❡st✉❞♦s ✧❙❡rá q✉❡ ❡①✐st❡ n ∈ Z∗ t❛❧ q✉❡
an
≡ 1(mod p)❄✧ ❖ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r✲
♠❛t✧✱ r❡s♣♦♥❞❡ ❡ss❛ ♣❡r❣✉♥t❛✳
❚❡♦r❡♠❛ ✶✻ ✭P❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮✿ ❙❡❥❛p✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ a∈N
❝♦♠ ♠❞❝✭a, p✮❂✶✱ ❡♥tã♦ ap−1 ≡1(mod p)
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡
ap−1 ≡1(mod p)⇔p|ap−1−1⇔p|ap−a, pois mdc(a, p) = 1
P♦rt❛♥t♦ ♣❛r❛ ♣r♦✈❛r ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ❜❛st❛ ♣r♦✈❛r q✉❡ p|ap −a✳
❱❛♠♦s ✉s❛r ✐♥❞✉çã♦ s♦❜r❡ a ♣❛r❛ ❞❡♠♦♥str❛r ❡ss❛ ♣r♦♣♦s✐çã♦✿
P❛r❛a = 1 ♦ r❡s✉❧t❛❞♦ é ✐♠❡❞✐❛t♦✳
❙✉♣♦♥❤❛ q✉❡ ❛ ♣r♦♣♦s✐çã♦ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ a✱ ✈❛♠♦s ✈❡r✐✜❝❛r ❛ ✈❛❧✐❞❛❞❡ ❞❛
♣r♦♣♦s✐çã♦ ♣❛r❛ a+ 1✿
(a+ 1)p
−(a+ 1) =ap+
p
1
ap−1+ · · ·+
p p−1
a+ 1−(a+ 1)
(a+ 1)p−(a+ 1) =ap−a+
p
1
ap−1+· · ·+
p p−1
a
é ❢á❝✐❧ ✈❡r q✉❡ s❡ p é ♣r✐♠♦✱ ❡♥tã♦ p |
p n
✱ ❝♦♠ 0< n < p✱ ❛ss✐♠✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦
❞❛ ✐♥❞✉çã♦ p|ap
−a✱ ♣♦rt❛♥t♦✱ ❛ ♣r♦♣♦s✐çã♦ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ a ♥❛t✉r❛❧✳
❆ s❡❣✉✐r ✈❡r❡♠♦s ✉♠ ❝♦r♦❧ár✐♦ ❞♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t q✉❡ ✈❛✐ r❡❧❛❝✐♦♥❛r ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦ Z∗
p ❝♦♠ ✉♠ s✉❜❣r✉♣♦ ❝í❝❧✐❝♦ ❞❡❧❡✳
❈♦r♦❧ár✐♦ ✸✳✶ ❙❡ a∈Z∗
p✳ ❙❡ 0< a < p✱ ❡♥tã♦ hai é ✉♠ s✉❜❣r✉♣♦ ❝í❝❧✐❝♦ ❞❡ Z∗p✳
P♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ♦r❞❡♠ ❞♦ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ Z∗
p ❡ ❛
❋✉♥çã♦ ❞❡ ❊✉❧❡r ❞❡ p✳
❚❡♦r❡♠❛ ✶✼ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❡♥tã♦ |Z∗
p |=φ(p)
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ p é ♣r✐♠♦ t❡♠♦s ♣❡❧❛✱ ❉❡✜♥✐çã♦ ✶✸✱ q✉❡
Z∗
p ={1,2, . . . , p−1}
❝♦♠ ✐ss♦✱ |Z∗
p |=p−1✳ P❡❧❛ ❉❡✜♥✐çã♦ ✶✹
φ(p) =p−1.
P♦rt❛♥t♦ |Z∗
p |=φ(p)
❱❡r❡♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s(Z∗
p,·)✿
❊①❡♠♣❧♦✿ (Z∗
3,·) ={1,2}✿ ❈❛❧❝✉❧❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✷ q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s ❛ ✶
♦✉ ✷ ♠ó❞✉❧♦ ✸ ♦❜t❡♠♦s✿
21 = 2≡2(mod 3) 22 = 4≡1(mod 3) 23 = 8≡2(mod 3) 24 = 16≡1(mod 3)
✳✳✳ ◆♦t❡ q✉❡✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✹✱2❣❡r❛ ♦ ❣r✉♣♦Z∗
3✳ P♦rt❛♥t♦✱ ✭Z∗3,·✮ é ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦✳ ⋄
❊①❡♠♣❧♦✿ (Z∗
5,·) ={1,2,3,4}✿ ❈❛❧❝✉❧❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✷ q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s
❛ ✶✱ ✷✱ ✸ ♦✉ ✹ ♠ó❞✉❧♦ ✺✱ ♦❜t❡♠♦s✿
21 = 2≡2(mod 5) 22 = 4≡4(mod 5) 23 = 8≡3(mod 5) 24 = 16≡1(mod 5)
✳✳✳
❈❛❧❝✉❧❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✸ q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s ❛ ✶✱ ✷✱ ✸ ♦✉ ✹ ♠ó❞✉❧♦ ✺ ♦❜t❡♠♦s✿
31 = 3≡3(mod 5) 32 = 9≡4(mod 5) 33 = 27≡2(mod 5) 34 = 81≡1(mod 5)
✳✳✳
❈❛❧❝✉❧❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✹ q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s ❛ ✶✱ ✷✱ ✸ ♦✉ ✹ ♠ó❞✉❧♦ ✺ ♦❜t❡♠♦s✿
41 = 4≡4(mod 5) 42 = 16≡1(mod 5) 43 = 64≡4(mod 5) 44 = 16≡1(mod 5)
✳✳✳ ◆♦t❡ q✉❡ h2i = (Z∗
5,·) ❡ h3i = (Z∗5,·)✱ ♦✉ s❡❥❛✱ 2 ❡ 3 sã♦ ❣❡r❛❞♦r❡s ❞❡ (Z∗5,·)✳ P♦r✲
t❛♥t♦✱ (Z∗
5,·) é ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦✳ ❖ ❣r✉♣♦h4i={1,4} é ✉♠ s✉❜❣r✉♣♦ ❞❡ (Z∗5,·)✳ ⋄
◆♦t❡ q✉❡ ♥❡♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ Z∗
p sã♦ ❣❡r❛❞♦r❡s ❞❡❧❡✱ ✈❛♠♦s ❞✐❢❡r❡♥❝✐❛r
❡ss❡s ❡❧❡♠❡♥t♦s q✉❡ sã♦ ❣❡r❛❞♦r❡s ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦ ✶✽ ❉✐③❡♠♦s q✉❡ g é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
p q✉❛♥❞♦ g é ✉♠ ❣❡r❛❞♦r ❞❡
Z∗
p✱ ♦✉ s❡❥❛✱
g é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡Z∗
p ⇔ hgi=Z
∗
p
❊①❡♠♣❧♦✿ ✷ ❡ ✸ sã♦ r❛í③❡s ♣r✐♠✐t✐✈❛s ❞❡ Z∗
5 ⋄
❚❡♦r❡♠❛ ✶✾ g é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
p s❡✱ ❡ s♦♠❡♥t❡ s❡✱ | hgi |=φ(p)✳
❉❡♠♦♥str❛çã♦✿ ✭⇒✮ ❙❡ g é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
p✱ ❡♥tã♦✱ ♣❡❧❛ ❉❡✜♥✐çã♦
✶✽✱ hgi = Z∗
p✱ ❛❧é♠ ❞✐ss♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✼✱ t❡♠♦s q✉❡ | Z∗p |= φ(p)✳ P♦rt❛♥t♦✱
| hgi |=φ(p)✳
✭⇐✮| hgi |= φ(p)⇒| hgi |={1,2, . . . , p−1} ⇒| hgi |=Z∗
p ⇒ g é r❛✐③ ♣r✐♠✐t✐✈❛
❞❡ Z∗
p
P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s q✉❡ ♦ ❣r✉♣♦ Z∗
p é ❝í❝❧✐❝♦✱ ❜❛st❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ r❛✐③
♣r✐♠✐t✐✈❛ ❞❡ Z∗
p✱ ♦✉ s❡❥❛✱ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❡❧❡♠❡♥t♦ g ∈ Z∗p t❛❧ q✉❡ O(g) =
ϕ(p)✳
❖❜s❡r✈❛çã♦ ✹ ◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❞❡♠♦♥str❛❞♦ ♥❛ Pr♦♣♦s✐çã♦ ✷ ❞❛ ♣á❣✐♥❛ ✻✳ ❈♦♠♦ |Z∗
p |=p−1✱ t❡♠♦s q✉❡✿ ❙❡ O(g) =n ❡ m∈Z✱ ❡♥tã♦
gm
≡1(mod p)⇔m |n
▼❛s✱ ♣❡❧♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ s❛❜❡♠♦s q✉❡
gp−1
≡1(mod p)
P♦rt❛♥t♦✱
gm ≡1(mod p)⇔p−1 = km
❆ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ ❢✉♥❝✐♦♥❛ ❝♦♠♦ ❝r✐tér✐♦ ♣❛r❛ s❛❜❡r♠♦s s❡ ✉♠ ❡❧❡♠❡♥t♦ é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
p✳ P❛r❛ ✐ss♦ ♣r♦❝❡❞❡♠♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s s❡
✉♠ ❡❧❡♠❡♥t♦ g ∈ Z∗
p é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗p✱ ❜❛st❛ ✈❡r✐✜❝❛r s❡ ❛❧❣✉♠ ❞✐✈✐s♦r
❞❡ p−1 é ❝♦♥❣r✉❡♥t❡ ❛ ✶ ♠ó❞✉❧♦ p✳ ❱❡r❡♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝♦♠♦
❢✉♥❝✐♦♥❛ ❡ss❡ ❝r✐tér✐♦✳
❊①❡♠♣❧♦✿ ❱❡r✐✜❝❛♥❞♦ s❡ ✷ é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
7✿ ❈♦♠♦ ϕ(7) = 6 ❡ 6 = 2·3
t❡♠♦s q✉❡
22 = 4≡4(mod 7) 23 = 8≡1(mod 7)
P♦rt❛♥t♦✱ ✷ ♥ã♦ é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
7 ⋄
❊①❡♠♣❧♦✿ ❱❡r✐✜❝❛♥❞♦ s❡ ✼ é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
11✿ ❈♦♠♦ ϕ(11) = 10❡ 10 = 2·5
t❡♠♦s
72 = 49≡5(mod 11) 75 = 16807≡10(mod 11)
P♦rt❛♥t♦✱ ✼ é r❛✐③ ♣r✐♠✐t✐✈❛ ❞❡ Z∗
11✿ ⋄
❆❣♦r❛ ✈❛♠♦s ❡st✉❞❛r ✉♠ ✐♠♣♦rt❛♥t❡ ❛❧❣♦r✐t♠♦ q✉❡ ❛✉①✐❧✐❛rá ♥♦s ❝á❧❝✉❧♦s ❞❡ ❧♦✲ ❣❛r✐t♠♦s ❉✐s❝r❡t♦s✳ ❊ss❡ ❛❧❣♦r✐t♠♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s✧ ❡ ✈❛✐ ♥♦s ❛❥✉❞❛r ❛ r❡s♦❧✈❡r ♦s s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s q✉❡ ❛♣❛r❡❝❡rã♦ ♥♦s ❝á❧❝✉❧♦s ❞♦ ♣ró①✐♠♦ ❈❛♣ít✉❧♦✳
✶✳✸ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s
❙❡❣✉♥❞♦ ❬✺❪✱ ◆♦ ♣r✐♠❡✐r♦ sé❝✉❧♦ ❞❛ ♥♦ss❛ ❡r❛✱ ♦ ♠❛t❡♠át✐❝♦ ❝❤✐♥ês ❙✉♥✲❚s✉ ♣r♦♣ôs ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿
◗✉❛❧ é ♦ ♥ú♠❡r♦ q✉❡ ❞❡✐①❛ r❡st♦s ✷✱ ✸ ❡ ✷ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r ✸✱ ✺ ❡ ✼❄
❆ r❡s♣♦st❛ ❞❛❞❛ ♣♦r ❙✉♥✲❚s✉ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛ ❢♦✐ ✷✸✳
❚r❛❞✉③✐♥❞♦ ❡♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✉♥✲❚s✉ ❡q✉✐✈❛❧❡ ❛ ♣r♦✲ ❝✉r❛r ❛s s♦❧✉çõ❡s ❞♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✿
X ≡2(mod 3)
X ≡3(mod 5)
X ≡2(mod 7).
▼❛✐s ❣❡r❛❧♠❡♥t❡✱ r❡s♦❧✈❡r❡♠♦s s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❞❛ ❢♦r♠❛✿
a1X ≡b1(mod m1)
a2X ≡b2(mod m2)
✳✳✳
arX ≡br(mod mr).
P❛r❛ q✉❡ t❛❧ s✐st❡♠❛ ♣♦ss✉❛ s♦❧✉çã♦✱ é ♥❡❝❡ssár✐♦ q✉❡ ♠❞❝(ai;mi)|bi✱ ♣❛r❛ t♦❞♦
i= 1, . . . , r✳ ❉❡ss❛ ❢♦r♠❛ ♦ s✐st❡♠❛ ❛❝✐♠❛ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❞❛ ❢♦r♠❛ X ≡c1(mod n1)
X ≡c2(mod n2)
✳✳✳
X ≡cr(mod nr).
P♦rt❛♥t♦ ✈❛♠♦s r❡s♦❧✈❡r ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❡♥❝♦♥tr❛♥❞♦ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❡q✉✐✈❛❧❡♥t❡ ❛tr❛✈és ❞♦ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r✳ ❖ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s é ✉♠ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✷✵ ✭❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s✮ ❖ s✐st❡♠❛
X ≡c1(mod n1)
X ≡c2(mod n2)
✳✳✳
X≡cr(mod nr).
♦♥❞❡ ♠❞❝(ni, nj) ❂ ✶✱ ♣❛r❛ t♦❞♦ ♣❛r ni;nj ❝♦♠ i 6= j✱ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦
♠ó❞✉❧♦ N =n1n2· · ·nr✳ ❚❛❧ s♦❧✉çã♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❝♦♠♦ s❡ s❡❣✉❡✿ X =N1y1c1+ · · ·+Nryrcr; ♦♥❞❡ Ni =N/ni ❡ yi é s♦❧✉çã♦ ❞❡ NiY ≡1mod ni, i= 1, . . . , r.
❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ♣r♦✈❛r q✉❡ X é s♦❧✉çã♦ s✐♠✉❧tâ♥❡❛ ❞♦
s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ❝♦♠♦ ni |Nj ✱ s❡ i6=j✱ ❡Niyi ≡1(modni)✱ t❡♠♦s q✉❡
X =N1y1c1+· · ·+Nryrcr ≡Niyici ≡ci(mod ni)
P♦r ♦✉tr♦ ❧❛❞♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ s❡ x′ é ♦✉tr❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛✱ ❡♥tã♦ x′ ≡
x(mod n1), ∀i, i= 1, . . . , r✳ ❉❡ ❢❛t♦✳ ❈♦♠♦ ♠❞❝(ni;nj) = 1✱ ♣❛r❛i6=j✱ s❡❣✉❡✲s❡ q✉❡
mmc[n1;. . .;nr] =n1· · ·nr =N❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡x≡x′(mod[n1, . . . , nr])⇒
x≡x′(mod n
i), i= 1, . . . , r✱ t❡♠♦s q✉❡ x≡x′(mod N).
❱❛♠♦s ✈❡r ✉♠ ❡①❡♠♣❧♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♣❡❧♦ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦s ❘❡st♦s✳
❊①❡♠♣❧♦✿ ◗✉❛❧ ♦ ♠❡♥♦r ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ ❞❡✐①❛ r❡st♦s ✶✱ ✸ ❡ ✺ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r ✺✱ ✼ ❡ ✾✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❄
❙♦❧✉çã♦✿ P❛r❛ r❡s♣♦♥❞❡r ❛ ♣❡r❣✉♥t❛ ❛❝✐♠❛✱ ❞❡✈❡♠♦s r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❞❡ ❝♦♥✲ ❣r✉ê♥❝✐❛s
x≡1(mod 5)
x≡3(mod 7)
x≡5(mod 9).
◆❡ss❡ ❝❛s♦✱ N = 5·7·9 = 315✱ ❧♦❣♦ n1 = 63, n2 = 45, n3 = 35✱ ❝♦♠ ✐ss♦ ✈❛♠♦s
r❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❝♦♥❣r✉ê♥❝✐❛s ✐s♦❧❛❞❛♠❡♥t❡✿
63Y ≡1(mod5)⇐y1 = 2
45Y ≡1(mod7)⇐y2 = 5
35Y ≡1(mod 9)⇐y3 = 8.
P♦rt❛♥t♦ ❛ s♦❧✉çã♦ ♠ó❞✉❧♦ N = 315 é ❞❛❞❛ ♣♦r✿
X =N1y1c1+N2y2c2+N3y3c3 = 63·2·1 + 45·5·3 + 35·8·5 = 2201
❈♦♠♦ 2201≡311(mod 315) ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❞❡s❡❥❛❞♦ é ✸✶✶ ⋄
✶✳✹ P❡rs♦♥❛❧✐❞❛❞❡s ▼❛t❡♠át✐❝❛s
✶✳✹✳✶ ▲❡♦♥❤❛r❞ ❊✉❧❡r
▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✜❣✉r❛ ✶✳✶✮ ❢♦✐✱ s❡♠ ❞ú✈✐❞❛✱ ✉♠ ❞♦s ♠❛✐♦r❡s ❡ ♠❛✐s ❢ért❡✐s ♠❛t❡✲ ♠át✐❝♦s ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳ ❊❧❡ ♥❛s❝❡✉ ♥❛ ❙✉íç❛✱ ♣❡rt♦ ❞❛ ❝✐❞❛❞❡ ❞❡ ❇❛s✐❧é✐❛✱ ✜❧❤♦
❋✐❣✉r❛ ✶✳✶✿ ❋♦♥t❡✿❤tt♣✿✴✴✇✇✇✲❤✐st♦r②✳♠❝s✳st✲❛♥❞✳❛❝✳✉❦✴P✐❝t❉✐s♣❧❛②✴❊✉❧❡r✳❤t♠❧
❞❡ ✉♠ ♠♦❞❡st♦ ♣❛st♦r ♣r♦t❡st❛♥t❡ q✉❡ ♥✉tr✐❛ ❛ ❡s♣❡r❛♥ç❛ ❞❡ q✉❡ s❡✉ ✜❧❤♦ s❡❣✉✐ss❡ ❛ ♠❡s♠❛ ❝❛rr❡✐r❛✳
❊✉❧❡r ♣♦ss✉í❛ ✉♠❛ ❣r❛♥❞❡ ❢❛❝✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❡ ❧í♥❣✉❛s ❡ ✉♠❛ ♣r♦✲ ❞✐❣✐♦s❛ ♠❡♠ór✐❛✱ ❛❧✐❛❞❛ ❛ ✉♠❛ ❡①tr❛♦r❞✐♥ár✐❛ ❤❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❡❢❡t✉❛r ♠❡♥t❛❧♠❡♥t❡ ❝♦♥t❛s ❝♦♠♣❧❡①❛s✱ ❤❛❜✐❧✐❞❛❞❡ ❡st❛ q✉❡ ❧❤❡ s❡r✐❛ ♠✉✐t♦ út✐❧ ♥♦ ✜♥❛❧ ❞❡ s✉❛ ✈✐❞❛✳ ❊✉❧❡r ❢♦✐ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s ♠❛✐s ❛t✐✈♦s ❞❛ ❤✐stór✐❛✳ ❊❧❡ ❡s❝r❡✈❡✉ ❧✐✈r♦s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ❞❛s ✈❛r✐❛çõ❡s✱ ♥♦ ❝á❧❝✉❧♦ ❞❛s ór❜✐t❛s ♣❧❛♥❡tár✐❛s✱ ❡♠ ❛rt✐❧❤❛r✐❛ ❡ ❜❛❧íst✐❝❛✱ ❡♠ ❛♥á✲ ❧✐s❡✱ s♦❜r❡ ❛ ❝♦♥str✉çã♦ ♥❛✈❛❧ ❡ ❞❡ ♥❛✈❡❣❛çã♦✱ s♦❜r❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ❧✉❛✳ ❉❡♣♦✐s ❞❡ s✉❛ ♠♦rt❡✱ ❡♠ ✶✼✽✸ ❛ ❆❝❛❞❡♠✐❛ ❞❡ ❙ã♦ P❡t❡rs❜✉r❣♦ ❝♦♥t✐♥✉♦✉ ❛ ♣✉❜❧✐❝❛r tr❛❜❛❧❤♦s ✐♥é❞✐t♦s ❞❡ ❊✉❧❡r ♣♦r ♠❛✐s ❞❡ ✺✵ ❛♥♦s ♠❛✐s✳
✶✳✹✳✷ P✐❡rr❡ ❞❡ ❋❡r♠❛t
❙❡❣✉♥❞♦ ❬✼❪✱ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭✜❣✉r❛ ✶✳✷✮ ❡r❛ ✉♠ ❛❞✈♦❣❛❞♦ ❢r❛♥❝ês ❞♦ P❛r❧❛✲ ♠❡♥t♦ ❞❡ ❚♦✉❧♦✉s❡✱ ♥❛ ❋r❛♥ç❛✱ ❡ ✉♠ ♠❛t❡♠át✐❝♦ ❛♠❛❞♦r✱ ❡❧❡ ❝♦♠✉♥✐❝❛✈❛✲s❡ ❝♦♠ ♦✉tr♦s ♠❛t❡♠át✐❝♦s ❞❡ s✉❛ é♣♦❝❛ ❛tr❛✈és ❞❡ ❝❛rt❛s✱ ❡ ♥❡❧❛s ❋❡r♠❛t ❞❡s❝r❡✈✐❛ s✉❛s ✐❞❡✐❛s✱ ❞❡s❝♦❜❡rt❛s ❡ ❛té ♣❡q✉❡♥♦s ❡♥s❛✐♦s s♦❜r❡ ❛ss✉♥t♦s ❞❡ s❡✉ ✐♥t❡r❡ss❡✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝❛♠♦s ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ❖ ❚❡♦r❡♠❛ ✶✻✱ ❞❛ ♣á❣✐♥❛ ✶✶✱ ❢♦✐ ❡♥✉♥❝✐❛❞♦ ♣♦r ❋❡r♠❛t ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p✱ t❡♠✲s❡ q✉❡ p ❞✐✈✐❞❡ ♦
♥ú♠❡r♦ ap−a✱ ♣❛r❛ t♦❞♦ a∈N✳
❋❡r♠❛t é ♠❛✐s ❧❡♠❜r❛❞♦ ♣❡❧♦ ♦ ✧Ú❧t✐♠♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✧✳ ❊st❡ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡
xn+yn =zn
◆ã♦ t❡♠ s♦❧✉çõ❡s ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦ ✐♥t❡✐r♦s ♣❛r❛ x✱ y ❡ z✱ q✉❛♥❞♦ n❃ ✷✳ ❋❡r♠❛t
❡s❝r❡✈❡✉✱ ♥❛ ♠❛r❣❡♠ ❞❛ tr❛❞✉çã♦ ❞❡ ❆r✐t❤♠❡t✐❝❛ ❞❡ ❉✐♦❢❛♥t♦ ❞❡ ❇❛❝❤❡t✳ ✧❊✉ ❞❡s❝♦❜r✐ ✉♠❛ ♣r♦✈❛ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ♥♦tá✈❡❧ q✉❡ ❡st❛ ♠❛r❣❡♠ é ♠✉✐t♦ ♣❡q✉❡♥❛ ♣❛r❛ ❝♦♥t❡r✳✧
❆❝r❡❞✐t❛✲s❡ ❛❣♦r❛ q✉❡ ❛ ✧♣r♦✈❛✧ ❞❡ ❋❡r♠❛t ❡st❛✈❛ ❡rr❛❞❛ ❡♠❜♦r❛ s❡❥❛ ✐♠♣♦ssí✈❡❧ t❡r ❛ ❝❡rt❡③❛ ❛❜s♦❧✉t❛✳ ❆ ✈❡r❞❛❞❡ ❞❛ ❛✜r♠❛çã♦ ❞❡ ❋❡r♠❛t ❢♦✐ ♣r♦✈❛❞❛ ❡♠ ❏✉♥❤♦ ❞❡
