❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❈✉❜♦ ▼á❣✐❝♦✿ Pr♦♣r✐❡❞❛❞❡s ❡ ❘❡s♦❧✉çõ❡s
❡♥✈♦❧✈❡♥❞♦ ➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s
▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s ●r✐♠♠
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✕ P❘❖❋▼❆❚ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s
Grimm, Luis Gustavo Hauff Martins
Cubo mágico : propriedades e resoluções envolvendo álgebra e Teoria de Grupos / Luis Gustavo Hauff Martins Grimm. - Rio Claro, 2016
82 f. : il., figs., tabs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientadora: Carina Alves
1. Álgebra. 2. Cubo de Rubik. 3. Grupos de permutação. 4. Comutadores e conjugados. 5. Proposta didática. I. Título. 512
G864c
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
▲✉✐s ●✉st❛✈♦ ❍❛✉✛ ▼❛rt✐♥s ●r✐♠♠
❈✉❜♦ ▼á❣✐❝♦✿ Pr♦♣r✐❡❞❛❞❡s ❡ ❘❡s♦❧✉çõ❡s ❡♥✈♦❧✈❡♥❞♦
➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❈❛r✐♥❛ ❆❧✈❡s ❖r✐❡♥t❛❞♦r❛
Pr♦❢✳ ❉r✳ ❈r✐st✐❛♥❡ ❆❧❡①❛♥❞r❛ ▲á③❛r♦ ❯◆❊❙P ✲ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ✲ ❇❛✉r✉
Pr♦❢✳ ❉r✳ ❆❣♥❛❧❞♦ ❏♦sé ❋❡rr❛r✐
❯◆❊❙P ✲ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ✲ ❇❛✉r✉
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ♠✐♥❤❛ ❡s♣♦s❛ ♣❡❧♦ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦✱ ❝♦♠♣❛♥❤✐❛ ♥♦s ❡st✉❞♦s ❡ ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♠ ♦ ▲❆❚❊❳✳
➚ ♣r♦❢❡ss♦r❛ ❈❛r✐♥❛ ❆❧✈❡s ♣❡❧❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❝♦♥tr✐❜✉✐çã♦ ♥❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦ ❝♦♥t✐♥✉❛❞❛ ❞❡ ♣r♦✜ss✐♦♥❛✐s q✉❡ ❛t✉❛♠ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳
❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞❛ s❡❝r❡t❛r✐❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❛t❡♥✲ çã♦ ❡ ❡♠♣❡♥❤♦ ❡♠ ❛❥✉❞❛r ♣r♦♥t❛♠❡♥t❡✳
❙❡ ✈♦❝ê ❢♦r ❝✉r✐♦s♦✱ ❡♥❝♦♥tr❛rá q✉❡❜r❛✲❝❛❜❡ç❛s ❡♠ t♦r♥♦ ❞❡ ✈♦❝ê✳ ❙❡ ✈♦❝ê ❡st✐✈❡r ❞❡t❡r♠✐♥❛❞♦✱ ✐rá r❡s♦❧✈ê✲❧♦s✳
❘❡s✉♠♦
❖ ❝✉❜♦ ♠á❣✐❝♦ é ✉♠ ❞♦s q✉❡❜r❛✲❝❛❜❡ç❛s ♠❛✐s ❢❛♠♦s♦s ❞♦ ♠✉♥❞♦✱ ❡ ❡♠ ❣❡r❛❧ ❛tr❛✐ ❛ ❛t❡♥çã♦ ❞❡ ♠✉✐t❛ ❣❡♥t❡✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❞♦s ♠❛t❡♠át✐❝♦s✳ ❖ ❞❡s❛✜♦✱ ❛s ❢♦r♠❛s✱ s✐♠❡tr✐❛s ❡ ♠♦✈✐♠❡♥t♦s ✐♥❞✉③❡♠ ❛ ✐❞❡✐❛ ❞❡ ❡st❛r♠♦s ❞✐❛♥t❡ ❞❡ ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✳ ❊ ♣♦❞❡♠♦s ✐r ❛❧é♠✳ ❆s ❛çõ❡s ❡ ♠♦✈✐♠❡♥t♦s ♥♦ ❝✉❜♦ ♠á❣✐❝♦ sã♦ ❡❧❡♠❡♥t♦s q✉❡ ❛t❡♥❞❡♠ ❛ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ❞❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❣r✉♣♦✱ ❛ss✐♠ ❝♦♠♦ t❛♠❜é♠ s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ✉♠ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s✳ ➚ ❧✉③ ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ❞♦s ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ✐r❡♠♦s ❛♥❛❧✐s❛r ❛❧❣✉♠❛s s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s ❝♦♠♦ ♦s ❝♦♠✉t❛❞♦r❡s ❡ ❝♦♥❥✉❣❛❞♦s✳ ❊①✐st❡♠ ✈ár✐♦s ❛❧❣♦r✐t♠♦s q✉❡ r❡s♦❧✈❡♠ ♦ ❝✉❜♦ ♠á❣✐❝♦ ❡ q✉❡ sã♦ ❢á❝❡✐s ❞❡ s❡r❡♠ ♦❜t✐❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ✐♥t❡r♥❡t✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❛❧é♠ ❞❡ tr❛③❡r ✉♠❛ ♣r♦♣♦st❛ ❞❡ r❡s♦❧✉çã♦✱ é ♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛r ✉♠ ❝❛♠✐♥❤♦ ♣❛r❛ ❛❧é♠ ❞❛ s✐♠♣❧❡s ♠❡♠♦r✐③❛çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❝♦♠♣r❡❡♥❞ê✲❧♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❡ r❡s♦❧✈❡r ✉♠ ❝✉❜♦ ♠á❣✐❝♦ é ❞❡ ♦r❞❡♠ ♠❛t❡♠át✐❝❛ ❡ ♥ã♦ ❡♠♣ír✐❝❛✳
❆❜str❛❝t
❚❤❡ ❘✉❜✐❦✬s ❈✉❜❡ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ❢❛♠♦✉s ♣✉③③❧❡ ♦❢ t❤❡ ✇♦r❧❞✱ ❛♥❞ ❣❡♥❡r❛❧❧② ❛t✲ tr❛❝ts t❤❡ ❛tt❡♥t✐♦♥ ♦❢ ♠❛♥② ♣❡♦♣❧❡✱ ❡s♣❡❝✐❛❧❧② ♠❛t❤❡♠❛t✐❝✐❛♥s✳ ❚❤❡ ❝❤❛❧❧❡♥❣❡✱ s❤❛♣❡s✱ s②♠♠❡tr✐❡s ❛♥❞ ♠♦✈❡♠❡♥ts ✐♥❞✉❝❡ t❤❡ ✐❞❡❛ ♦❢ ❜❡✐♥❣ ✐♥ ❢r♦♥t ♦❢ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t✳ ❆♥❞ ✇❡ ❝❛♥ ❣♦ ❢✉rt❤❡r✳ ❚❤❡ ❛❝t✐♦♥s ❛♥❞ ♠♦✈❡♠❡♥ts ✐♥ t❤❡ ♠❛❣✐❝ ❝✉❜❡ ❛r❡ ❡❧❡♠❡♥ts t❤❛t ♠❡❡t ❛❧❧ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ str✉❝t✉r❡ ♦❢ ❛ ❣r♦✉♣✱ ❛s ✇❡❧❧ ❛s r❡❧❛t❡ t♦ ❛ ❣r♦✉♣ ♦❢ ♣❡r♠✉t❛t✐♦♥s✳ ■♥ ❧✐❣❤t ♦❢ t❤❡ ●r♦✉♣ ❚❤❡♦r② ❛♥❞ P❡r♠✉t❛t✐♦♥s ❣r♦✉♣s ✇❡ ✇✐❧❧ ❡①❛♠✐♥❡ s♦♠❡ s❡q✉❡♥❝❡s ♦❢ ♠♦✈❡♠❡♥ts s✉❝❤ ❛s ❝♦♠♠✉t❛t♦rs ❛♥❞ ❝♦♥❥✉❣❛t❡s✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❛❧❣♦r✐t❤♠s t❤❛t s♦❧✈❡ t❤❡ ♠❛❣✐❝ ❝✉❜❡ ❛♥❞ ✇❤✐❝❤ ❛r❡ ❡❛s② t♦ ♦❜t❛✐♥✱ ❢♦r ❡①❛♠♣❧❡✱ ❛t t❤❡ ■♥t❡r♥❡t✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ❜❡②♦♥❞ t♦ s❤♦✇ ❛ r❡s♦❧✉t✐♦♥✱ ✐s t♦ ♣r♦✈✐❞❡ ❛ ♣❛t❤ ❜❡②♦♥❞ s✐♠♣❧❡ ♠❡♠♦r✐③❛t✐♦♥ ♦❢ ❛♥ ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✐t✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ❢♦r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ s♦❧✈✐♥❣ ❛ ❘✉❜✐❦✬s ❈✉❜❡ ✐s ♠❛t❤ ❛♥❞ ♥♦t ❡♠♣✐r✐❝❛❧✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❱✐sã♦ ❡①♣❧♦❞✐❞❛ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦3×3×3. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✷ ▼❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ 3×3×3. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✸ ❚✐♣♦s ❡ ❢♦r♠❛t♦s ❞❡r✐✈❛❞♦s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ❋❛❝❡s ❞♦ ❝✉❜♦3×3×3 ❡ s✉❛ ♣❧❛♥✐✜❝❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✺ ◆♦♠❡♥❝❧❛t✉r❛ ❞♦s ❝✉❜✐♥❤♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ ❊✐①♦s ❞❡ r♦t❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❊①❡♠♣❧♦s ❞❡ ♣♦s✐çã♦ ❡ ♦r✐❡♥t❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✽ ❈ó❞✐❣♦s ❞❛s r♦t❛çõ❡s ❞❛s ❢❛❝❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶ ◗✉❛❞r❛❞♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞♦ ❣r✉♣♦GQ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✷ ❈♦♠♣♦s✐çã♦Rr◦R1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✸ ❈♦♠♣♦s✐çã♦R1 ◦Rr✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✹ ▼❛❝r♦sF R ❡ RF✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳✺ ▼❛❝r♦F4✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✹✳✶ ❈✉❜✐♥❤♦s ❛❢❡t❛❞♦s ♣❡❧❛ ♠❛❝r♦S✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✹✳✷ ❈✉❜✐♥❤♦s ❛❢❡t❛❞♦s ♣❡❧♦ ♠♦✈✐♠❡♥t♦ F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✸ ❊❢❡✐t♦ ❣❡r❛❞♦ ♣❡❧❛s ♠❛❝r♦sS ❡T✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✾
✷ ❖ ❈✉❜♦ ▼á❣✐❝♦ ✷✶
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ ❡ s❡✉ ♠❡❝❛♥✐s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❋♦r♠❛t♦s ❡ ✈❛r✐❛çõ❡s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ▼♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ▼❛❝r♦ ♦✉ s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦ ✷✾
✸✳✶ ❚❡♦r✐❛ ❇ás✐❝❛ ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ●r✉♣♦s ❛❞✐t✐✈♦s ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦s ❞❡ ❝❧❛ss❡s ❞❡ r❡st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❈❧❛ss❡s ▲❛t❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ●r✉♣♦ ❞❡ ❘✉❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ❙✉❜❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺✳✶ ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ✉♠ s✉❜❝♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✻ ❙✉❜❣r✉♣♦s ◆♦r♠❛✐s ❡ ●r✉♣♦s ◗✉♦❝✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✼ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✽ P❡r♠✉t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✽✳✶ Pr♦❞✉t♦ ❞❡ ❈✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✽✳✷ ❘❡♣❡t✐çã♦ ❞❡ ❝✐❝❧♦s ❡ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✾ ❆ss✐♥❛t✉r❛ ❞❡ ✉♠❛ P❡r♠✉t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✾✳✶ P❛r✐❞❛❞❡ ❞❡ ❈✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹ ❆♣❧✐❝❛çõ❡s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çã♦ ♥♦ ❈✉❜♦
▼á❣✐❝♦ ✺✺
✹✳✶ ❈❛❧❝✉❧❛♥❞♦ ❛ ♦r❞❡♠ ❞❡ ✉♠❛ ♠❛❝r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷ P❛r✐❞❛❞❡ ❞♦s ♠♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✸ ❈❛❧❝✉❧❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ♣♦s✐çõ❡s ❞♦ ❝✉❜♦ ♠á❣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✹ ❈♦♠✉t❛❞♦r❡s ❡ ❝♦♥❥✉❣❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✺ ❘❡s♦❧✈❡♥❞♦ ♦ ❈✉❜♦ ▼á❣✐❝♦ ✻✸
✺✳✶✳✷ ▼ét♦❞♦ r❡❛❧✐③á✈❡❧ ♣♦r ❤✉♠❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✷ ❘❡s♦❧✈❡♥❞♦ ♦ ❈✉❜♦ ▼á❣✐❝♦ ♥♦ ♠❡♥♦r t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✳✸ ❘❡s♦❧✈❡♥❞♦ ♦ ❈✉❜♦ ▼á❣✐❝♦ ❝♦♠ ❛ ♠❡♥♦r ♠❡♠♦r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✺✳✸✳✶ ✶❛ ❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛ ❢❛❝❡ ✐♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✸✳✷ ✷❛ ❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛s ❢❛❝❡s ❛❞❥❛❝❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✸✳✸ ✸❛ ❊t❛♣❛✿ ❢♦r♠❛r ✉♠❛ ❝r✉③ ♥❛ ❢❛❝❡ ♦♣♦st❛ à ✐♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✸✳✹ ✹❛ ❊t❛♣❛✿ ♣♦s✐❝✐♦♥❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✸✳✺ ✺❛ ❊t❛♣❛✿ ♦r✐❡♥t❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✸✳✻ ❘❡s✉♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✻ ❆♣❧✐❝❛çã♦ ❊s❝♦❧❛r ✼✸
✻✳✶ ❆♣r❡s❡♥t❛çã♦ ❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✷ ❈ó❞✐❣♦s ❞❡ ❝❛❞❛ ♠♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✸ ❘❡s♦❧✉çã♦ ❞♦ ❈✉❜♦ ▼á❣✐❝♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❝❛♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✶ ❊t❛♣❛ ✶✿ ❋♦r♠❛r ✉♠❛ ❝r✉③ ❜r❛♥❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✷ ❊t❛♣❛ ✷✿ P♦s✐❝✐♦♥❛r ♦s ❝❛♥t♦s ❜r❛♥❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✻✳✸✳✸ ❊t❛♣❛ ✸✿ ❘❡s♦❧✈❡r ❛ ❝❛♠❛❞❛ ❞♦ ♠❡✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✻✳✸✳✹ ❊t❛♣❛ ✹✿ ❋♦r♠❛r ✉♠❛ ❝r✉③ ❛♠❛r❡❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✸✳✺ ❊t❛♣❛ ✺✿ ❘❡s♦❧✈❡r ❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✸✳✻ ❊t❛♣❛ ✻✿ ❖r✐❡♥t❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❝❛♥t♦ ❞❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✳ ✼✽ ✻✳✸✳✼ ❊t❛♣❛ ✼✿ P♦s✐❝✐♦♥❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ❛r❡st❛ ❞❛ ❝❛♠❛❞❛ s✉♣❡r✐♦r ✳ ✼✽
✶ ■♥tr♦❞✉çã♦
❉✉r❛♥t❡ ❞é❝❛❞❛s ♦ ❞❡s❛✜♦ ❞❛ r❡s♦❧✉çã♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❝✉❜♦ ❞❡ ❘✉❜✐❦✱ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ s❡✉ ✐♥✈❡♥t♦r✱ ✐♥tr✐❣♦✉ ♠✉✐t❛ ❣❡♥t❡✳ ❊r♥ö ❘✉❜✐❦ ♥❛s❝❡✉ ❡♠ ✶✸ ❞❡ ❥✉❧❤♦ ❞❡ ✶✾✹✹ ❡♠ ❇✉❞❛♣❡st❡✱ ❍✉♥❣r✐❛ ❡ ♥❛ ❞é❝❛❞❛ ❞❡ ✼✵ ❞❛✈❛ ❛✉❧❛s ❞❡ ❛rq✉✐✲ t❡t✉r❛ ♥❛ ❙❝❤♦♦❧ ❢♦r ❈♦♠♠❡r❝✐❛❧ ❆rt✐sts✳ ❊♠ ✶✾✼✹ t❡✈❡ ❛ ✐❞❡✐❛ ❞❡ ❝♦♥str✉✐r ✉♠ ❝✉❜♦ ❝❛♣❛③ ❞❡ r♦t❛❝✐♦♥❛r s✉❛s ❢❛❝❡s ♣❛r❛ q✉❡ ❡♥tã♦ ♣✉❞❡ss❡ ✐❧✉str❛r ♠❡❧❤♦r ♣❛r❛ s❡✉s ❛❧✉♥♦s ♦ ❝♦♥❝❡✐t♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧❛r ❡r❛ ❞❡ ♠❛❞❡✐r❛ ❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡✐s ❢❛❝❡s ❞♦ ❝✉❜♦ ❡r❛ ♣✐♥t❛❞❛ ❞❡ ✉♠❛ ❝♦r ❞✐❢❡r❡♥t❡✳ ❈❛❞❛ ❢❛❝❡ é ❞✐✈✐❞✐❞❛ ❡♠ ✾ ❝✉❜✐♥❤♦s ♠❡♥♦r❡s t♦t❛❧✐③❛♥❞♦ ✷✻ ❝✉❜♦s ✈✐sí✈❡✐s ❡ ✉♠ ✈✐rt✉❛❧ q✉❡ s❡ ❡①✐st✐ss❡ ❡st❛r✐❛ ❧♦❝❛❧✐③❛❞♦ ♥♦ ✐♥t❡r✐♦r ❞♦ ❝✉❜♦✳ ❊st❡ ✷✼♦ ❝✉❜✐♥❤♦ ✈✐rt✉❛❧ é ❥✉st❛♠❡♥t❡ ♦ ♠❡❝❛♥✐s♠♦ q✉❡ ♣r❡♥❞❡ ❛s ✻ ❢❛❝❡s✳ ◆♦ ♠❡s♠♦ ❛♥♦ ❞❛ ❝r✐❛çã♦✱ ♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ❣❛♥❤♦✉ ♦ ♣rê♠✐♦ ❛❧❡♠ã♦ ❞❡ ❥♦❣♦ ❞♦ ❛♥♦✳ ❊r♥ö ❘✉❜✐❦ ♥ã♦ t✐♥❤❛ ❡①❛t❛♠❡♥t❡ ❛ ✐♥t❡♥çã♦ ❞❡ ❝r✐❛r ✉♠ q✉❡❜r❛✕❝❛❜❡ç❛ ❝✉❥❛ s♦❧✉çã♦ ❝♦♥s✐st❡ ❡♠ ❞❡✐①❛r ❝❛❞❛ ✉♠❛ ❞❛s ❢❛❝❡s s♦♠❡♥t❡ ❝♦♠ ✉♠❛ ❝♦r✳ ❖ ♣ró✲ ♣r✐♦ ❝r✐❛❞♦r ❞♦ ❝✉❜♦ ❞❡♠♦r♦✉ ❝❡r❝❛ ❞❡ ✉♠ ♠ês ♣❛r❛ r❡♠♦♥tá✲❧♦ à s✉❛ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧✳ ❊♠ ✶✾✽✵ ♦ ❝✉❜♦ ❝♦♠❡ç♦✉ ❛ s❡r ♣r♦❞✉③✐❞♦ ❞❡ ❢♦r♠❛ ✐♥❞✉str✐❛❧ ❡ ❡st✐♠❛✕s❡ q✉❡ ❞❡s❞❡ ❡♥tã♦ ❥á t❡♥❤❛♠ s✐❞♦ ✈❡♥❞✐❞♦s ♠❛✐s ❞❡ ✸✺✵ ♠✐❧❤õ❡s ❞❡ ✉♥✐❞❛❞❡s ❡♠ t♦❞♦ ♦ ♠✉♥❞♦✳
■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ✐r❡♠♦s tr❛③❡r ♥♦çõ❡s s♦❜r❡ s❡✉ ♠❡❝❛♥✐s♠♦ ❞❡ ❢✉♥❝✐✲ ♦♥❛♠❡♥t♦ ❡ q✉❛✐s ♠♦✈✐♠❡♥t♦s sã♦ ♣❡r♠✐t✐❞♦s ❡❢❡t✉❛r✳ ❊st❛ ♣r✐♠❡✐r❛ ❛♣r❡s❡♥t❛çã♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ q✉❡ é ❡ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ✉♠ ❝✉❜♦ ♠á❣✐❝♦✳ ❊✱ ❛❧é♠ ❞✐ss♦✱ ♦s ♠♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s✱ ❜❡♠ ❝♦♠♦ ❞♦s ❝✉❜✐♥❤♦s✱ sã♦ ♦ ♦❜❥❡t✐✈♦ ❛❧✈♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♣♦✐s sã♦ ❡st❡s ♦s ❡❧❡♠❡♥t♦s q✉❡ ✐r❡♠♦s ❡st✉❞❛r ♠❛✐s ❛❞✐❛♥t❡✳
◆♦ ❈❛♣ít✉❧♦ ✸ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✳ ❆ ♣❛rt✐r ❞❡❧❡s ✐❞❡♥t✐✜❝❛r❡♠♦s ♦ ❝✉❜♦ ♠á❣✐❝♦ ❝♦♠♦ ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✱ ❛ss✐♠ ❝♦♠♦ ❛s ❛çõ❡s ❡ ♦s ♠♦✈✐♠❡♥t♦s ♥♦ ❝✉❜♦ s❡rã♦ ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ♦♣❡r❛çõ❡s q✉❡ ❛t❡♥❞❡♠ às ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ✈✐st❛s✳
◆♦ ❈❛♣ít✉❧♦ ✹ ✐r❡♠♦s ❛♣❧✐❝❛r ♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❡♥t❡♥❞❡r♠♦s ❛❧❣✉♥s ♠♦✈✐♠❡♥t♦s✱ ❡st❛❞♦s ❡ ❛❧❣♦r✐t♠♦s q✉❡ r❡s♦❧✈❡♠ ♦ ❝✉❜♦ ♠á❣✐❝♦✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❧❝✉❧❛r❡♠♦s ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❝♦♥✜❣✉r❛çõ❡s q✉❡ ♦ ❝✉❜♦ ♠á❣✐❝♦ ♣♦❞❡ ❛ss✉♠✐r✱ ❥✉♥✲ t❛♠❡♥t❡ ❝♦♠ ❛ ❥✉st✐✜❝❛t✐✈❛ ❞❡ q✉❡ ❛❧❣✉♠❛s ❝♦♥✜❣✉r❛çõ❡s sã♦ ✐♠♣♦ssí✈❡✐s ❞❡ s❡r❡♠
✷✵ ■♥tr♦❞✉çã♦
❛t✐♥❣✐❞❛s✳
✷ ❖ ❈✉❜♦ ▼á❣✐❝♦
❖ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❢❛♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ ♦ ❝✉❜♦ ♠á❣✐❝♦ ❞❡♣❡♥❞❡♠ ❞❡ ❝♦♥❤❡❝❡r♠♦s ❝❡rt♦s ❝♦♥❝❡✐t♦s q✉❡ ♦ ❝❡r❝❛♠✱ ❝♦♠♦ s✉❛ ❡str✉t✉r❛✱ ❝ó❞✐❣♦ ❞♦s ❡❧❡♠❡♥t♦s ✭❝✉❜✐♥❤♦s✱ ❢❛❝❡s ❡ ♠♦✈✐♠❡♥t♦s✮ ❡ ❛té ♠❡s♠♦ s✉❛ ♠❛♥✐♣✉❧❛çã♦✳ ❱❡r❡♠♦s q✉❡ ♦ ♠♦❞♦ ❝♦♠♦ ♦ s❡❣✉r❛♠♦s é ❝r✉❝✐❛❧ ♣❛r❛ ❝❤❡❣❛r♠♦s à s✉❛ r❡s♦❧✉çã♦✳ ❆♦ ❧♦♥❣♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ❛ ❞❡s❝r✐çã♦ ❞❡ t♦❞♦s ♦s ❝♦♥❝❡✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ✐♥✐❝✐❛r♠♦s ♥♦ss♦ ❡st✉❞♦ s♦❜r❡ ♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳
✷✳✶ ❊❧❡♠❡♥t♦s ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ ❡ s❡✉ ♠❡❝❛♥✐s♠♦
Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❞❡s❝r❡✈❡r ❡ ❝❧❛ss✐✜❝❛r t♦❞♦s ♦s ✷✻ ❝✉❜✐♥❤♦s q✉❡ ❝♦♠♣õ❡♠ ♦ ❝✉❜♦ ♠á❣✐❝♦✳ ◆❡♠ t♦❞❛s ❛s ❢❛❝❡s ❞♦s ❝✉❜✐♥❤♦s sã♦ ✈✐sí✈❡✐s✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ s✉❛ ♣♦s✐çã♦ ♥♦ ❝✉❜♦ ❡❧❡s ❛♣r❡s❡♥t❛♠ ✶✱ ✷ ♦✉ ✸ ❢❛❝❡s ✈✐sí✈❡✐s✳ P♦❞❡♠♦s q✉❛♥t✐✜❝❛r ❡ ❝❧❛ss✐✜❝❛r ♦s ❝✉❜✐♥❤♦s ❞❡ ✸ ♠❛♥❡✐r❛s✿• ✻ ❝✉❜✐♥❤♦s ❝❤❛♠❛❞♦s ❞❡ ❝❡♥tr❛✐s ♣♦r ❡st❛r❡♠ ♥♦ ❝❡♥tr♦ ❞❛s ❢❛❝❡s ❞♦ ❝✉❜♦✱ ♦♥❞❡
só ✉♠❛ ❢❛❝❡ é ✈✐sí✈❡❧❀
• ✶✷ ❝✉❜✐♥❤♦s ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛✱ ♦✉ ❞❡ ♠❡✐♦✱ ❡ q✉❡ ❝♦♠♣õ❡♠ ❛ ❛r❡st❛ ❞♦ ❝✉❜♦✱
♦♥❞❡ ❞✉❛s ❢❛❝❡s sã♦ ✈✐sí✈❡✐s❀
• ✽ ❝✉❜✐♥❤♦s ❝❤❛♠❛❞♦s ❞❡ ❝❛♥t♦✱ ♦✉ ❞❡ q✉✐♥❛✱ ❡ q✉❡ sã♦ ♦s ✈ért✐❝❡s ❞♦ ❝✉❜♦✱ ♦♥❞❡
três ❞❡ s✉❛s ❢❛❝❡s sã♦ ✈✐sí✈❡✐s✳
❋✐❣✉r❛ ✷✳✶✿ ❱✐sã♦ ❡①♣❧♦❞✐❞❛ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦3×3×3.
✷✷ ❖ ❈✉❜♦ ▼á❣✐❝♦
❆ ✜❣✉r❛ ✷✳✶ ♠♦str❛ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ 3×3×3❞❡ ♠♦❞♦ ❛ ✈✐s✉❛❧✐③❛r♠♦s s✉❛ ♠♦♥✲
t❛❣❡♠✳
❖s ❝✉❜✐♥❤♦s ❝❡♥tr❛✐s sã♦ ♦s q✉❡ ❞❡t❡r♠✐♥❛♠ q✉❛❧ s❡rá ❛ ❝♦r ❞❛ ❢❛❝❡ ❞♦ ❝✉❜♦ ❡ ♦ ú♥✐❝♦ ♠♦✈✐♠❡♥t♦ q✉❡ ❡❧❡s r❡❛❧✐③❛♠ é ♦ ❞❡ r♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ q✉❡ ♦s ♣r❡♥❞❡ ❛♦ ♠❡❝❛♥✐s♠♦ ❝❡♥tr❛❧✳ ❉✐③❡♠♦s q✉❡ ❡st❡s ❝✉❜✐♥❤♦s sã♦ ✜①♦s✳ ❆ ❋✐❣✉r❛ ✷✳✷ ✐❧✉str❛ ♦ ♠❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ♦♥❞❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♦ ❝✉❜✐♥❤♦ ❝❡♥tr❛❧ é ✜①♦ ❡ s❡✉ ú♥✐❝♦ ♠♦✈✐♠❡♥t♦ ♣❡r♠✐t✐❞♦ é ♦ ❞❡ r♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ q✉❡ ♦ ♣r❡♥❞❡✳
❋✐❣✉r❛ ✷✳✷✿ ▼❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ❞❡ ✉♠ ❝✉❜♦ ♠á❣✐❝♦3×3×3.
✷✳✷ ❋♦r♠❛t♦s ❡ ✈❛r✐❛çõ❡s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦
❊①✐st❡♠ ❞❡③❡♥❛s ❞❡ t✐♣♦s✱ ❢♦r♠❛t♦s ❡ t❛♠❛♥❤♦s ❞❡ ❝✉❜♦ ♠á❣✐❝♦✳ ❖ ♣ró♣r✐♦ ❘✉❜✐❦ q✉❛♥❞♦ ✐❞❡❛❧✐③♦✉ ♦ ❝✉❜♦ ♣❡♥s♦✉ ❡♠ ✉♠ ❝✉❜♦2×2×2✱ ❡♥tr❡t❛♥t♦ ❛ ✐❞❡✐❛ ❞❡ ❝♦♥str✉✐r
❛ ❝♦♥✜❣✉r❛çã♦✱ q✉❡ s❡ t♦r♥♦✉ ❛ ♠❛✐s ❢❛♠♦s❛✱ s✉r❣✐✉ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❡❧❡ q✉❡r✐❛ ♠❛♥t❡r ♦s ❝❡♥tr♦s ✜①♦s✳ ❖ ❝♦♥❝❡✐t♦ ❞♦ q✉❡❜r❛ ❝❛❜❡ç❛ s❡ ❡①♣❛♥❞✐✉ ❡ ♠✉✐t♦s ❞❡✐①❛r❛♠ ❞❡ t❡r ♦ ❢♦r♠❛t♦ ❝ú❜✐❝♦✱ s❡♥❞♦ ✉t✐❧✐③❛❞❛s ♦✉tr❛s ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ❝♦♠♦ ✉♠ t❡tr❛❡❞r♦ ♦✉ ❡♥tã♦ ✉♠ ❞♦❞❡❝❛❡❞r♦✳ ❊①✐st❡♠ t❛♠❜é♠ ❛q✉❡❧❡s ❝♦♠ ❢♦r♠❛s ✐rr❡❣✉❧❛r❡s✳ ❆ ❋✐❣✉r❛ ✷✳✸ tr❛③✱ ❞❡♥tr❡ ❛s ❞❡③❡♥❛s✱ ✻✵ t✐♣♦s ❡ ❢♦r♠❛t♦s ❞❡r✐✈❛❞♦s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳
❖ ❢♦r♠❛t♦ ♠❛✐s ❝♦♠✉♠ é ♦ ❝✉❜♦ 3×3×3 ❞❡ ✺✱✼ ❝♠ ❞❡ ❛r❡st❛ ❡ ❞❡ ❢❛❝❡s ♣✐♥t❛❞❛s
♥❛s ❝♦r❡s ❜r❛♥❝❛✱ ❛♠❛r❡❧❛✱ ❛③✉❧✱ ✈❡r❞❡✱ ✈❡r♠❡❧❤♦ ❡ ❧❛r❛♥❥❛✳ ❆ ❋✐❣✉r❛ ✷✳✹ ✐❧✉str❛ ❡st❛ ❝♦♥✜❣✉r❛çã♦✳
❋♦r♠❛t♦s ❡ ✈❛r✐❛çõ❡s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ✷✸
❋✐❣✉r❛ ✷✳✸✿ ❚✐♣♦s ❡ ❢♦r♠❛t♦s ❞❡r✐✈❛❞♦s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳
❋✐❣✉r❛ ✷✳✹✿ ❋❛❝❡s ❞♦ ❝✉❜♦ 3×3×3 ❡ s✉❛ ♣❧❛♥✐✜❝❛çã♦✳
❚❛❜❡❧❛ ✷✳✶✿ ◆♦♠❡s ❡ ❝ó❞✐❣♦s ❞❛s ❢❛❝❡s P♦rt✉❣✉ês ■♥❣❧ês ❈ó❞✐❣♦ ❋r❡♥t❡ ❋r♦♥t F
❚rás ❇❛❝❦ B
❉✐r❡✐t❛ ❘✐❣❤t R
❊sq✉❡r❞❛ ▲❡❢t L
❈✐♠❛ ❯♣ U
❇❛✐①♦ ❉♦✇♥ D
P❛r❛ ❢❛❝✐❧✐t❛r ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦s ♠♦✈✐♠❡♥t♦s ❞♦s ❝✉❜✐♥❤♦s ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ♥♦♠❡♥❝❧❛t✉r❛ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡❧❡s ✉t✐❧✐③❛♥❞♦ ❛ ♠❡s♠❛ r❡❢❡rê♥❝✐❛ ❝✐t❛❞❛ ♥❛ s❡çã♦ ❛♥t❡✲ r✐♦r✱ ✐st♦ é✱ ❛ ❢❛❝❡ ❞❛ ❢r❡♥t❡✳ ❆ ♥♦♠❡♥❝❧❛t✉r❛ é ❜❛s❡❛❞❛ ♥❛ ✐♥t❡rs❡❝çã♦ ❞❛s ❢❛❝❡s ❡♠ q✉❡ ♦ ❝✉❜✐♥❤♦ s❡ ❡♥❝♦♥tr❛✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❝✉❜✐♥❤♦ ❞❡ ❛r❡st❛ q✉❡ ❡st✐✈❡r ♥❛ ✐♥t❡rs❡çã♦ ❞❛ ❢❛❝❡ ❞❛ ❢r❡♥t❡ ✭F✮ ❝♦♠ ❛ ❢❛❝❡ ❞❡ ❝✐♠❛ ✭U✮ s❡rá ❝❤❛♠❛❞♦ ❞❡F U ❡ ♦ ❝✉❜✐♥❤♦ ❞❡ ❝❛♥t♦
q✉❡ ❡st✐✈❡r ♥❛ ✐♥t❡rs❡❝çã♦ ❞❛ ❢❛❝❡ ❞❡ trás ✭B✮✱ ❝♦♠ ❛ ❢❛❝❡ ❞❛ ❞✐r❡✐t❛ ✭R✮ ❡ ❝♦♠ ❛ ❢❛❝❡
✷✹ ❖ ❈✉❜♦ ▼á❣✐❝♦
♦s ❝✉❜✐♥❤♦s ❞♦ ❝✉❜♦ ♠á❣✐❝♦✳
❋✐❣✉r❛ ✷✳✺✿ ◆♦♠❡♥❝❧❛t✉r❛ ❞♦s ❝✉❜✐♥❤♦s✳
❖❜s❡r✈❛çã♦ ✷✳✶✳ ◆ã♦ ❤á ❞✐st✐♥çã♦ ❡♥tr❡ ❛s ♦r❞❡♥s ❝♦♠ q✉❡ ❛ ♥♦t❛çã♦ é ❡s❝r✐t❛✱ ♦✉ s❡❥❛✱ F RD=F DR=RF D=RDF =F RD=F DR.
✷✳✸ ▼♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s
❖ ♠❡❝❛♥✐s♠♦ ✐♥t❡r♥♦ ♣❡r♠✐t❡ q✉❡ ❝❛❞❛ ❢❛❝❡ t❡♥❤❛ ❛ ❧✐❜❡r❞❛❞❡ ❞❡ r♦t❛çã♦ ❞❡ 0◦ ❛ 360◦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ q✉❡ ✜①❛ ♦ ❝✉❜✐♥❤♦ ❝❡♥tr❛❧ ❛♦ ♠❡❝❛♥✐s♠♦ ✐♥t❡r♥♦✱ t❛♥t♦ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦ q✉❛♥t♦ ❛♥t✐✲❤♦rár✐♦ t♦♠❛♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ s❡♠♣r❡ ❛ ❢❛❝❡ q✉❡ ✜❝❛ à ❢r❡♥t❡ ❞❡ q✉❡♠ ♠❛♥✐♣✉❧❛ ♦ ❝✉❜♦✳
❋✐❣✉r❛ ✷✳✻✿ ❊✐①♦s ❞❡ r♦t❛çã♦✳
❖ ♠♦✈✐♠❡♥t♦ s❡rá ❝♦♠♣❧❡t♦ ✭✈✐❞❡ ❋✐❣✉r❛ ✷✳✻✮ q✉❛♥❞♦ ❛ ❢❛❝❡ ❢♦r r♦t❛❝✐♦♥❛❞❛ ❞❡
▼♦✈✐♠❡♥t♦s ❞❛s ❢❛❝❡s ✷✺
❞❡t❡r♠✐♥❛❞❛ ❢❛❝❡✱ q✉❛tr♦ ✈❡③❡s r❡♣❡t✐❞❛ ♦ ♠♦✈✐♠❡♥t♦ ❝♦♠♣❧❡t♦ ♦ ❝✉❜♦ ♥ã♦ é ❛❧t❡r❛❞♦✱ ♦✉ s❡❥❛✱ é ✉♠❛ ♦♣❡r❛çã♦ ♥❡✉tr❛✳
◆♦s ♠ét♦❞♦s ♠❛✐s tr❛❞✐❝✐♦♥❛✐s ❞❡ r❡s♦❧✉çã♦ ❞♦ ❝✉❜♦ ✉t✐❧✐③❛♠✲s❡ ❝ó❞✐❣♦s ♣❛r❛ ✐❞❡♥✲ t✐✜❝❛r ❝❛❞❛ ♠♦✈✐♠❡♥t♦ ❞❡ r♦t❛çã♦ ❞❛s ❢❛❝❡s✳ ❊st❡s ❝ó❞✐❣♦s s❡❣✉❡♠ ❛ ♠❡s♠❛ ❧ó❣✐❝❛ ❞❛ ♥♦♠❡♥❝❧❛t✉r❛ ❞❛s ❢❛❝❡s✱ ✐st♦ é✱ ❛ ♣r✐♠❡✐r❛ ❧❡tr❛ ♠❛✐ús❝✉❧❛ ❞♦ ♥♦♠❡ ❞♦ ❧❛❞♦✱ ❡♠ ✐♥❣❧ês✳ P❛r❛ ❞✐st✐♥❣✉✐r ❛ r♦t❛çã♦ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦ ♦✉ ❛♥t✐✲❤♦rár✐♦✱ ❣❡r❛❧♠❡♥t❡ sã♦ ✉t✐❧✐③❛❞♦s ❞♦✐s sí♠❜♦❧♦s✿ ✉♠❛ ❛♣óstr♦❢❡ s♦❜r❡s❝r✐t❛ ♥❛ ❧❡tr❛ ♦✉ ❛ ❧❡tr❛ s❡❣✉✐❞❛ ❞❡ ✉♠i♠✐♥ús❝✉❧♦✳
P♦r ❡①❡♠♣❧♦✱ s❡ q✉✐s❡r♠♦s r❡❛❧✐③❛r ✉♠ ♠♦✈✐♠❡♥t♦ ❞❛ ❢❛❝❡ ❞✐r❡✐t❛ ✉♠❛ ✈❡③ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❡s❝r❡✈❡♠♦s✶ R′
♦✉ Ri✳
❈❛❞❛ ❝✉❜✐♥❤♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦ ♣♦ss✉✐ ✉♠❛ ♣♦s✐çã♦ ❡ ✉♠❛ ♦r✐❡♥t❛çã♦✳
• ❉❡✜♥✐♠♦s ❝♦♠♦ ♣♦s✐çã♦✱ ♦ ❧♦❝❛❧ ❝♦rr❡t♦ ❡♠ q✉❡ ♦ ❝✉❜✐♥❤♦ ❞❡✈❡ s❡ ❡♥❝♦♥tr❛r ❡♠
r❡❧❛çã♦ ❛♦s ❝✉❜✐♥❤♦s ❞❡ ❝❡♥tr♦✳
• ❉❡✜♥✐♠♦s ❝♦♠♦ ♦r✐❡♥t❛çã♦ ❛ ❝♦♥✜❣✉r❛çã♦ ❝♦rr❡t❛ q✉❡ ❛s ❝♦r❡s ❞❡ ✉♠ ❞❡t❡r♠✐✲
♥❛❞♦ ❝✉❜✐♥❤♦ ❞❡✈❡ s❡ ❡♥❝♦♥tr❛r ❡♠ r❡❧❛çã♦ ❛♦s ❝✉❜✐♥❤♦s ❞❡ ❝❡♥tr♦✳
❋✐❣✉r❛ ✷✳✼✿ ❊①❡♠♣❧♦s ❞❡ ♣♦s✐çã♦ ❡ ♦r✐❡♥t❛çã♦✳
❯♠ ❝✉❜✐♥❤♦ ❞❡ ❛r❡st❛ ♣♦❞❡ ❡st❛r ♥❛ s✉❛ ♣♦s✐çã♦ ❝♦rr❡t❛✱ ♣♦ré♠ ❝♦♠ s✉❛ ♦r✐❡♥t❛çã♦ ✐♥✈❡rt✐❞❛✳ ❖ ♠❡s♠♦ ♣♦❞❡ ♦❝♦rr❡r ❝♦♠ ✉♠ ❝✉❜✐♥❤♦ ❞❡ q✉✐♥❛✳ ◆♦t❡ ♣♦r ❡st❛s ❞✉❛s
✷✻ ❖ ❈✉❜♦ ▼á❣✐❝♦
r❛③õ❡s ✉♠ ❝✉❜✐♥❤♦ ❞❡ ❛r❡st❛ ♣♦ss✉✐ ✶✷ ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❛ss✉♠✐r✱ ✶ ❝♦rr❡t❛ ❡ ✶✶ ❡rr❛❞❛s✱ ❛ss✐♠ ❝♦♠♦ s✉❛s ❝♦r❡s ♣♦❞❡♠ ❛ss✉♠✐r ✷ ♦r✐❡♥t❛çõ❡s ❞✐❢❡r❡♥t❡s✱ ✶ ❝♦rr❡t❛ ❡ ✶ ❡rr❛❞❛✳ ❆♥❛❧♦❣❛♠❡♥t❡ ✉♠ ❝✉❜✐♥❤♦ ❞❡ q✉✐♥❛ ♣♦ss✉✐ ✽ ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❛ss✉♠✐r✱ ✶ ❝♦rr❡t❛ ❡ ✼ ❡rr❛❞❛s✱ ❛ss✐♠ ❝♦♠♦ s✉❛s ❝♦r❡s ♣♦❞❡♠ ❛ss✉♠✐r ✸ ♦r✐❡♥t❛çõ❡s ❞✐❢❡r❡♥t❡s✱ ✶ ❝♦rr❡t❛ ❡ ✷ ❡rr❛❞❛s✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s t❛♥t♦ ❞❡ ♣♦s✐çã♦ ❡ ♦r✐❡♥t❛çã♦ ❝♦rr❡t♦s ❡ ✐♥❝♦rr❡t♦s ♥❛ ✜❣✉r❛ ✷✳✼✳
✷✳✹ ▼❛❝r♦ ♦✉ s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s
❋✐❣✉r❛ ✷✳✽✿ ❈ó❞✐❣♦s ❞❛s r♦t❛çõ❡s ❞❛s ❢❛❝❡s✳
❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s é ❞❡s❝r✐t❛ ♣❡❧❛ s❡q✉ê♥❝✐❛ ❞❡ ❝ó❞✐❣♦s ❞❛s r♦t❛çõ❡s q✉❡ s❡ ❞❡s❡❥❛ ❡❢❡t✉❛r✱ ❡s❝r✐t❛ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ♥❛ ♠❡s♠❛ ♦r❞❡♠ ❡♠ q✉❡ ❛s ♦♣❡r❛çõ❡s ❞❡✈❡♠ s❡r r❡❛❧✐③❛❞❛s ❡ t♦♠❛♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❛ ❢❛❝❡ q✉❡ ❡stá à ❢r❡♥t❡ ❞❡ q✉❡♠ ♦ ♠❛♥✉s❡✐❛✳ ❆ ❡st❛ s❡q✉ê♥❝✐❛ ❞❛r❡♠♦s ♦ ♥♦♠❡ ❞❡ ♠❛❝r♦✳ ◗✉❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ♦✉ ✉♠❛ ♠❛❝r♦ ❢♦r ❛♣❧✐❝❛❞❛ ♥♦ ❝✉❜♦ ✉♠❛ ✐♠♣♦rt❛♥t❡ r❡❣r❛ ❞❡✈❡ s❡r s❡❣✉✐❞❛✿ ✉♠❛ ✈❡③ ✐♥✐❝✐❛❞❛ ❛ s❡q✉ê♥❝✐❛ ❛ ❢❛❝❡ ❢r♦♥t❛❧ ❞❡ r❡❢❡rê♥❝✐❛ ♥ã♦ ♣♦❞❡ s❡r ❛❧t❡r❛❞❛ ❛té q✉❡ ❡❧❛ s❡❥❛ ❝♦♥❝❧✉í❞❛✳ ◗✉❛❧q✉❡r r♦t❛çã♦ ❞♦ ♣ró♣r✐♦ ❝✉❜♦ ❡♠ q✉❛❧q✉❡r ❞✐r❡çã♦ ❡ s❡♥t✐❞♦ ✐rá ✐♥t❡r✲ ❢❡r✐r ♥❛ s❡q✉ê♥❝✐❛✳ P♦r ❡st❛ r❛③ã♦✱ q✉❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s é ❡❢❡t✉❛❞❛✱ ❛s ú♥✐❝❛s r♦t❛çõ❡s q✉❡ ❞❡✈❡♠ s❡r ❛♣❧✐❝❛❞❛s s❡rã♦ s♦♠❡♥t❡ ♥❛s ❢❛❝❡s r❡s♣❡❝t✐✈❛s ❛♦ ❝ó✲ ❞✐❣♦✳ ❆ ❋✐❣✉r❛ ✷✳✽ ✐❧✉str❛ ♦s ❝ó❞✐❣♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♥♦s ❛❧❣♦r✐t♠♦s ❞❡ s♦❧✉çã♦ ❞♦ ❝✉❜♦✳
❆❣♦r❛ ✈❛♠♦s s✉♣♦r q✉❡ q✉❡r❡♠♦s ❡❢❡t✉❛r q✉❛tr♦ ♠♦✈✐♠❡♥t♦s s❡❣✉✐❞♦s ♥❛ s❡❣✉✐♥t❡ ♦r❞❡♠✿ ✉♠❛ r♦t❛çã♦ ❞❛ ❢❛❝❡ s✉♣❡r✐♦r ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦✱ ✉♠❛ r♦t❛çã♦ ❞❛ ❢❛❝❡ ❞✐r❡✐t❛ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦✱ ✉♠❛ r♦t❛çã♦ ❞❛ ❢❛❝❡ s✉♣❡r✐♦r ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❡ ✉♠❛ r♦t❛çã♦ ❞❛ ❢❛❝❡ ❞✐r❡✐t❛ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✳ ❖ ❝ó❞✐❣♦ ♣❛r❛ ❡st❛ ♠❛❝r♦ ❙ s❡rá✿
▼❛❝r♦ ♦✉ s❡q✉ê♥❝✐❛s ❞❡ ♠♦✈✐♠❡♥t♦s ✷✼
❈❛s♦ s❡ q✉❡✐r❛ ❛♣❧✐❝❛r ✉♠ ♠❡s♠♦ ♠♦✈✐♠❡♥t♦ X ♠❛✐s ❞❡ ✉♠❛ ✈❡③✱ ✉t✐❧✐③❛✲s❡ ❛ s❡✲
❣✉✐♥t❡ ♥♦t❛çã♦✿ S =XX =X2✱ T =XXX =X3✱ ❡t❝✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ s❡ ❡❢❡t✉❛r
❞✉❛s r♦t❛çõ❡s ❝♦♥s❡❝✉t✐✈❛s ❞❛ ❢❛❝❡ s✉♣❡r✐♦r ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦ ❡s❝r❡✈❡✲s❡✿ S =U2✳
❈♦♠♦ é ♣♦ssí✈❡❧ ♣❡r❝❡❜❡r✱ ❛ ❝♦❞✐✜❝❛çã♦ ❞♦s ♠♦✈✐♠❡♥t♦s é ❡ss❡♥❝✐❛❧✱ ❛ ✜♠ ❞❡ t♦r♥❛r t❛♥t♦ ❛ ❧❡✐t✉r❛ q✉❛♥t♦ ❛ ♣ró♣r✐❛ ❡①❡❝✉çã♦ ❞❛s ♠❛❝r♦s ♠❛✐s s✐♠♣❧❡s ❡ á❣✐❧✷✳ ◆♦ ❈❛♣ít✉❧♦ ✺ ✈♦❧t❛r❡♠♦s ❛ tr❛t❛r ❞♦s ❝ó❞✐❣♦s ❡ s❡q✉ê♥❝✐❛s q✉❡ ❝♦♠♣õ❡♠ ♦s ❛❧❣♦r✐t♠♦s q✉❡ ❧❡✈❛♠ à s♦❧✉çã♦ ❞♦ ❝✉❜♦✳
✸ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛❜♦r❞❛r ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ➪❧❣❡❜r❛✱ q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥✲ tr❛❞♦s ♥❛s r❡❢❡rê♥❝✐❛s ❬✶❪✱ ❬✷❪✱ ❬✸❪ ❡ ❬✹❪ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ❡ ❣r✉♣♦s ❞❡ ♣❡r♠✉t❛çã♦✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ r❡❧❛❝✐♦♥❛r ♠♦✈✐♠❡♥t♦s ❡s♣❡❝í✜❝♦s ❞♦ ❝✉❜♦ ♠á❣✐❝♦ ❝♦♠ t❛✐s ❝♦♥❝❡✐t♦s✳ ❆ ✐❞❡✐❛ é ✉t✐❧✐③❛r ❡st❛s ❞✉❛s t❡♦r✐❛s ♣❛r❛ ❛✉①✐❧✐❛r ❛ ♦❜t❡♥çã♦ ❞❡ ♠❛❝r♦s q✉❡ ♣♦ss✐❜✐❧✐t❡♠✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝✉❜♦ ❡st❡❥❛ ❡♠❜❛r❛❧❤❛❞♦✱ r❡t♦r♥á✕❧♦ à ♣♦s✐çã♦ ♦r✐❣✐♥❛❧✱ ✐st♦ é✱ ❛s ✻ ❢❛❝❡s ❡st❛r❡♠ t♦❞❛s ❝♦♠ s✉❛s r❡s♣❡❝t✐✈❛s ❝♦r❡s✳
✸✳✶ ❚❡♦r✐❛ ❇ás✐❝❛ ❞❡ ●r✉♣♦s
❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠ ❝♦♥❥✉♥t♦ G ♥ã♦ ✈❛③✐♦✱ ❝♦♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♦♣❡r❛çã♦(∗) t❛❧ q✉❡
G×G→G
(a, b)7→a∗b
s❡rá ❝❤❛♠❛❞♦ ❞❡ ❣r✉♣♦ ❡ ❞❡♥♦t❛♠♦s ♣♦r (G,∗) s❡ ❛s três s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♣❛r❛ t❛❧
♦♣❡r❛çã♦ ❢♦r❡♠ s❛t✐s❢❡✐t❛s✿ ✶✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿
a∗(b∗c) = (a∗b)∗c, ∀a, b, c∈G.
✷✳ ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿
∃e∈Gt❛❧ q✉❡e∗a=a∗e=a, ∀a∈G.
✸✳ ❊❧❡♠❡♥t♦ s✐♠étr✐❝♦✿
∀a∈G, ∃a′ ∈Gt❛❧ q✉❡ a∗a′ =a′∗a=e.
❯♠ ❣r✉♣♦ é ❝❤❛♠❛❞♦ ❞❡ ❛❜❡❧✐❛♥♦ ♦✉ ❝♦♠✉t❛t✐✈♦ s❡ ❛♣r❡s❡♥t❛ t❛♠❜é♠ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
✹✳ ❈♦♠✉t❛t✐✈✐❞❛❞❡✿
a∗b =b∗a, ∀a, b∈G.
✸✵ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
❊①❡♠♣❧♦ ✸✳✶✳ ●r✉♣♦s✿
✶✳ (Z,+)✱ (Q,+)✱ (R,+)✱ (C,+) sã♦ ❣r✉♣♦s ❛❞✐t✐✈♦s ❡ (Q\ {0},·)✱ (R\ {0},·) ❡ (C\ {0},·)sã♦ ❣r✉♣♦s ♠✉❧t✐♣❧✐❝❛t✐✈♦s✳ ◆♦t❡ q✉❡ t♦❞♦s ❡❧❡s sã♦ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s✱
♣♦✐s ❡♠ t♦❞♦s ❡❧❡s ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ é ✈á❧✐❞❛✳
✷✳ P❛r❛ ❛♣❧✐❝❛r ♣r♦♣r✐❡❞❛❞❡s ❞❛ ➪❧❣❡❜r❛ ♥♦ ❝✉❜♦ ♠á❣✐❝♦✱ ✈❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❣❡✲ ♦♠étr✐❝♦✱ ♦♥❞❡ ❡s❝r❡✈❡r❡♠♦s ♦ ❣r✉♣♦ GQ r❡❢❡r❡♥t❡ às s✐♠❡tr✐❛s ❡s♣❛❝✐❛✐s ❞❡ ✉♠
q✉❛❞r❛❞♦✳
❙❡❥❛ Q1Q2Q3Q4 ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦ ❝❡♥tr♦ ❡stá ♥❛ ♦r✐❣❡♠ O ❞♦s ❡✐①♦s ❝❛rt❡s✐❛✲
♥♦s✳ ❈❤❛♠❛r❡♠♦s ❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣❡❧❛s ❞✐❛❣♦♥❛✐s ❞♦ q✉❛❞r❛❞♦ ❡ ♣♦r s✉❛s ♠❡❞✐❛tr✐③❡s ❞❡ d1✱ d2✱ r ❡ s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✸✳✶✳
❋✐❣✉r❛ ✸✳✶✿ ◗✉❛❞r❛❞♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞♦ ❣r✉♣♦ GQ✳
❆s tr❛♥s❢♦r♠❛çõ❡s ❡s♣❛❝✐❛✐s ✭s✐♠❡tr✐❛s✮ q✉❡ ♣r❡s❡r✈❛♠ ❛ ❢♦r♠❛ ❞♦ q✉❛❞r❛❞♦ ♣♦✲ ❞❡♠ ♦❝♦rr❡r t❛♥t♦ ♥♦ ♣❧❛♥♦ q✉❛♥t♦ ♥♦ ❡s♣❛ç♦✿
• ❚r❛♥s❢♦r♠❛çõ❡s ♣❧❛♥❛s✿
❉❡♥♦t❛r❡♠♦s ♣♦r id✱ Rπ
2✱ Rπ ❡ R 3π
2 ❛s r♦t❛çõ❡s ❡♠ t♦r♥♦ ❞❛ ♦r✐❣❡♠ O ❞♦
q✉❛❞r❛❞♦✱ ❞❡ â♥❣✉❧♦s0✱ π
2✱π❡
3π
2 ✱ ♥♦ s❡♥t✐❞♦ ❛♥t✐✕❤♦rár✐♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
• ❚r❛♥s❢♦r♠❛çõ❡s ❡s♣❛❝✐❛✐s✿
❉❡♥♦t❛r❡♠♦s ♣♦r R1✱ R2✱ Rr ❡ Rs ❛s r♦t❛çõ❡s ❞❡ â♥❣✉❧♦ π ❡♠ r❡❧❛çã♦ ❛♦s
❡✐①♦sd1✱ d2✱ r ❡ s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆ss✐♠ GQ = n
id, Rπ
2, Rπ, R 3π
2 , R1, R2, Rr, Rs o
❡ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s é ✉♠ ❣r✉♣♦✳
◆♦t❡ q✉❡ ♦ ❣r✉♣♦GQ ♥ã♦ é ❛❜❡❧✐❛♥♦✱ ♣♦✐s ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ♥ã♦ é ✈❡r✐✜✲
●r✉♣♦s ❛❞✐t✐✈♦s ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦s ❞❡ ❝❧❛ss❡s ❞❡ r❡st♦s ✸✶
❋✐❣✉r❛ ✸✳✷✿ ❈♦♠♣♦s✐çã♦Rr◦R1✳
▲♦❣♦ Rr◦R1 =Rπ
2✳
❋✐❣✉r❛ ✸✳✸✿ ❈♦♠♣♦s✐çã♦R1◦Rr✳
▲♦❣♦ R1◦Rr =R3π
2 ✳
❱❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡ ♦ ●r✉♣♦ ❞❡ ❘✉❜✐❦ ♥ã♦ é ❛❜❡❧✐❛♥♦✳ ❊♠ ♣❛rt✐❝✉❧❛r ❡st❡ é ✉♠ ❞♦s ♠♦t✐✈♦s ♣❡❧♦ q✉❛❧ ♦ ❈✉❜♦ ❞❡ ❘✉❜✐❦ ❛♣r❡s❡♥t❛ ❛❧t♦ ♥í✈❡❧ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ❡♠ s❡r r❡s♦❧✈✐❞♦✳
✸✳✷ ●r✉♣♦s ❛❞✐t✐✈♦s ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦s ❞❡ ❝❧❛ss❡s ❞❡
r❡st♦s
❖ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ r❡st♦ ♠ó❞✉❧♦m✱ é ♦ ❝♦♥❥✉♥t♦ q✉♦❝✐❡♥t❡ ❞❡Z♣❡❧❛ r❡❧❛çã♦
❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ♠ó❞✉❧♦ m✳ ■st♦ é
Zm ={0,1,2, . . . , m−1},∀m >1;m∈Z.
❙❡♥❞♦ ❛ss✐♠0é ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ✐♥t❡✐r♦s ❝ô♥❣r✉♦s ❛ ✵✱ ♠ó❞✉❧♦m✱1♣♦r t♦❞♦s
♦s ✐♥t❡✐r♦s ❝ô♥❣r✉♦s ❛ ✶✱ ♠ó❞✉❧♦ m✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳
❆❣♦r❛ s❡❥❛ ❛ ♦♣❡r❛çã♦ (+) ❞❡ ❛❞✐çã♦ ♠ó❞✉❧♦ ♠ s♦❜r❡ Zm✱ ❛ q✉❛❧ ✈❛❧❡ ❛ ❛ss♦❝✐❛t✐✲
✈✐❞❛❞❡ ❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ q✉❡ é ❞❡✜♥✐❞❛ ❞❡✜♥✐❞❛ ♣♦r
a+b=a+b✱ ♦♥❞❡ a, b∈Zm.
✸✷ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
a+ 0 =a+ 0 =a.
P♦rt❛♥t♦✱ 0é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✳
a+m−a=a+ (m−a) =m = 0✱ ❥á q✉❡ m≡0 (♠♦❞m)
P♦rt❛♥t♦✱ −a=m−a é ♦ ❡❧❡♠❡♥t♦ ♦♣♦st♦✳
❆ss✐♠✱ (Zm,+) é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ m > 1✱ q✉❡ é ❝❤❛♠❛❞♦ ❞❡
❣r✉♣♦ ❛❞✐t✐✈♦ ❞❡ ❝❧❛ss❡s ❞❡ r❡st♦ ♠ó❞✉❧♦ ♠✳ ◆♦t❡ q✉❡ ❛ ♦r❞❡♠ ❞❡st❡ ❣r✉♣♦ é ♠✳
❆❣♦r❛ s❡❥❛ t❛♠❜é♠ ❛ ♦♣❡r❛çã♦(·)❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó❞✉❧♦ ♠ ❡♠Zm ♦♥❞❡ t❛♠❜é♠
sã♦ ✈á❧✐❞❛s ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ q✉❡ é ❞❡✜♥✐❞❛ ♣♦r✿
a·b=ab✱ ♦♥❞❡a, b∈Zm.
❊♠ r❡❧❛çã♦ à ❡st❛ ♦♣❡r❛çã♦ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✿
a·1 =a·1 =a.
P♦rt❛♥t♦ 1é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✳
❆❣♦r❛ é ♣r❡❝✐s♦ ♥♦t❛r q✉❡ ♦ ❡❧❡♠❡♥t♦0♥ã♦ ❡stá ❞❡✜♥✐❞♦ ♥♦ ❝♦♥❥✉♥t♦Zm❝♦♠ ❛ ♦♣❡✲
r❛çã♦ (·)✱ ♣♦✐s ❡❧❡ ♥ã♦ ❛♣r❡s❡♥t❛ ✉♠ ✐♥✈❡rs♦ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó❞✉❧♦ ♠✱ ❧♦❣♦ ❞❡✈❡♠♦s
❝♦♥s✐❞❡r❛r ❛♣❡♥❛s ♦ ❝♦♥❥✉♥t♦ Zm− {0}✳ ❊♥tr❡t❛♥t♦✱ ♠❡s♠♦ ♦❜s❡r✈❛❞❛ ❡st❛ r❡str✐çã♦✱
♥ã♦ ❤á ❣❛r❛♥t✐❛s ❞❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ r❡st❛♥t❡ s❡rá ✉♠ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❈♦♠ ❡❢❡✐t♦✱ t❡♠♦s q✉❡ ❛ r❡str✐çã♦ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó❞✉❧♦ ✹ ❛♦s ❡❧❡♠❡♥t♦s ❞❡ Z4− {0}✱ ♥❡♠
s❡q✉❡r é ✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡ ❡st❡ ❝♦♥❥✉♥t♦✱ ✉♠❛ ✈❡③ q✉❡ 2·2 = 4 = 0✳
◆❡st❡ s❡♥t✐❞♦✱ ♣r♦✈❛r❡♠♦s q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó❞✉❧♦ m ❛♣❧✐❝❛❞❛ ❛♦s ❡❧❡♠❡♥t♦s ❞❡
Z∗
m =Zm− {0} é ✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡ ❡st❡ ❝♦♥❥✉♥t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱m é ✉♠ ♥ú♠❡r♦
♣r✐♠♦✳
Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s s✉♣♦r✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ m ♥ã♦ é ♣r✐♠♦✳ ❈♦♠♦ m > 1✱
♣♦❞❡♠♦s ♦❜t❡r ❞♦✐s ✐♥t❡✐r♦s a, b > 1 t❛✐s q✉❡ ab = m✳ ▲♦❣♦ ab = m✳ ❊ ♥❡st❡ ❝❛s♦✱
❝♦♠♦ ab=a·b ❡m = 0✱ ❡♥tã♦ t❡rí❛♠♦s q✉❡ a·b = 0✱ ♦ q✉❡ é ❛❜s✉r❞♦ ♣❡❧❛ ❤✐♣ót❡s❡✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ✈❛♠♦s s✉♣♦r✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ab = 0 ♣❛r❛ ❛❧❣✉♠ a, b ∈ Z∗m✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡ t❡♠♦s q✉❡ab≡0 (♠♦❞m)✳ ▲♦❣♦✱m |ab✱ ❡ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡m é
♣r✐♠♦✱ ❡♥tã♦ m|a ♦✉ m|b✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ✈❛♠♦s s✉♣♦r ♦ ♣r✐♠❡✐r♦ ❝❛s♦✱
❈❧❛ss❡s ▲❛t❡r❛✐s ✸✸
♦ q✉❡ é ❛❜s✉r❞♦✱ ❥á q✉❡ ♣♦r ❤✐♣ót❡s❡ a∈Z∗m✳
P♦r ✜♠✱ ✈❛♠♦s ♠♦str❛r q✉❡✱ s❡ m é ♣r✐♠♦✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó❞✉❧♦ m✱ q✉❛♥❞♦
r❡str✐t❛ ❛♦s ❡❧❡♠❡♥t♦s ❞❡ Z∗m✱ ❢❛③ ❞❡ss❡ ❝♦♥❥✉♥t♦ ✉♠ ❣r✉♣♦✳ P❛r❛ ✐st♦✱ ❜❛st❛ ♠♦str❛r
q✉❡ ∀ a ∈ Z∗m✱ ♣♦❞❡✲s❡ ❡♥❝♦♥tr❛r b ∈ Zm∗ t❛❧ q✉❡ a·b = 1✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ a ∈ Z∗m✱
❡♥tã♦ a ♥ã♦ é ♠ú❧t✐♣❧♦ ❞❡m✱ ❡ ❝♦♠♦m é ♣r✐♠♦✱ ❡♥tã♦mdc(m, a) = 1✳ ▲♦❣♦✱ ♣♦❞❡♠♦s
❡s❝r❡✈❡r mx0 +ay0 = 1✱ ♣❛r❛ ❝♦♥✈❡♥✐❡♥t❡s ✐♥t❡✐r♦s x0 ❡ y0 ✭✐❞❡♥t✐❞❛❞❡ ❞❡ ❇❡③♦✉t✮✳
❘❡❞✉③✐♥❞♦✲s❡ ❡ss❛ ✐❣✉❛❧❞❛❞❡ ♠ó❞✉❧♦ m t❡♠♦s✿
mx0+ay0 = 1 =m·x0+a·y0 =a·y0 = 1
♦♥❞❡ ♦❜t❡♠♦s q✉❡ y0✱ q✉❡ ♣❡rt❡♥❝❡ ❛ Z∗m✱ é ♦ ✐♥✈❡rs♦ ❞❡a✳
❆ss✐♠✱ (Z∗m,·) é ✉♠ ❣r✉♣♦✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ m é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❱❡❥❛♠♦s ✉♠
❡①❡♠♣❧♦✿
❊①❡♠♣❧♦ ✸✳✷✳ ❯s❛♥❞♦ ♦ r❛❝✐♦❝í♥✐♦ ✉t✐❧✐③❛❞♦ ♥❛ ❞❡♠♦s♥tr❛çã♦ ❛❝✐♠❛✱ ✈❛♠♦s ❞❡t❡r✲ ♠✐♥❛r ♦ ✐♥✈❡rs♦ ❞❡ 4 ❡♠ Z∗5✳ ▲♦❣♦✱ ✉♠❛ ♣♦ssí✈❡❧ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❉✐♦❢❛♥t✐♥❛
5x0+ 4y0 = 1 ♣♦❞❡ s❡r ♦ ♣❛r (x0, y0) = (1,−1)✱ ♦✉ s❡❥❛✱ y0 =−1✳ P♦rt❛♥t♦ ♦ ✐♥✈❡rs♦
❞❡ 4é −1 = 4✱ ♣♦✐s4≡ −1 (♠♦❞5)✳
✸✳✸ ❈❧❛ss❡s ▲❛t❡r❛✐s
❙❡❥❛G✉♠ ❣r✉♣♦ ❡ s❡❥❛H ✉♠ s✉❜❣r✉♣♦ ❞❡G✳ ➚ r❡s♣❡✐t♦ ❞❡G✱ ❞❡✜♥✐♠♦s ❛ r❡❧❛çã♦
❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ∼
E ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
y∼
E x⇔ ∃h ∈H t❛❧ q✉❡ y=xh.
P♦r ❞❡✜♥✐çã♦✱ ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ q✉❡ ❝♦♥té♠ ♦ ❡❧❡♠❡♥t♦ x é ♦ ❝♦♥❥✉♥t♦
n
y∈g |y ∼
E x o
= {xh|h∈H}❀ ❞❡♥♦t❛r❡♠♦s ❡st❡ ❝♦♥❥✉♥t♦ ♣♦r xH ❡ ♦ ❝❤❛♠❛r❡✲
♠♦s ❞❡ ❝❧❛ss❡ ❧❛t❡r❛❧ à ❡sq✉❡r❞❛ ❞❡ H ❡♠ G q✉❡ ❝♦♥té♠ x✳ ❈❛s♦ ♥ã♦ ❤❛❥❛ ❝♦♥❢✉sã♦✱
s✐♠♣❧❡s♠❡♥t❡ ❛ ❝❤❛♠❛r❡♠♦s ❞❡ ❝❧❛ss❡ ❧❛t❡r❛❧ ❞❡ x à ❡sq✉❡r❞❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ H é ❛
❝❧❛ss❡ ❧❛t❡r❛❧ ❞♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❡ à ❡sq✉❡r❞❛✳ ❖❜s❡r✈❡ q✉❡ y∈xH ⇔yH =xH✳
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ❛✐♥❞❛ ❞❡✜♥✐r ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✿
y∼
D x⇔ ∃h∈H t❛❧ q✉❡ y=hx.
❉❡st❛ ❢♦r♠❛ ♦❜t❡♠♦s ❛s ❝❧❛ss❡s ❧❛t❡r❛✐s à ❞✐r❡✐t❛ ❞❡ H ❡♠ G✳ P♦rt❛♥t♦ ❛ ❝❧❛ss❡
❧❛t❡r❛❧ ❞❡ x à ❞✐r❡✐t❛ é Hx={hx |h∈H}✳
❉❡✜♥✐çã♦ ✸✳✷✳ ❆ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❧❛t❡r❛✐s à ❡sq✉❡r❞❛ é ♦ í♥❞í❝❡ ❞❡ H ❡♠ G✳ ❉❡♥♦t❛r❡♠♦s ❡❧❡ ♣♦r (G:H)✳
❖❜s❡r✈❛çã♦ ✸✳✶✳ ❖ í♥❞✐❝❡ ❞❡ H ❡♠ G t❛♠❜é♠ é ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❛s
✸✹ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
ϕ :{❝❧❛ss❡s ❧❛t❡r❛✐s à ❡sq✉❡r❞❛} → {❝❧❛ss❡s ❧❛t❡r❛✐s à ❞✐r❡✐t❛}
xH →Hx−1
❉❡✜♥✐çã♦ ✸✳✸✳ ❉❛❞❛ ✉♠❛ ♣❛rt✐çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✱ ✉♠ s✐st❡♠❛ ❞❡ r❡♣r❡s❡♥t❛♥t❡s é ✉♠ ❝♦♥❥✉♥t♦ {xα}α∈Γ q✉❡ t❡♠ ❡①❛t❛♠❡♥t❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ❞❛ ♣❛rt✐çã♦✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ q✉❛❧q✉❡r s✐st❡♠❛ ❞❡ ♣r❡s❡♥t❛♥t❡s ❞❛s ❝❧❛ss❡s ❧❛t❡r❛✐s à ❡sq✉❡r❞❛ ❞❡ H ❡♠ G é ✐❣✉❛❧ ❛ (G:H)✳
Pr♦♣♦s✐çã♦ ✸✳✶✳ ❚♦❞❛s ❛s ❝❧❛ss❡s ❧❛t❡r❛✐s ❞❡ H ❡♠ G tê♠ ❛ ♠❡s♠❛ ❝❛r❞✐♥❛❧✐❞❛❞❡✱
✐❣✉❛❧ à ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ H✳
❉❡♠♦♥str❛çã♦✳ ❆ ❢✉♥çã♦
H →xH h 7→xh
é ❝❧❛r❛♠❡♥t❡ ✉♠❛ ❜✐❥❡çã♦✳
❚❡♦r❡♠❛ ✸✳✶✳ ✭❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡✮ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ H ✉♠ s✉❜❣r✉♣♦ ❞❡ G✳ ❊♥tã♦ t❡r❡♠♦s |G|=|H|(G:H)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ♦r❞❡♠ ❡ ♦ í♥❞✐❝❡ ❞❡ H ❞✐✈✐❞❡♠
❛ ♦r❞❡♠ ❞❡ G✳
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ à ❡sq✉❡r❞❛∼
E ❡♠ G✳ ❉❡st❛
❢♦r♠❛ ✐r❡♠♦s ♦❜t❡r ✉♠❛ ♣❛rt✐çã♦ ❞❡G❡♠ ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r
♠♦str❛ q✉❡ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❝❧❛ss❡s t❡♠♦s |H| ❡❧❡♠❡♥t♦s✳ ❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱ ♦
♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s é (G:H)✱ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ |G|=|H|(G:H)
❈♦r♦❧ár✐♦ ✸✳✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ s❡❥❛ α ∈ G✳ ❊♥tã♦ ❛ ♦r❞❡♠ ❞❡ α ❞✐✈✐❞❡ ❛
♦r❞❡♠ ❞❡ G✳
❉❡♠♦♥str❛çã♦✳ P♦r ❞❡✜♥✐çã♦✱O(α) =|hαi|✳ P♦r ✜♠ ❛♣❧✐❝❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡
❛♦ s✉❜❣r✉♣♦ hαi✳ ◆♦t❡ q✉❡✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❡st❡ ❝♦r♦❧ár✐♦ ♠♦str❛ q✉❡ α|G|=e ❈♦r♦❧ár✐♦ ✸✳✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠ ♣r✐♠❛✳ ❊♥tã♦G é ❝í❝❧✐❝♦✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ α ∈ G\ {e} ❡ ❝♦♥s✐❞❡r❡ hαi ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r α✳ P❡❧♦
❚❡♦r❡♠❛ ❞❡ ▲❛❣r❛♥❣❡✱ |hαi| ❞✐✈✐❞❡ |G| ❡ ♣♦rt❛♥t♦ |hαi|=|G|✱ ♣♦✐s |G| é ♣r✐♠♦✳ ▲♦❣♦
G=hGi✳
✸✳✹ ●r✉♣♦ ❞❡ ❘✉❜✐❦
❈♦♥❢♦r♠❡ ✈✐st♦ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ❝❤❛♠❛♠♦s ❞❡ R✱L✱ F✱ B✱U ❡ D✱ ♦s ♠♦✈✐♠❡♥t♦s ♥♦
s❡♥t✐❞♦ ❤♦rár✐♦ ❞❛s ❢❛❝❡s ❞✐r❡✐t❛✱ ❡sq✉❡r❞❛✱ ❢r❡♥t❡✱ trás✱ ❝✐♠❛ ❡ ❜❛✐①♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♠♦✈✐♠❡♥t♦s ♣❡r♠✐t✐❞♦s ♥♦ ❝✉❜♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛♦ ❡♠❜❛r❛❧❤á✲❧♦✱ ❝❤❛♠❛r❡♠♦s ❞❡ ●r✉♣♦ R ❞❡ ❘✉❜✐❦✳ ❱❛♠♦s ✈❡r✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞❛s três ❝♦♥❞✐çõ❡s
●r✉♣♦ ❞❡ ❘✉❜✐❦ ✸✺
✶✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿
❆♦ r❡❛❧✐③❛r♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ q✉❛❧q✉❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡ ✸ ♠♦✈✐♠❡♥t♦s ❣❡♥ér✐❝♦s
X✱ Y✱ Z✱ ✈❡r✐✜❝❛♠♦s q✉❡✿ X(Y Z) = (XY)Z✳
❆♦ ♠❛♥✐♣✉❧❛r ♦ ❝✉❜♦ ❛ ♣r♦✈❛ ❞❡st❛ ❝♦♥❞✐çã♦ é ❡✈✐❞❡♥t❡ ❡ ❛ ❝♦♥s✐❞❡r❛r❡♠♦s ❝♦♠♦ tr✐✈✐❛❧✳
✷✳ ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿
❆ ❡①✐stê♥❝✐❛ ❞♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ é ✈❡r✐✜❝❛❞❛ ♣❡❧♦ ♠♦✈✐♠❡♥t♦ ✏♥ã♦ ❢❛③❡r ♥❛❞❛✑ ❡♠ q✉❛❧q✉❡r ❞❛s ✻ ❢❛❝❡s✳ ❈♦♠ ✐ss♦ ❣❛r❛♥t✐♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ I ❞❡ q✉❛❧q✉❡r ✉♠❛
❞❛s s❡✐s ❢❛❝❡s ❞♦ ❝✉❜♦ ♦✉ ❞❡ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s✳
✸✳ ❊❧❡♠❡♥t♦ ✐♥✈❡rs♦✿
❖ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ♥♦ ●r✉♣♦R❞❡ ❘✉❜✐❦ s✐❣♥✐✜❝❛ ❞❡s❢❛③❡r ❛ s❡q✉ê♥❝✐❛✱ ❞❡ ✉♠ ♦✉
♠❛✐s ♠♦✈✐♠❡♥t♦s✱ r❡❛❧✐③❛❞❛✳ ❊ ❛q✉✐ ❤á ❞✉❛s ✐♠♣♦rt❛♥t❡s ❝♦♥s✐❞❡r❛çõ❡s ❛ s❡r❡♠ ❢❡✐t❛s✿ s❡ ♦ ♠♦✈✐♠❡♥t♦ ❢♦✐ ❞❡ ❛♣❡♥❛s ✉♠❛ ❢❛❝❡ ♦✉ ❞❡ ✉♠❛ ♠❛❝r♦✳ ❙❡❥❛ ✉♠ ♠♦✲ ✈✐♠❡♥t♦ ❣❡♥ér✐❝♦ X✱ ✉♠❛ r♦t❛çã♦ ❞❡ ✉♠❛ ❞❛s ✻ ❢❛❝❡s ❞❡ 90◦ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦✱ ❡♥tã♦ s❡✉ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ s❡rá ♦ ♠♦✈✐♠❡♥t♦✱ t❛♠❜é♠ ❞❡ 90◦✱ ❞❡st❛ ♠❡s♠❛ ❢❛❝❡✱ ♣♦ré♠ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✳ ❉❡♥♦t❛r❡♠♦s ❡st❡ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ X ❝♦♠♦
s❡♥❞♦ X−1 ♦✉ s✐♠♣❧❡s♠❡♥t❡ X′✳ ❈❛s♦ s❡❥❛ ❢❡✐t♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s✱ ❡♥tã♦ ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ s❡rá ❛ r❡❛❧✐③❛çã♦ ❞♦s ✐♥✈❡rs♦s ❞♦s ♠♦✈✐♠❡♥t♦s✱ ♣♦ré♠ ♥❛ ♦r❞❡♠ ✐♥✈❡rs❛✱ ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳
❊①❡♠♣❧♦ ✸✳✸✳ ❙❡❥❛ ✉♠❛ ♠❛❝r♦ S =F R′
U✱ ❡♥tã♦ s❡✉ ✐♥✈❡rs♦ s❡rá S−1 =U′
RF′
✳
❖❜s❡r✈❛çã♦ ✸✳✷✳ ❖ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❡♠ ✉♠ ❣r✉♣♦ é ú♥✐❝♦✱ ♣♦r ✐ss♦ q✉❡ ♥♦ ❝❛s♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦ é ♥❡❝❡ssár✐♦ q✉❡ ❡❧❡ ❡st❡❥❛ ❜❡♠ ❞❡✜♥✐❞♦ ♣❛r❛ q✉❡ s❡ ❡✈✐t❡ ❞✉♣❧✐❝✐❞❛❞❡ q✉❛♥❞♦ s❡ ❞❡s❡❥❛ ♦❜t❡r ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♠♦✈✐♠❡♥t♦ ♦✉ ♦ ❞❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦s✳ P♦r ❡st❡ ♠♦t✐✈♦✱ ❞❡✈❡♠♦s t❡r ❝❧❛r♦ q✉❡ r❡❛❧✐③❛r ❛♣❡♥❛s ✉♠ ♠♦✈✐♠❡♥t♦ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ s❡ r❡❛❧✐③❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♠♦✈✐♠❡♥t♦s✱ ❡ ❝♦♠ ✐ss♦ ❣❛r❛♥t✐✲ ♠♦s ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡♥tr♦ ❞♦ ●r✉♣♦ R ❞❡ ❘✉❜✐❦✳
❆♣❡s❛r ❞❡ ✈❛❧❡r ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ❡♥tr❡ ♠♦✈✐♠❡♥t♦s ❡♠ ❢❛❝❡s ♦♣♦st❛s✱ ❛ ♠❡s♠❛ ♥ã♦ é ✈❡r✐✜❝❛❞❛ ❡♥tr❡ ♠♦✈✐♠❡♥t♦s ❡♠ ❢❛❝❡s ❛❞❥❛❝❡♥t❡s✳ ❖s ♠♦✈✐♠❡♥t♦s F R ❡ RF ♥ã♦ sã♦ ✐❣✉❛✐s✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✸✳✹✳
❆❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♠♦✈✐♠❡♥t♦s ❝♦♠✉t❛t✐✈♦s sã♦✿
✸✻ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
❋✐❣✉r❛ ✸✳✹✿ ▼❛❝r♦sF R ❡ RF✳
P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ●r✉♣♦ R ❞❡ ❘✉❜✐❦ ♥ã♦ é ❛❜❡❧✐❛♥♦✳ ❊st❡ ❢❛t♦ t♦r♥❛ ♦
❝✉❜♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ q✉❡❜r❛✲❝❛❜❡ç❛s ❞✐❢í❝✐❧ ❞❡ s❡r r❡s♦❧✈✐❞♦✳
✸✳✺ ❙✉❜❣r✉♣♦s
❉❡✜♥✐çã♦ ✸✳✹✳ ❙❡❥❛(G,∗) ✉♠ ❣r✉♣♦✱ ❝❤❛♠❛r❡♠♦s ❞❡ s✉❜❣r✉♣♦ ❞❡G✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ H ❞❡ G ♥ã♦ ✈❛③✐♦ ✭❞❡♥♦t❛r❡♠♦s ♣♦r H < G✮ q✉❛♥❞♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ G✱ ❡st❡
s✉❜❝♦♥❥✉♥t♦ H é ✉♠ ❣r✉♣♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s s❛t✐s❢❡✐t❛s ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ h ❞❡ H✿
✶✳ ❋❡❝❤❛♠❡♥t♦✿
h1∗h2 ∈H, ∀h1, h2 ∈H.
✷✳ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿
h1∗(h2∗h3) = (h1∗h2)∗h3, ∀h1, h2, h3 ∈H.
✸✳ ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿
∃eh ∈H t❛❧ q✉❡eh∗h=h∗eh =h, ∀h∈H.
✹✳ ❊❧❡♠❡♥t♦ s✐♠étr✐❝♦✿
∀h∈H, ∃h‘∈H t❛❧ q✉❡h∗h‘=h‘∗h=eh.
❖❜s❡r✈❛çã♦ ✸✳✸✳ ❆ ❝♦♥❞✐çã♦ ✷ é s❡♠♣r❡ s❛t✐s❢❡✐t❛✱ ♣♦✐s ❛ ✐❣✉❛❧❞❛❞❡ g1 ∗(g2 ∗g3) =
❙✉❜❣r✉♣♦s ✸✼
❖❜s❡r✈❛çã♦ ✸✳✹✳ ❖ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ eh ❞❡ H é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐❣✉❛❧ ❛♦ ❡❧❡♠❡♥t♦
♥❡✉tr♦ e ❞❡G✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ e ❡eh✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ❡❧❡♠❡♥t♦s ♥❡✉tr♦s ❞❡G ❡ H✳
❈♦♠♦
eh∗eh =eh =eh∗e
❡ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦ é ✈á❧✐❞♦ ❡♠ r❡❧❛çã♦ à ♦♣❡r❛çã♦ ∗✱ ❡♥tã♦ e=eh✳
❖❜s❡r✈❛çã♦ ✸✳✺✳ ❉❛❞♦h∈H✱ ♦ ✐♥✈❡rs♦ ❞❡h❡♠Hé ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐❣✉❛❧ ❛♦ ✐♥✈❡rs♦
❞❡ h ❡♠ G✳ ❉❡ ❢❛t♦✱ s❡❥❛ ✉♠ ❡❧❡♠❡♥t♦ b ∈H ❡ ✐♥❞✐q✉❡♠♦s ♣♦r b‘ ❡ b‘
h s❡✉s s✐♠étr✐❝♦s
❡♠ G❡ H✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♠♦✱ ♣♦ré♠✱
b‘h∗b=eh =e=b‘∗b
❡♥tã♦ b‘
h =b‘✳
◆❛ ♣rát✐❝❛✱ ♣❛r❛ ✈❡r✐✜❝❛r q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦H é ✉♠ s✉❜❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦ G✱
❜❛st❛ ✉s❛r♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿
Pr♦♣♦s✐çã♦ ✸✳✷✳ ❙❡❥❛(G,∗)✉♠ ❣r✉♣♦✳ P❛r❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦✱ ♥ã♦ ✈❛③✐♦✱ H ⊂Gs❡❥❛
✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ a∗b‘ s❡❥❛ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ H s❡♠♣r❡
q✉❡ a ❡ b ♣❡rt❡♥❝❡r❡♠ ❛ ❡st❡ ❝♦♥❥✉♥t♦✳
❉❡♠♦♥str❛çã♦✳ (⇒) ❙❡ a, b ∈ H✱ ❡♥tã♦ a∗b‘
h ∈ H ❥á q✉❡✱ ♣♦r ❤✐♣ót❡s❡✱ (H,∗) é ✉♠
❣r✉♣♦✳ ▼❛s ♣❡❧❛ ♦❜s❡r✈❛çã♦ ✸✳✹ t❡♠♦s q✉❡ b‘
h =b‘ ❡✱ ♣♦rt❛♥t♦✱ a∗b‘∈H✳
(⇐) ❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱ ❍ é ♥ã♦ ✈❛③✐♦✱ ❡♥tã♦ é ♣♦ssí✈❡❧ ❝♦♥s✐❞❡r❛r♠♦s ✉♠ ❡❧❡✲
♠❡♥t♦ q✉❛❧q✉❡r x0 ∈ H✳ P♦r ❡st❡ ❢❛t♦✱ ❡ ♣❡❧❛ ❤✐♣ót❡s❡ t❡♠♦s q✉❡ x0 ∗x‘0 = e ∈ H✳
❈♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ ✉♠ ❡❧❡♠❡♥t♦ b ∈ H✱ ❞❛ ❤✐♣ót❡s❡ ❡ ❞❛ ❝♦♥❝❧✉sã♦ ❛♥t❡r✐♦r s❡❣✉❡
q✉❡✿
e∗b‘ =b‘∈H.
❆❣♦r❛ ✈❛♠♦s ♠♦str❛r q✉❡ H é ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ à ♦♣❡r❛çã♦ ∗✳ ❈♦♠ ❡❢❡✐t♦✱ s❡
a, b ∈ H✱ ❡♥tã♦ ♣❡❧❛ ❝♦♥❝❧✉sã♦ ❛♥t❡r✐♦r✱ a, b‘ ∈ H✳ ❉❡ ♦♥❞❡✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛
❤✐♣ót❡s❡✱ t❡♠♦s q✉❡✿
a∗(b‘)‘=a∗b∈H.
❆ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡♠H é ✐♠❡❞✐❛t❛✱ ♣♦✐s s❡a, b, c∈H✱ ❡♥tã♦a, b, c∈G❡✱ ♣♦rt❛♥t♦ a∗(b∗c) = (a∗b)∗c✭✉♠❛ ✈❡③ q✉❡ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ é ✈á❧✐❞❛ ❡♠ ●✮✳
❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✿
❊①❡♠♣❧♦ ✸✳✹✳ ❙❡ G é ✉♠ ❣r✉♣♦✱ ❡♥tã♦ {e} ❡ G sã♦ s✉❜❣r✉♣♦s ❞❡ G✱ ❝❤❛♠❛❞♦s ❞❡
✸✽ ➪❧❣❡❜r❛ ♥♦ ❈✉❜♦ ▼á❣✐❝♦
❊①❡♠♣❧♦ ✸✳✺✳ (2Z,+) é ✉♠ s✉❜❣r✉♣♦ ❞❡ (Z,+)✳ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✱ s❡ n é ✉♠
✐♥t❡✐r♦ q✉❛❧q✉❡r✱ (nZ,+) é ✉♠ s✉❜❣r✉♣♦ ❞❡ (Z,+)✳
❊①❡♠♣❧♦ ✸✳✻✳ {id, Rπ} ❡ nid, Rπ
2, Rπ, R 3π
2 o
sã♦ s✉❜❣r✉♣♦s ❞❡ GQ✳
✸✳✺✳✶ ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ✉♠ s✉❜❝♦♥❥✉♥t♦
❯s❛r❡♠♦s ❛q✉✐ ❛ ♥♦t❛çã♦ ♠✉❧t✐♣❧✐❝❛t✐✈❛✱ ♣♦r s❡r ♠❛✐s s✐♠♣❧❡s ❡ ❞❡ ✉s♦ ♠❛✐s ❢r❡q✉❡♥t❡ ❡♠ ❚❡♦r✐❛ ❞♦s ●r✉♣♦s✳
❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ G✳ ❯t✐❧✐③❛r❡♠♦s ❛ s❡❣✉✐♥t❡
♥♦t❛çã♦✿
< S >={a1·a2·...·an;n∈N, ai ∈S ou a−i 1 ∈S}
◗✉❛♥❞♦ ❙ é ✜♥✐t♦ ❞❛ ❢♦r♠❛ S ={a1, a2, ..., am}é ❝♦♠✉♠ ❞❡♥♦t❛r♠♦s s✐♠♣❧❡s♠❡♥t❡
♣♦r✿
hSi=ha1·a2·...·ami ♥♦ ❧✉❣❛r ❞❡hSi=h{a1, a2, ..., am}i
◆♦t❡ q✉❡ s❡ g ∈G ❡♥tã♦
hgi={...,(g−1)2, g−1, e, g, g2, ...}
❊ ❛✐♥❞❛✱ s❡ ✉s❛r♠♦s ❛ ♥♦t❛çã♦ g−r ♣❛r❛ (g−1)r, r∈N t❡r❡♠♦s✿ hgi={gt;t∈Z}
Pr♦♣♦s✐çã♦ ✸✳✸✳ ❙❡❥❛ ❙ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❣r✉♣♦ G✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ hSi é ✉♠
s✉❜❣r✉♣♦ ❞❡ G✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x, y ∈ hSi t❡♠♦s✿
x=a1·a2·...·an ❝♦♠ ai ∈S ♦✉ a−i 1 ∈S
y=b1·b2·...·bm ❝♦♠ bi ∈S ♦✉ b−i 1 ∈S
▲♦❣♦✱ x·y=a1·a2·...·an·b1·b2·...·bm ❡ x−1 =a−n1·a−n−11·...·a−21·a−11 t❛♠❜é♠
❡stã♦ ❡♠ hSi✳
❉❡✜♥✐çã♦ ✸✳✺✳ ❙❡ S é ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞♦ ❣r✉♣♦ G✱ ❡♥tã♦ hSi é ❝❤❛♠❛❞♦
s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r S✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ g ❞♦ ❣r✉♣♦ G✱ ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r g é hgi=
