ÁLGEBRA E O CUBO DE RUBIK

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ tr✐â♥❣✉❧♦ ▼✐♥❡✐r♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ ◆❛t✉r❛✐s ❡ ❊❞✉çã♦

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

➪❧❣❡❜r❛ ❡ ♦ ❈✉❜♦ ❞❡ ❘✉❜✐❦

❘♦❜s♦♥ ●✉✐♠❛rã❡s

❘❡❧❛tór✐♦ ♣❛r❛ ♦ ❊①❛♠❡ ●❡r❛❧ ❞❡ ◗✉❛❧✐✜❝❛çã♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❆♠♦r✐♠ ❙✐❧✈❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ tr✐â♥❣✉❧♦ ▼✐♥❡✐r♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ ◆❛t✉r❛✐s ❡ ❊❞✉❝❛çã♦

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

➪❧❣❡❜r❛ ❡ ♦ ❈✉❜♦ ❞❡ ❘✉❜✐❦

❘♦❜s♦♥ ●✉✐♠❛rã❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❆♠♦r✐♠ ❙✐❧✈❛

C a t a l o g a ç ã o n a f o n t e : B i b l i o t e c a d a U n i v e r s i d a d e F e d e r a l d o T r i â n g u l o M i n e i r o

Lara, Robson Guimarães de Miranda

L325a Álgebra e o Cubo de Rubik / Robson Guimarães de Miranda Lara. -- 2016.

66 f. : il., fig.

Dissertação (Mestrado Profissional em Matemática em Rede Na- cional) -- Universidade Federal do Triângulo Mineiro, Uberaba, MG, 2016

Orientador: Prof. Dr. Leonardo Amorim Silva

1. Matemática - Estudo e ensino. 2. Álgebra. 3. Teoria dos grupos. 4. Cubo mágico. I. Silva, Leonardo Amorim. II. Universidade Federal do Triângulo Mineiro. III. Título.

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✱ ♠❡ ❞❛♥❞♦ ❢♦rç❛ ♣❛r❛ ❛t✐♥❣✐r ♠❡✉s ♦❜❥❡t✐✈♦s ❡ r❡❛❧✐③❛r ♠❡✉s s♦♥❤♦s✳

❆♦s ♠❡✉s ♣❛✐s ❡ ✐r♠ã♦s ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ♥❛ t♦r❝✐❞❛✱ ❛♣♦✐❛♥❞♦ ❡ ❛❝r❡❞✐t❛♥❞♦ ♥❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ♣r♦❥❡t♦✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ t♦❞♦ ❡ss❡ ♣❡rí♦❞♦✳

❆♦s ❛♠✐❣♦s ❲②s♥❡r ▼❛① ❡ ◆❛tá❧✐❛ ●♦♥ç❛❧✈❡s✱ ♣❛r❝❡✐r♦s ♥❡ss❡ s♦♥❤♦✱ ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ✈✐❛❣❡♠ ❡ ❞❡ s❛❧❛ ❞❡ ❛✉❧❛✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ❆♠♦r✐♠ ❙✐❧✈❛✱ ♣♦r t♦❞♦ ❛♣♦✐♦✱ ❞❡❞✐❝❛çã♦✱ t❡♠♣♦ ❡ ♣❛❝✐ê♥❝✐❛✳ Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ s❡ t♦r♥❛ ❝♦♠ ❝❡rt❡③❛ ✉♠ ❡①❡♠♣❧♦✱ ✉♠❛ r❡❢❡rê♥❝✐❛✳

✧❆ ♠❡♥t❡ q✉❡ s❡ ❛❜r❡ ❛ ✉♠❛ ♥♦✈❛ ✐❞❡✐❛ ❥❛♠❛✐s ✈♦❧t❛rá ❛♦ s❡✉ t❛♠❛♥❤♦ ♦r✐❣✐♥❛❧✳✧

❘❡s✉♠♦

❊ss❛ ❞✐ss❡rt❛çã♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♠♦str❛r ❝♦♠♦ ❛ ♠❛t❡♠át✐❝❛ ❛tr❛✈és ❞❡ s✉❛s ✐♥ú✲ ♠❡r❛s t❡♦r✐❛s q✉❡ ♣❛r❛ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ♥✉♥❝❛ s❛❡♠ ❞♦ ❝❛♠♣♦ ❞❛ ❛❜str❛çã♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s✱ ♣♦❞❡ s❡r ❛ss♦❝✐❛❞❛ ❛ ✉♠ ❜r✐♥q✉❡❞♦ ♠✉♥❞✐❛❧♠❡♥t❡ ❢❛♠♦s♦✱ ♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳ ▼♦str❛r❡♠♦s q✉❡ ♦ ❝✉❜♦ é ✉♠ ❣r✉♣♦ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡ ✉s❛✲ r❡♠♦s ❛ t❡♦r✐❛ ❞♦s ❣r✉♣♦s ♣❛r❛ ❛♥❛❧✐s❛r♠♦s q✉❛❧ é✱ r❡❛❧♠❡♥t❡✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ✈❛❧✐❞❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ❡♠ s✉❛ r❡s♦❧✉çã♦✳

❆❜str❛❝t

❚❤✐s ❞✐ss❡rt❛t✐♦♥ ❛✐♠s t♦ s❤♦✇ ❤♦✇ ♠❛t❤❡♠❛t✐❝s t❤r♦✉❣❤ ❤✐s ♥✉♠❡r♦✉s t❤❡♦r✐❡s✱ t❤❛t ❢♦r t❤❡ ✈❛st ♠❛❥♦r✐t② ♦❢ st✉❞❡♥ts ♥❡✈❡r ❧❡❛✈❡ t❤❡ ❛❜str❛❝t✐♦♥ ✜❡❧❞✱ s✉❝❤ ❛s ❣r♦✉♣ t❤❡♦r②✱ ❝❛♥ ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ✇♦r❧❞ ❢❛♠♦✉s t♦②✱ t❤❡ ❘✉❜✐❦✬s ❈✉❜❡✳ ■t ❝❛♥ ❜❡ ♥♦t❡❞ t❤❛t t❤❡ ❝✉❜❡✬s ♠♦✈❡♠❡♥ts ❝❤❛♥❣❡ t❤❡ s❡tt✐♥❣s ♦❢ t❤❡ ❢❛❝❡s ❜✉t r❡t❛✐♥ t❤❡ ♦✈❡r❛❧❧ s❤❛♣❡ ♦❢ t❤❡ ❝✉❜❡✱ t❤❡r❡❢♦r❡ ✇❡ ❝❛♥ r❡♣r❡s❡♥t s✉❝❤ ♠♦✈❡♠❡♥ts ❛s ♣❡r♠✉t❛t✐♦♥s✳ ❋✐♥❛❧❧②✱ ✇❡ s❛✇ t❤❛t t❤❡ s❡t ♦❢ ❛❧❧ ❢❛❝❡s ♣❡r♠✉t❛❄♦♥s ♦❢ t❤❡ ❘✉❜✐❦✬s ❝✉❜❡ ❢♦r♠ ❛ ❣r♦✉♣ ❛♥❞ ✉s❡❞ t❤❡ ❣r♦✉♣ t❤❡♦r② t♦ ❛♥❛❧②③❡ ✇❤❛t ✐s t❤❡ ❛❝t✉❛❧ ❛♠♦✉♥t ♦❢ ✈❛❧✐❞ s♦❧✉t✐♦♥s t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ✐ts r❡s♦❧✉t✐♦♥✳❲r✐t✐♥❣ ❛♥ ❡✣❝✐❡♥t ❛❜str❛❝t ✐s ❤❛r❞ ✇♦r❦✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✼

✷ ❘❡❧❛çõ❡s✱ ❆♣❧✐❝❛çõ❡s ❡ ❖♣❡r❛çõ❡s ✶✾

✷✳✶ ❘❡❧❛çã♦ ❇✐♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✶ ❘❡❧❛çã♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❘❡❧❛çõ❡s ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ❘❡❧❛çã♦ ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✷ P❛rt✐çã♦ ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✸ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸ ●r✉♣♦s ✷✺

✸✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷ ●❡r❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ●r✉♣♦s ❞❡ ❙✐♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✹ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹✳✶ ❖ ❙✐♥❛❧ ❞❡ ✉♠ ❍♦♠♦♠♦r✜s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✺ ❖ ●r✉♣♦ ❆❧t❡r♥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✻ ❆çõ❡s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✹ ❈✉❜♦ ❞❡ ❘✉❜✐❦ ✹✺

✹✳✵✳✶ ❆s ❝♦♥✜❣✉r❛çõ❡s ❞♦ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✶ ❈♦♥✜❣✉r❛çõ❡s ❱á❧✐❞❛s ❞♦ ❈✉❜♦ ❞❡ ❘ú❜✐❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✸

✶ ■♥tr♦❞✉çã♦

❖ ❝✉❜♦ ❞❡ ❘✉❜✐❦ ♦✉ ❝✉❜♦ ♠á❣✐❝♦ ❢♦✐ ❝r✐❛❞♦ ♥♦ ❞✐❛ ✶✾ ❞❡ ♠❛✐♦ ❞❡ ✶✾✼✹ ♣❡❧♦ ❡s❝✉❧t♦r ❡ ♣r♦❢❡ss♦r ❞❡ ❛rq✉✐t❡t✉r❛ ❤ú♥❣❛r♦ ❊r♥õ ❘✉❜✐❦✳ ❘✉❜✐❦✱ ❛♦s ✷✾ ❛♥♦s✱ tr❛❜❛❧❤❛✈❛ ❡♠ ✉♠ ♠♦❞❡❧♦ tr✐❞✐♠❡♥s✐♦♥❛❧ q✉❡ ♦ ❛❥✉❞❛r✐❛ ❛ tr❛❜❛❧❤❛r ❝♦♠ ♦ ❡♥s✐♥♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❛♦s s❡✉s ❛❧✉♥♦s✳

❖ ❜r✐♥q✉❡❞♦ ❢♦✐ ♣❛t❡♥t❡❛❞♦ ❡♠ ✶✾✼✼✱ ❡ ❡♠ s❡❣✉✐❞❛ ❧❛♥ç❛❞♦s ♥♦ ♠❡r❝❛❞♦✳ ❍♦❥❡✱ ❡①✐st❡♠ ✈ár✐❛s ✈❡rsõ❡s ❞❡st❡ ❜r✐♥q✉❡❞♦✱ ♣♦r ❡①❡♠♣❧♦ (2×2×2)✱ ♦r✐❣✐♥❛❧ (3×3×3) ❡ (5×5×5)✳

❖ ❜r✐♥q✉❡❞♦ ❢♦✐ ✐♥✐❝✐❛❧♠❡♥t❡ ❜❛t✐③❛❞♦ ♣♦r ❈✉❜♦ ▼á❣✐❝♦✱ ♣❡❧♦ ♣ró♣r✐♦ ❘✉❜✐❦✱ ♠❛s ❛ ❧❡✐ ❞❡ ♣❛t❡♥t❡s ❞❛ ❍✉♥❣r✐❛✱ ♥❛ é♣♦❝❛ r❡❣✐❞❛ ♣♦r ✉♠ ❣♦✈❡r♥♦ ❝♦♠✉♥✐st❛✱ ♥ã♦ ♣❡r♠✐t✐❛ ❛ ❛♠♣❧✐❛çã♦ ❞♦s r❡❣✐str♦s ❡♠ ❝❛rát❡r ✐♥t❡r♥❛❝✐♦♥❛❧✳ P♦r ✐ss♦✱ q✉❛♥❞♦ ❛ ■❞❡❛❧ ❚♦②s ❢♦✐ r❡❣✐str❛r ♦ ❜r✐♥q✉❡❞♦✱ t❡✈❡ ❞❡ ♠✉❞❛r ♦ ♥♦♠❡ ♣❛r❛ ❝✉❜♦ ❞❡ ❘✉❜✐❦✳ ❈♦♠ ♦ ❧❛♥ç❛♠❡♥t♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦✱ s✉r❣✐r❛♠ t❛♠❜é♠ ♦s ♣r✐♠❡✐r♦s ❝❛♠♣❡♦♥❛t♦s ❞❡ r❡s♦❧✉çã♦ ❞♦ ❞❡s❛✜♦✳ ❯♠❛ ❡st✉❞❛♥t❡ ✈✐❡t❝♦♥❣✉❡ ❞❡ ✶✻ ❛♥♦s ❣❛♥❤♦✉ ♦ ♣r✐♠❡✐r♦ ❝❛♠♣❡♦♥❛t♦ ♠✉♥❞✐❛❧ ❞❡ ❝✉❜♦ ♠á❣✐❝♦✱ q✉❡ ❛❝♦♥t❡❝❡✉ ❡♠ ❇✉❞❛♣❡st❡ ❡♠ ✶✾✽✷✳ ❊❧❛ r❡s♦❧✈❡✉ ♦ ❥♦❣♦ ❡♠ ✷✷✱✾✺ s❡❣✉♥❞♦s✳ ❖ ❛t✉❛❧ r❡❝♦r❞✐st❛ é ▲✉❝❛s ❊tt❡r✱ ❞❡ ✶✹ ❛♥♦s✱ ❝♦♠ ♦ t❡♠♣♦ ❞❡ ✹✱✾✵✹ s❡❣✉♥❞♦s✱ ♠❛s ✐st♦ ❡♥tr❡ ❤✉♠❛♥♦s✱ ♦ ♠❡❧❤♦r t❡♠♣♦ ♣❡rt❡♥❝❡ ❛ ✉♠ r♦❜ô ❝r✐❛❞♦ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ✭✷✱✸✾ s❡❣✉♥❞♦s✮✳ ❊①✐st❡♠ ♠❛✐s ❞❡ ✹✵ q✉❛tr✐❧❤õ❡s ❞❡ ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s ❡♠ ✉♠ ❝✉❜♦ ♠á❣✐❝♦ ✭sã♦ ❡①❛t❛♠❡♥t❡ ✹✸✳✷✺✷✳✵✵✸✳✷✼✹✳✽✺✻✳✵✵✵ ❝♦♠❜✐♥❛çõ❡s✮✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ✉♠❛ ♣❡ss♦❛ ♣❡❣❛r ✉♠ ❝✉❜♦ ♠á❣✐❝♦ ❡ ✜③❡r ✉♠❛ ❥♦❣❛❞❛ ♣♦r s❡❣✉♥❞♦✱ ❡❧❛ ❞❡♠♦r❛rá ♣❡❧♦ ♠❡♥♦s ✶✳✹✵✵ tr✐❧❤õ❡s ❞❡ ❛♥♦s ♣❛r❛ ❢❛③❡r t♦❞❛s ❛s ♠♦✈✐♠❡♥t❛çõ❡s ♣♦ssí✈❡✐s✳ ❉❡s❞❡ ❛ ✐♥✈❡♥çã♦ ❞♦ ❝✉❜♦ ♠á❣✐❝♦✱ ❡♠ ✶✾✼✹✱ ❡st✉❞✐♦s♦s t❡♥t❛♠ ❞❡s❝♦❜r✐r ♦ ♠í♥✐♠♦ ♥❡❝❡ssár✐♦ ❞❡ ❥♦❣❛❞❛s ♣❛r❛ ❝♦♠♣❧❡t❛r ♦ ❞❡s❛✜♦✳ ❊♠ ❥✉❧❤♦ ❞❡ ✷✵✶✵✱ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❝♦♠♣✉t❛❞♦r✱ ✉♠ ❣r✉♣♦ ❞❡ ♣❡sq✉✐s❛❞♦r❡s ❝❤❡❣♦✉ à ❝♦♥❝❧✉sã♦✿ ♦ ❥♦❣♦ só ❝♦♥s❡❣✉❡ s❡r r❡s♦❧✈✐❞♦ ❝♦♠ ✉♠ ♠í♥✐♠♦ ❞❡ ✷✵ ♠♦✈✐♠❡♥t❛çõ❡s✳ ❖ ♠❛✐s ❝♦♠♣❧❡①♦ ❝✉❜♦ ♠á❣✐❝♦ ❡①✐st❡♥t❡ é ♦ ❝✉❜♦ 17×17×17q✉❡ ❢♦✐ r❡s♦❧✈✐❞♦ ❡♠ ✼ ❤♦r❛s✱ ✸✷ ♠✐♥✉t♦s ❡ ✹✻ s❡❣✉♥❞♦s✱

❞✐✈✐❞✐❞❛s ❡♠ ✺ ❞✐❛s ♣♦r ❑❡♥♥❡t❤ ❇r❛♥❞♦♥✳

❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦ é ✉s❛r ❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ♣❛r❛ ❢❛③❡r ✉♠❛ ❞✐s❝✉ssã♦ ❛ r❡s♣❡✐t♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ♣♦ssí✈❡✐s q✉❡ ✉♠ ❝✉❜♦ ❞❡ ❘✉❦✐❦ ♣♦❞❡ ❛ss✉♠✐r✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❢❛❧❛r❡♠♦s s♦❜r❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ♠❛✐s ❜ás✐❝♦s ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ◆♦ ❈❛♣ít✉❧♦ ✸✱ ❞❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ t❡♦r✐❛ ❞❡ ❣r✉♣♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s

✶✽ ■♥tr♦❞✉çã♦

✷ ❘❡❧❛çõ❡s✱ ❆♣❧✐❝❛çõ❡s ❡ ❖♣❡r❛çõ❡s

✷✳✶ ❘❡❧❛çã♦ ❇✐♥ár✐❛

❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ❝❤❛♠❛♠♦s ❞❡ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦✱ ❡

❞❡♥♦t❛♠♦s ♣♦r A×B ✭❧ê✲s❡✿ ❆ ❝❛rt❡s✐❛♥♦ ❇✮ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♣❛r❡s

♦r❞❡♥❛❞♦s (x, y) ❝♦♠ x∈A ❡ y ∈B.

A×B ={(x, y)|x∈A ❡ y∈B} ✭✷✳✶✮

❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ❝❤❛♠❛♠♦s ❞❡ r❡❧❛çã♦ ❞❡A ❡♠ B✱ t♦❞♦

s✉❜❝♦♥❥✉♥t♦ R ❞❡ A×B✳

P❛r❛ ✐♥❞✐❝❛r q✉❡ (a, b) ∈ R✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ aRb✳ ❙❡ (a, b) 6∈ R✱ ❡s❝r❡✈❡r❡♠♦s a 6❘ b✳

❖s ❝♦♥❥✉♥t♦sA ❡ B sã♦ ❝❤❛♠❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡✱ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ ❡ ❝♦♥✲

❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ ❞❡ R✳

❙❡❥❛ R ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B✳ ❈❤❛♠❛✲s❡ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ r❡❧❛çã♦ R✱ ✉♠

s✉❜❝♦♥❥✉♥t♦ ❞❡ A✱ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s x ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s ❡①✐st❡

✉♠ ❡❧❡♠❡♥t♦ y ♣❡rt❡♥❝❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ B✱ t❛❧ q✉❡xRy✳

D(R) = {x∈A| ∃y∈B :xRy} ✭✷✳✷✮

❈❤❛♠❛✲s❡ ■♠❛❣❡♠ ❞❡ ✉♠❛ r❡❧❛çã♦R✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡B✱ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s

♦s ❡❧❡♠❡♥t♦s y ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s ❡①✐st❡ ✉♠ x ♣❡rt❡♥❝❡♥t❡A✱ t❛❧ q✉❡xRy✳ Im(R) = {y∈B | ∃x∈A:xRy} ✭✷✳✸✮

❊①❡♠♣❧♦ ✷✳✶✳ ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s A={1,2,3}❡ B ={1,2,3,4,5}✱ t❡♠♦s q✉❡✿

A×B ={(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,2),(2,3),(2,4),(2,5),(3,1),(3,2),

(3,3),(3,4),(3,5)}✳ ◗✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ❞♦ ❝♦♥❥✉♥t♦ A×B é ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B✳ ❆ s❡❣✉✐r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ r❡❧❛çã♦ ❞❡A❡♠B✳ R1 =∅;R2 ={(1,1),(2,2),(3,3)};

R3 ={(1,3),(2,3),(3,3)};

❙❡ A = B = Z✱ A ×B é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡

♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡ s❡ A = B = R✱ A×B é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♣❛r❡s

♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s✳

✷✵ ❘❡❧❛çõ❡s✱ ❆♣❧✐❝❛çõ❡s ❡ ❖♣❡r❛çõ❡s

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛ ❘ ✉♠❛ r❡❧❛çã♦ ❞❡ ❆ ❡♠ ❇✱ ❝❤❛♠❛✲s❡ r❡❧❛çã♦ ✐♥✈❡rs❛ ❞❡ ❘✱ ❡ ✐♥❞✐❝❛✲s❡ ♣♦r R−1 _{❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ❇ ❡♠ ❆✿}

R−1 _{={(y, x)∈B×A: (x, y)∈R} ✭✷✳✹✮} Pr♦♣r✐❡❞❛❞❡s ✷✳✶✳ ❉❡❝♦rr❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ r❡❧❛çã♦ ✐♥✈❡rs❛ ❛s ♣r♦♣r✐✲ ❡❞❛❞❡s s❡❣✉✐♥t❡s✿

P1✿ D(R−1_{) = Im(R)❀}

P2✿ Im(R−1_{) =D(R)❀}

P3✿ (R−1₎−1 _=R

✷✳✶✳✶ ❘❡❧❛çã♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦

❉❡✜♥✐çã♦ ✷✳✹✳ ◗✉❛♥❞♦ A = B ❡ R é ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B✱ ❞✐③✲s❡ q✉❡ R é ✉♠❛

r❡❧❛çã♦ s♦❜r❡ A ♦✉✱ ❛✐♥❞❛✱ R é ✉♠❛ r❡❧❛çã♦ ❡♠ A✳

❉❡✜♥✐çã♦ ✷✳✺✳ ❉❛❞❛ ✉♠❛ r❡❧❛çã♦ R✱ ❞✐③❡♠♦s q✉❡ ❛ r❡❧❛çã♦ é✿

❘❡✢❡①✐✈❛✿ q✉❛♥❞♦ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ A s❡ r❡❧❛❝✐♦♥❛ ❝♦♥s✐❣♦ ♠❡s♠♦✳ ❖✉ s❡❥❛✱ q✉❛♥❞♦

♣❛r❛ t♦❞♦ x∈A✱ xRx✳

❙✐♠étr✐❝❛✿ s❡ yRx s❡♠♣r❡ q✉❡ xRy✱ ♦✉ s❡❥❛✱ s❡ xRy ❡♥tã♦ yRx✳

❆♥t✐✲s✐♠étr✐❝❛✿ s❡ x = y✱ s❡♠♣r❡ q✉❡ xRy ❡ yRx✳ ❖✉ s❡❥❛✱ s❡ xRy ❡ yRx✱ ❡♥tã♦ x=y✳

❚r❛♥s✐t✐✈❛✿ s❡ xRz s❡♠♣r❡ q✉❡ xRy ❡ yRz✳ ❖✉ s❡❥❛✱ s❡ xRy ❡ yRz ❡♥tã♦ xRz✳

✷✳✷ ❘❡❧❛çõ❡s ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛

✷✳✷✳✶ ❘❡❧❛çã♦ ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛

❉❡✜♥✐çã♦ ✷✳✻✳ ❯♠❛ r❡❧❛çã♦ ❘ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ❆ ♥ã♦ ✈❛③✐♦ é ❝❤❛♠❛❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ❆ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❘ é r❡✢❡①✐✈❛✱ s✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛✳ ❖✉ s❡❥❛✱ ❘ ❞❡✈❡ ❝✉♠♣r✐r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✐✲ ❙❡ ① ∈ ❆✱ ❡♥tã♦ ①❘①❀

✐✐✲ ❙❡ ①✱② ∈ ❆ ❡ ①❘② ❡♥tã♦ ②❘①❀

✐✐✐✲ ❙❡ ①✱②✱③ ∈ ❆ ❡ ①❘② ❡ ②❘③✱ ❡♥tã♦ ①❘③✳

❊①❡♠♣❧♦ ✷✳✷✳ ❆ ❘❡❧❛çã♦ R={(a, a),(b, b),(c, c),(a, b),(b, a)}é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✲

❘❡❧❛çõ❡s ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛ ✷✶

❉❡✜♥✐çã♦ ✷✳✼✳ ❙❡❥❛ R ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ A✳ P❛r❛ ❝❛❞❛ a ∈ A✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s x ∈ A t❛✐s q✉❡ xRa ❝❤❛♠❛✲s❡ ❝❧❛ss❡ ❞❡

❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ a ❡ ✐♥❞✐❝❛✲s❡ ♣♦r ¯a✳ ❖✉ s❡❥❛✱

¯

a={x∈A|xRa} ✭✷✳✺✮

❉❡✜♥✐çã♦ ✷✳✽✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♠ó❞✉❧♦ R s❡rá ✐♥❞✐❝❛❞♦ ♣♦r A/R ❡ ❝❤❛♠❛❞♦ ❝♦♥❥✉♥t♦✲q✉♦❝✐❡♥t❡ ❞❡ A ♣♦r R✳

❊①❡♠♣❧♦ ✷✳✸✳ ◆❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛R ={(a, a),(b, b),(c, c),(a, b),(b, a)}t❡♠♦s✿

¯

a = {a, b}; ¯_{b = {a, b};} ¯

c = {c};

❛ss✐♠✱ A/R ={¯a,c¯}={{a, b},{c}}✳

❊①❡♠♣❧♦ ✷✳✹✳ ❈♦♥s✐❞❡r❡ ❛ r❡❧❛ççã♦ s♦❜r❡ Z ❞❛❞❛ ♣♦r ①❘② ↔ ① ≡ ② ✭♠♦❞ ♠✮✱ ∀ ①✱② ∈ Z✱ ♠ ❃ ✶✳ ❊♥tã♦ ❘ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❘❡❧❛çã♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✳ ❆ r❡❧❛çã♦ R ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m ✭m∈Z ❡m >1✮ s♦❜r❡ Z é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

✭✐✮ ❙❡♥❞♦ ❛∈Z✱ ❡❢❡t✉❛♠♦s ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ ❛ ♣♦r m✱ ♦❜t❡♥❞♦ ♦ q✉♦❝✐❡♥t❡ q

❡ ♦ r❡st♦ r✳ ❚❡♠♦s✿

a=mq+r ❡ 0≤r < m ✭✷✳✻✮

❡ ❞❛í ✈❡♠✿

a−r =qm ✭✷✳✼✮

P♦rt❛♥t♦✿

a≡r (mod m) ✭✷✳✽✮ ¯

a= ¯r ✭✷✳✾✮

❈♦♥❝❧✉í♠♦s q✉❡ ¯a ❂ ④①∈ Z✴①≡ ❛✭♠♦❞ ♠✮⑥ é ✉♠❛ ❝❧❛ss❡ ✐❣✉❛❧ ❛ r¯✱ ❡♠ q✉❡ r é ♦

r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r m✳ ❝♦♠♦ r∈ {0,1,2, ..., m−1}✱ ✈❡♠✿

Z/R= ¯0,¯1,¯2, ..., m−1 ✭✷✳✶✵✮

✭✐✐✮ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛♠ ❞✉❛s ❝❧❛ss❡s✱ r¯❡ ¯s✱ ✐❣✉❛✐s ❡♠ {¯0,¯1,¯2, ..., m−1}✱ r❡✲

♣r❡s❡♥t❛❞❛s ♣♦r ❡❧❡♠❡♥t♦s r ❡ s✱ ❞✐❣❛♠♦s r < s✳ ❊♥tã♦✿

¯

✷✷ ❘❡❧❛çõ❡s✱ ❆♣❧✐❝❛çõ❡s ❡ ❖♣❡r❛çõ❡s

❉❡r¯= ¯ss❡❣✉❡ q✉❡r ≡s(mod m)❡ ♣♦rt❛♥t♦m|s−r✱ ❆❜s✉r❞♦✱ ♣♦✐s0< s−r < m✱

❡ ✐ss♦ é ✐♠♣♦ssí✈❡❧✳

❈♦♥❝❧✉í♠♦s q✉❡ {¯0,¯1,¯2, ..., m−1} é ❝♦♥st✐t✉í❞♦ ♣♦r ❡①❛t❛♠❡♥t❡ m ❡❧❡♠❡♥t♦s ❞✐s✲

t✐♥t♦s ❞♦✐s ❛ ❞♦✐s✱ ♦✉ s❡❥❛✿

Zm =Z/R={¯0,¯1,¯2, ..., m−1} ✭✷✳✶✷✮ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡❥❛ ❘ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ❆ ❡ s❡❥❛♠ ❛✱ ❜ ∈ ❆✳ ❆s

s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ aRb

✭✐✐✮ a∈¯b

✭✐✐✐✮ b∈¯a

✭✐✈✮ ¯a= ¯b

❉❡♠♦♥str❛çã♦✿ ❉❡✈❡♠♦s ♣r♦✈❛r (i)⇒(ii)⇒(iii)⇒(iv)⇒ ✭✐✮✳

(i)⇒(ii)✿ ➱ ❞❡❝♦rrê♥❝✐❛ ❞❡ ❞❡✜♥✐çã♦ ❞❡ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

(ii)⇒(iii)✿ ❈♦♠♦a∈¯b✱ ❡♥tã♦aRb✱ ❧♦❣♦ ♣❡❧❛ s✐♠❡tr✐❛ ❞❡R✱bRa✱ ❡ ♣♦rt❛♥t♦b∈¯a✳

(iii) ⇒ (iv)✿ P♦r ❤✐♣ót❡s❡ b ∈ ¯a✱ ♦✉ s❡❥❛✱ bRa✱ ❧♦❣♦ aRb✳ ❚❡♠♦s q✉❡ ♣r♦✈❛r q✉❡

¯

a⊂¯b ❡¯b ⊂a¯✳

P❛r❛ ♣r♦✈❛r ❛ ♣r✐♠❡✐r❛ ❞❡ss❛s ✐♥❝❧✉sõ❡s✱ t♦♠❡♠♦s x∈a¯✳ ❊♥tã♦xRa ❡✱ ❧❡✈❛♥❞♦ ❡♠

❝♦♥t❛ q✉❡ aRb✱ ❝♦♥❝❧✉í♠♦s ♣♦r tr❛♥s✐t✐✈✐❞❛❞❡ ❞❡ R✱ q✉❡xRb✳ ❉❛í x∈¯b ❡ ¯a⊂¯b✳

❆♥❛❧♦❣❛♠❡♥t❡ s❡ ♣r♦✈❛¯_b⊂¯a✳

(iv) ⇒ (i)✿ ❈♦♠♦ a ∈ ¯a ❡ b ∈ ¯b✱ ♦s ❝♦♥❥✉♥t♦s a¯ ❡ ¯b ♥ã♦ sã♦ ✈❛③✐♦s✳ ❚♦♠❡♠♦s ✉♠ x ∈ ¯a = ¯b✳ ❊♥tã♦ xRa ❡ xRb✳ ❉❛í✱ ♣❡❧❛ s✐♠❡tr✐❛ ❞❡ R✱ aRx ❡ xRb✳ ❆ tr❛♥s✐t✐✈✐❞❛❞❡

❞❡R ❣❛r❛♥t❡ ❡♥tã♦ q✉❡ aRb✳

✷✳✷✳✷ P❛rt✐çã♦ ❞❡ ✉♠ ❈♦♥❥✉♥t♦

❉❡✜♥✐çã♦ ✷✳✾✳ ❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ❝❧❛ss❡ F ❞❡ s✉❜❝♦♥✲

❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❞❡ A✱ é ✉♠❛ ♣❛rt✐çã♦ ❞❡ A s❡✱ ❡ s♦♠❡♥t❡ s❡✿

✭✐✮ ❉♦✐s ❡❧❡♠❡♥t♦s q✉❛✐sq✉❡r ❞❡ F ♦✉ sã♦ ✐❣✉❛✐s ♦✉ sã♦ ❞✐s❥✉♥t♦s❀

✭✐✐✮ ❆ ✉♥✐ã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ F é ✐❣✉❛❧ ❛♦ ❝♦♥❥✉♥t♦ A✳

Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡R é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦A✱ ❡♥tã♦ A/R

❘❡❧❛çõ❡s ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛ ✷✸

❉❡♠♦♥str❛çã♦✿

❙❡❥❛a¯∈A/R✳ ❈♦♠♦R é r❡✢❡①✐✈❛✱ aRa❡✱ ♣♦rt❛♥t♦✱a∈¯a✳ ❆ss✐♠ ¯a6=∅♣❛r❛ t♦❞♦

¯

a∈A/R✳

❙❡❥❛♠ ¯a✱ ¯b ∈ A/R t❛✐s q✉❡ ¯a∩¯b 6= ∅✳ Pr♦✈❛r❡♠♦s q✉❡ ¯a = ¯b✳ ❉❡ ❢❛t♦✱ s❡❥❛ y∈a¯∩¯b✳ ❊♥tã♦ y∈¯a ❡ y∈¯b ❡✱ ♣♦rt❛♥t♦yRa ❡yRb✳ ❉❛íaRy ❡ yRb ❡ ♣♦rt❛♥t♦aRb✳

❆ ♣r♦♣♦s✐çã♦ ✷✳✶ ❣❛r❛♥t❡ ❡♥tã♦ q✉❡ ¯a= ¯b✳

Pr♦✈❡♠♦s q✉❡∪a∈A¯a=A✳

P❛r❛ ❝❛❞❛ a ∈A✱ t❡♠♦s ¯a ⊂A✱ ♣♦rt❛♥t♦ ∪a∈A¯a⊂ A✳ ❙❡♥❞♦ x ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧✲ q✉❡r ❞❡ A✱ ❡♥tã♦ xRx✳ ❉❛í✱x∈x¯ ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱x∈ ∪a∈A¯a✱ ❛ss✐♠✱ A⊂ ∪a∈A¯a✳

Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ F é ✉♠❛ ♣❛rt✐çã♦ ♥♦ ❝♦♥❥✉♥t♦ A✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❞❡

❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ A✱ t❛❧ q✉❡ A/R=F✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ R ❛ r❡❧❛çã♦ s♦❜r❡ A ❛ss✐♠ ❞❡✜♥✐❞❛✿ xRy s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∃

E ∈ F t❛❧ q✉❡ x ∈ E ❡ y ∈ E✱ ♦✉ s❡❥❛✱ x ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ y q✉❛♥❞♦ ❡①✐st❡ ✉♠

❝♦♥❥✉♥t♦ E ❞❛ ♣❛rt✐çã♦ F ❛♦ q✉❛❧ ♣❡rt❡♥❝❡♠ x ❡ y✳ Pr♦✈❛r❡♠♦s q✉❡ R é r❡❧❛çã♦ ❞❡

❡q✉✐✈❛❧ê♥❝✐❛✳ ❚❡♠♦s✿

✭✐✮ P❛r❛ t♦❞♦ x ❡♠ A ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ E ⊂ A t❛❧ q✉❡ E ∈ F ❡ x ∈ E✱

♣♦rt❛♥t♦ xRx✳

✭✐✐✮ ❙❡x❡ysã♦ ❡❧❡♠❡♥t♦s q✉❛✐sq✉❡r ❞❡At❛✐s q✉❡xRy✱ ❡♥tã♦x, y ∈E✱ ♣❛r❛ ❛❧❣✉♠ E ∈F✳ ❖❜✈✐❛♠❡♥t❡✱ ❡♥tã♦✱ y, x∈E✳ ▲♦❣♦ yRx✳

✭✐✐✐✮ ❙❡❥❛♠ x✱ y ❡ z ❡❧❡♠❡♥t♦s q✉❛✐sq✉❡r ❞❡ A t❛✐s q✉❡ xRy ❡ yRz✳ ■ss♦ s✐❣♥✐✜❝❛

q✉❡ x, y ∈ E ❡ y, z ∈D✱ ♣❛r❛ ❝♦♥✈❡♥✐❡♥t❡s D ❡ E ∈ F✳ ▲♦❣♦✱ y ∈ E ❡ y ∈D✳ ❈♦♠♦

❞♦✐s ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r ❞❡ F q✉❡ ♥ã♦ sã♦ ❞✐s❥✉♥t♦s sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐❣✉❛✐s✱ ❡♥tã♦ E = D✳ ❉❡st❡ ❢❛t♦ ❞❡❝♦rr❡ q✉❡ x ❡ z ♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❧❛ss❡ F ❞❡

♦♥❞❡ xRz✳

✷✳✷✳✸ ❋✉♥çõ❡s

❉❡✜♥✐çã♦ ✷✳✶✵✳ ❙❡❥❛ f ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B✳ ❉✐③❡♠♦s q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❞❡ A ❡♠ B ❡ ❞❡♥♦t❛♠♦s ♣♦r ❢✿❆ → ❇ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ x∈A ❡①✐st❡ ✉♠ ú♥✐❝♦ y∈B t❛❧ q✉❡ (x, y)∈f✳

❙❡f é ✉♠❛ ❢✉♥çã♦ ❞❡A ❡♠ B✱ ❡s❝r❡✈❡r❡♠♦s✿ y =f(x)✳ ✭❧ê✲s❡✿ y é ❛ ✐♠❛❣❡♠ ❞❡x

♣❡❧❛ ❢✉♥çã♦ f✳✮

❉❡✜♥✐çã♦ ✷✳✶✶✳ ❯♠❛ ❢✉♥çã♦ f ❞❡ A ❡♠ B é ✐♥❥❡t✐✈❛✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ q✉❛✐sq✉❡r q✉❡

s❡❥❛♠ x1 ❡ x2 ❞❡ A✱ s❡ x1 6= x2✱ ❡♥tã♦ f(x1) 6= f(x2)✳ P♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ❢✉♥çã♦ ✐♥❥❡t✐✈❛ ❞❡ ♠❛♥❡✐r❛ ❡q✉✐✈❛❧❡♥t❡ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ✉♠❛ ❢✉♥çã♦ f ❞❡ A ❡♠ B é

✐♥❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1 ❡ x2 ❞❡ A✱ s❡ f(x1) = f(x2)✱ ❡♥tã♦

✷✹ ❘❡❧❛çõ❡s✱ ❆♣❧✐❝❛çõ❡s ❡ ❖♣❡r❛çõ❡s

❉❡✜♥✐çã♦ ✷✳✶✷✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦f ❞❡A ❡♠B é s♦❜r❡❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

♣❛r❛ t♦❞♦ y∈ B ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x∈ A t❛❧ q✉❡ f(x) =y✳ ◆♦t❡♠♦s q✉❡ f :A →B

é s♦❜r❡❥❡t♦r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Im(f) =B✳

❉❡✜♥✐çã♦ ✷✳✶✸✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f ❞❡ A ❡♠ B é ❜✐❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f é ✐♥❥❡t✐✈❛ ❡ s♦❜r❡❥❡t✐✈❛✳ ❊ss❛ ❞❡✜♥✐çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ ✉♠❛ ❢✉♥çã♦ f ❞❡ A ❡♠ B

é ❜✐❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ y ∈ B✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ x∈A t❛❧ q✉❡ f(x) = y✳

❉❡✜♥✐çã♦ ✷✳✶✹✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡ A ❡♠ B ❜✐❥❡t✐✈❛✳ ❉❡✜♥✐♠♦s ❝♦♠♦ ❛ ✐♥✈❡rs❛

❞❛ ❢✉♥çã♦ f✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r f−1_{✱ ❛ ❢✉♥çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ y ∈ B ✉♠ ú♥✐❝♦ x ∈} ❆✳

❉❡✜♥✐çã♦ ✷✳✶✺✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s f : A → B ❡ g : B → C✱ ❡♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r

✉♠❛ ♥♦✈❛ ❢✉♥çã♦f◦g :A→C✱ ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡f ❡g✱ ♣♦r (f◦g)(x) =

g(f(x))✳

❖❜s❡r✈❛çã♦ ✷✳✶✳ ❊♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞♦s ❈❛♣ít✉❧♦s ✸ ❡ ✹✱ ❡s❝r❡✈❡♠♦s (f ◦g) ♣❛r❛

❞❡♥♦t❛r g(f(x)) ❡♠ ✈❡③ ❞❡ (f ◦g) = f(g(x)) ✳ ◆♦ ❡♥t❛♥t♦✱ ❞❡s❞❡ q✉❡ s❡❥❛♠ ❝♦♥s✐s✲

✸ ●r✉♣♦s

✸✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s

❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ G ♠✉♥✐❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ⋆ é ✉♠ ❣r✉♣♦

q✉❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s s❡❣✉✐♥t❡s sã♦ s❛t✐s❢❡✐t❛s✿

✭✐✮ ❉❛❞♦s q✉❛✐sq✉❡rx✱ y∈G✱x ⋆ y ∈G✱ ♦✉ s❡❥❛✱ ♦ ❣r✉♣♦ é ❢❡❝❤❛❞♦ ♣❛r❛ ❛ ♦♣❡r❛çã♦ ⋆✳

✭✐✐✮ ❚❡♠♦s q✉❡x ⋆(y ⋆ z) = (x ⋆ y)⋆ z ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈G✱ ♦✉ s❡❥❛✱ ❛ ♦♣❡r❛çã♦ ⋆ é ❛ss♦❝✐❛t✐✈❛✳

✭✐✐✐✮ ❊①✐st❡ e∈ G✱ ❝❤❛♠❛❞♦ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ t❛❧ q✉❡ x ⋆ e=e ⋆ x =x✱ ♣❛r❛ t♦❞♦ x∈G✳

✭✐✈✮ ❉❛❞♦ q✉❛❧q✉❡rx∈G✱ ❡①✐st❡ x−1 _{∈Gt❛❧ q✉❡ x ⋆ x}−1 _=x−1 _{⋆ x=e✳}

❉✐r❡♠♦s q✉❡ ✉♠ ❣r✉♣♦(G, ⋆)é ❝♦♠✉t❛t✐✈♦ ♦✉ ❛❜❡❧✐❛♥♦ s❡x⋆y =y⋆x♣❛r❛ q✉❛✐sq✉❡r x, y ∈G✳

Pr♦♣r✐❡❞❛❞❡s ✸✳✶✳ ❙❡ (G, ⋆) é ✉♠ ❣r✉♣♦✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮ ❖ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ❣r✉♣♦ é ú♥✐❝♦✳

✭✐✐✮ ❉❛❞♦ x∈G✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ x−1 _{t❛❧ q✉❡ x ⋆ x}−1 _=x−1_{⋆ x=e✳} ✭✐✐✐✮ ❚❡♠♦s q✉❡ (x−1₎−1 _=x✳

✭✐✈✮ (x ⋆ y)−1 _=y−1 _{⋆ x}−1_✳

✭✈✮ ❱❛❧❡♠ ❛s ❧❡✐s ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦ ❛ ❞✐r❡✐t❛ ❡ ❛ ❡sq✉❡r❞❛✱ ✐st♦ é✱ ❞❛❞♦s x, y, z∈G

x ⋆ y =x ⋆ z⇒y=z ❡ y ⋆ x=z ⋆ x⇒y=z

✭✈✐✮ ❉❛❞♦sa, b∈G✱ ❛s ❡q✉❛çõ❡s ❧✐♥❡❛r❡sa ⋆ x=b ❡x ⋆ a=b tê♠ ú♥✐❝❛s s♦❧✉çõ❡s ❡♠ G✳

✷✻ ●r✉♣♦s

❉❡♠♦♥str❛çã♦✿

✭✐✮ ❙❡ e1 ❡ e2 sã♦ ❡❧❡♠❡♥t♦s ♥❡✉tr♦s ❞❡ ✭G✱⋆✮✱ ❡♥tã♦✿

e1 ⋆ e2 ❂e2✱ ✭♣♦✐s e1 é ❡❧❡♠❡♥t♦ ♥❡✉tr♦✮

e1 ⋆ e2 ❂e1✱ ✭♣♦✐s e2 é ❡❧❡♠❡♥t♦ ♥❡✉tr♦✮ ▲♦❣♦✱ e1❂e2

✭✐✐✮ ❙❡ y1 ❡y2 sã♦ ✐♥✈❡rs♦s ❞❡x✱ ❡♥tã♦x ⋆ y1 ❂y1 ⋆ x❂e✱ ❡ x ⋆ y2 ❂y2 ⋆ x❂e✱ ❞❡ss❡ ♠♦❞♦✱

y1 =e ⋆ y1 = (y2⋆ x)⋆ y1

= y2⋆(x ⋆ y1)

= y2⋆ e

= y2 ■st♦ é✱ y1 ❂ y2

✭✐✐✐✮❉❛❞♦ x ∈ G✱ ✉♠ ❡❧❡♠❡♥t♦ y ∈ Gé✱ ♣♦r ❞❡✜♥✐çã♦ ♦ ✐♥✈❡rs♦ ❞❡ x ♦✉ ✈✐❝❡✲✈❡rs❛✱

q✉❛♥❞♦✿

x ⋆ y❂y ⋆ x❂e

❈♦♠♦ x ⋆ x−1 _❂x−1 _{⋆ x ❂e✱ ❡♥tã♦ x ❂ ✭x}−1₎−1_✳ ✭✐✈✮ ❱❛♠♦s ♠♦str❛r q✉❡

✭x ⋆ y✮⋆ ✭x−1 _{⋆ y}−1_)❂✭x−1 _{⋆ y}−1_{)✭x ⋆ y✮ ❂ e}

❯s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❞❛ ♦♣❡r❛çã♦ ❡♠ G✱ ♣♦❞❡✲s❡ ♦♠✐t✐r ♦s ♣❛rê♥t❡s❡s ♥❛

❡q✉❛çã♦ ❛❝✐♠❛✱ ❞❡ ♠♦❞♦ q✉❡✿

✭x ⋆ y✮ ⋆ ✭x−1 _{⋆ y}−1_{) ❂x ⋆ y ⋆ x}−1 _{⋆ y}−1 _{❂ x ⋆ e ⋆ x}−1 _{❂ e} ✭✈✮ ❈♦♠♦ ❡①✐st❡ x1 ∈ G t❛❧ q✉❡ x1 ⋆ x❂ e❂ x ⋆ x1✱ t❡♠♦s✿

x ⋆ y ❂ x ⋆ z ⇒ x1 ⋆✭x ⋆y✮ ❂ x1 ⋆ ✭x ⋆ z✮✱ ✭♦♣❡r❛♥❞♦ à ❡sq✉❡r❞❛ ❝♦♠x1✮

⇒✭x1 ⋆x✮⋆y ❂ ✭x1 ⋆x✮⋆z ✭♣♦✐s⋆ é ❛ss♦❝✐❛t✐✈❛✮

⇒ e⋆y ❂e ⋆ z ✭♣♦✐sx1⋆x ❂e✮✳

■st♦ é✱ y ❂z✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♠♦str❛✲s❡ q✉❡y ⋆ x ❂z ⋆ x ✐♠♣❧✐❝❛ ❡♠y ❂ z✳

✭✈✐✮ ❱❛♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çã♦ a ⋆ x ❂ b❀ ♦

♦✉tr♦ ❝❛s♦ é tr❛t❛❞♦ s✐♠✐❧❛r♠❡♥t❡✳ ❙❡❥❛ a1 ∈ G✱ ❝♦♠ a1 ⋆ a ❂ e✳ ▲♦❣♦ ♦ ❡❧❡♠❡♥t♦ x1 ❂ a1 ⋆ b ∈G é t❛❧ q✉❡✿

a ⋆ ✭a1 b✮ ❂ ✭a ⋆ a1✮⋆ b ❂ e ⋆ b ❂b

■st♦ é✱x1 é ✉♠❛ s♦❧✉çã♦ ❞❡a ⋆ x❂b✳ ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡y1 ∈Gs❡❥❛ ♦✉tr❛ s♦❧✉çã♦✳ P♦r ✐ss♦✱ a ⋆ x1 ❂ b ❡ a ⋆ y1 ❂ b✱ ♦✉ s❡❥❛ a ⋆ x1 ❂ a ⋆ y1✱ ▲♦❣♦✱ ♣♦r ✭✈✮✱ t❡♠♦s x1 ❂

❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✷✼

❊①❡♠♣❧♦s ✸✳✶✳ 1−❖s ❝♦♥❥✉♥t♦s (Z,+)✱(Q,+)✱ (R,+) ❡(C,+) sã♦ ❡①❡♠♣❧♦s ❞❡ ❣r✉✲

♣♦s ❛❜❡❧✐❛♥♦s ❝♦♠ ❛ s♦♠❛ ✉s✉❛❧✳

2−P❛r❛ ❝❛❞❛n∈N✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❞✉❛s ♦♣❡r❛çõ❡s ♣❛r❛ ♦ ❝♦♥❥✉♥t♦Zn ={¯0,¯1,¯2, ..., n−1} ❞❛❞❛s ♣♦r ¯a+ ¯b = a+b ❡ ¯a¯b = ab✳ Pr✐♠❡✐r❛♠❡♥t❡ ♦❜s❡r✈❡♠♦s q✉❡ ❛s ♦♣❡r❛çõ❡s sã♦

❜❡♠ ❞❡✜♥✐❞❛s✿ s❡❥❛♠ a1, a2, b1, b2 ∈Zn t❛✐s q✉❡ a¯1 = ¯a2 ❡b¯1 = ¯b2✱ t❡♠♦s q✉❡

a1 =a2+n·(k1) ❡ b1 =b2+n·(k2) ✭✸✳✶✮ ❝♦♠ k1, k2 ∈Z✳ ❙♦♠❛♥❞♦a1 ❝♦♠ b1✱ ♦❜t❡♠♦s

a1+b1 =a2+b2+n·(k1+k2). ❖✉ s❡❥❛

(a1+b1)≡(a2+b2) mod n ⇔ a1+b1 =a2 +b2. ▲♦❣♦✱

¯

a1+ ¯b1 =a1+b1 =a2+b2 = ¯a2+ ¯b2 P❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ t❡♠♦s✿ ❞❡ ✸✳✶ t❡♠♦s q✉❡

a1.b1 = (a2+n·(k1))(b2+n·(k2)) =a2.b2+n·(a2.k2 +b2k2+nk1k2). ❉❡ss❡ ♠♦❞♦

a1.b1 ≡a2.b2 mod n ⇔ a1.b1 =a2.b2. P♦rt❛♥t♦✱

¯

a1.b¯1 =a1.b1 =a2.b2 = ¯a2.b¯2.

❆ss✐♠ ❞❛❞♦s q✉❛✐sq✉❡r a,¯ ¯b ∈Zn✱ t❡♠♦s q✉❡ a+b ∈ Zn✱ a,¯ ¯b ∈ Zn✱ t❡♠♦s q✉❡ a¯+ ¯b =

a+b =b+a= ¯b+ ¯a✳ ❚❡♠♦s t❛♠❜é♠ q✉❡

¯

a+ (¯b+ ¯c) = ¯a+a+b

= a+ (b+c) = (a+b) +c

= a+b+ ¯c

= (¯a+ ¯b) + ¯c, ∀a,¯ ¯b,c¯∈Zn.

❉❛❞♦ ¯a ∈ Zn✱ t❡♠♦s q✉❡ ¯a+ ¯0 = a+ 0 = ¯a✳ P♦r ú❧t✐♠♦✱ ❞❛❞♦ ¯a ∈ Zn t❡♠♦s q✉❡

¯

x = n−a ∈ Zn ❡ q✉❡ a¯+ ¯x = ¯a+n−a = a+n−a = ¯n = ¯0✳ ▲♦❣♦✱ G = (Zn,+) é ✉♠ ❣r✉♣♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡✜♥✐❞❛ ❛❝✐♠❛✳

❆✐♥❞❛ ❡♠Zn ❝♦♠♦ ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡ ❞❛❞♦s q✉❛✐sq✉❡r ¯a,¯b∈ Zn✱ t❡♠♦s q✉❡ ab∈ Zn✳ ❋❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛♠♦s q✉❡ ¯a.(¯b.c¯) = (¯a.¯b).¯c✱ ∀a,¯ ¯b,¯c ∈ Zn✱ q✉❡ ¯a.¯b = ¯b.¯a✱ q✉❡

¯

a.¯1 = ¯a✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r ❛✐♥❞❛ q✉❡✱ ❞❛❞♦ ¯a ∈Zn✱ ❡①✐st❡¯b ∈Zn t❛❧ q✉❡ ¯a.¯b = ¯1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ mdc(a, n) = 1✳ ▲♦❣♦✱ ♦ ❝♦♥❥✉♥t♦

✷✽ ●r✉♣♦s

❉❡✜♥✐çã♦ ✸✳✷✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❣r✉♣♦ G✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ H ❞❡ G é

✉♠ s✉❜❣r✉♣♦ ❞❡ G q✉❛♥❞♦ H✱ ❝♦♠ ❛ ♦♣❡r❛çã♦ ✐♥❞✉③✐❞❛ ❞❡ G✱ t❛♠❜é♠ é ✉♠ ❣r✉♣♦✳

❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ H < G ♣❛r❛ ✐♥❞✐❝❛r q✉❡ H é s✉❜❣r✉♣♦ ❞❡ G✳

❊①❡♠♣❧♦s ✸✳✷✳ 1− ❙♦❜ ❛s ❛❞✐çõ❡s ✉s✉❛✐s✱ t❡♠♦s q✉❡

Z<Q<R<C.

❡ s♦❜ ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ✉s✉❛✐s

Q∗ <R∗ <C∗.

2− ❖❜s❡r✈❡ q✉❡Q ⊂R✱ ♣♦r❡♠ (Q, .) ♥ã♦ é s✉❜❣r✉♣♦ ❞❡ (R,+) ♣♦✐s ❛s ♦♣❡r❛çõ❡s sã♦

❞✐st✐♥t❛s✳

❚❡♦r❡♠❛ ✸✳✶✳ ❙❡❥❛ H ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ ✉♠ ❣r✉♣♦ G✳ ❊♥tã♦✱ H é ✉♠

s✉❜❣r✉♣♦ ❞❡ G s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠❛ ❞❛s ❝♦♥❞✐çõ❡s s❡❣✉✐♥t❡s é s❛t✐s❢❡✐t❛✿

✭✐✮ h1.h2 ∈H ❡ h1−1 ∈H✱ ♣❛r❛ t♦❞♦ h1, h2 ∈H✳ ✭✐✐✮ h1.h−21 ∈H✱ ♣❛r❛ t♦❞♦ h1, h2 ∈H✳

❉❡♠♦♥str❛çã♦✿ ❙❡ H é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ ❡♥tã♦ H t❛♠❜é♠ é ✉♠ ❣r✉♣♦ ❡✱ ♣♦r

✐ss♦ ❛s ❝♦♥❞✐çõ❡s ✭✶✮ ❡ ✭✷✮ sã♦ ❝❧❛r❛♠❡♥t❡ s❛t✐s❢❡✐t❛s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡

H s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✶✮✳ ▲♦❣♦✱ ♣❛r❛ q✉❛❧q✉❡r h ∈ H✱ t❡♠♦s q✉❡ h−1 _{∈ H✳ ❆ss✐♠✱}

e❂h✳h−1 _{∈ H✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ H ❁ G✳ ❋✐♥❛❧♠❡♥t❡✱ s❡ H s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✷✮✱} ❡♥tã♦ ❞❛❞♦s h1✱ h2 ∈ H✱

e ❂ h2✳h−1

2 ∈H → h−21 ❂e✳h−21 ∈ H✳ ❈♦♠ ✐ss♦✱

h1✳h2 ❂h1 ✳ ✭h−21)−1 ∈H

P♦rt❛♥t♦✱ H é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✳

❉❡✜♥✐çã♦ ✸✳✸✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❯♠ s✉❜❣r✉♣♦ H ❞❡ G ❝❤❛♠❛✲s❡ ◆♦r♠❛❧ q✉❛♥❞♦ ghg−1 _{∈H,∀g ∈G ❡ ∀h∈H,}

♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

gHg−1 _={ghg−1 _{| ∀h∈H ❡ ∀g ∈G} ⊂H.}

◆♦t❛çã♦✿ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ N ✁G ♣❛r❛ ✐♥❞✐❝❛r q✉❡ N é s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G✳ ❊①❡♠♣❧♦ ✸✳✶✳ ❙❡ Gé ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ t♦❞♦ s✉❜❣r✉♣♦ H ❞❡ G é ♥♦r♠❛❧✳

●❡r❛❞♦r❡s ✷✾

❚❡♦r❡♠❛ ✸✳✷✳ ❙❡❥❛ H ✉♠ s✉❜❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦G✳ ❊♥tã♦✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦

❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ H✁G✳

✭✐✐✮ gHg−1 _{=H✱∀g ∈G✳} ✭✐✐✐✮ gH =Hg✱∀g ∈G✳

❉❡♠♦♥str❛çã♦✿ ✭✐✮ ⇒ ✭✐✐✮ P♦r ❤✐♣ót❡s❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ t❡♠✲s❡ ♥❛t✉r❛❧♠❡♥t❡ ❛

✐♥❝❧✉sã♦ gHg−1 _{⊂H✳ ❆❣♦r❛✱ ❞❛❞♦h∈H✱}

h=g−1_(ghg−1_{)g ∈H✱}

♣♦✐sghg−1 _{∈H✁G✳ ■ss♦ ♥♦s ❞✐③ q✉❡ H ⊂gHg}−1 _{❡✱ ♣♦rt❛♥t♦gHg}−1 _=H✳

✭✐✐✮ ⇒ ✭✐✐✐✮ P❛r❛ g ∈ G✱ s❡❥❛ x ∈ gH✱ ❞✐❣❛♠♦s x = gh ♣❛r❛ ❛❧❣✉♠ h ∈ H✳ ▲♦❣♦✱

♣♦r ❤✐♣ót❡s❡✱

xg−1 _=ghg−1 _∈gHg−1 _=H ✐st♦ é✱ xg−1 _{= h1✱ ❝♦♠ h}

1 ∈ H✳ P♦rt❛♥t♦ x= h1g ∈ Hg✱ ❞❡ ♠♦❞♦ q✉❡ gH ⊂Hg✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣r♦✈❛✲s❡ q✉❡ Hg ⊂gH✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ Hg =gH✳

✭✐✐✐✮ ⇒ ✭✐✮ ❙❡❥❛♠ g ∈G ❡ h∈H✳ ❈♦♠♦ gH =Hg ❡ gh∈Hg✱ s❡❣✉❡ q✉❡ gh=h2g ♣❛r❛ ❛❧❣✉♠ h2 ∈H✱ ♦✉ s❡❥❛✱ ghg−1 =h2 ∈H✳ P♦rt❛♥❞♦H✁G✳

✸✳✷ ●❡r❛❞♦r❡s

❉❡✜♥✐çã♦ ✸✳✹✳ ❙❡❥❛ (G, .) ✉♠ ❣r✉♣♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ♠✉❧t✐♣❧✐❝❛t✐✈❛✳ ❉❛❞♦s a ∈ G ❡ n ∈Z✱ ❞❡✜♥❡✲s❡ n✲és✐♠❛ ♣♦tê♥❝✐❛ ❞❡ a✱ an_{✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿}

an =



 

 

e s❡ n = 0, an−1_{.a s❡ n >0,}

(a−n₎−1 _{s❡ n <0.}

❙❡ ❛ ♦♣❡r❛çã♦ ❡♠G❢♦r ❛❞✐t✐✈❛✱ ❡♥tã♦ ❞❡✜♥✐✲s❡ ♠ú❧t✐♣❧♦ ❞❡a✱ n.a✱ ❛♦ ✐♥✈és ❞❡ ♣♦tê♥❝✐❛

❞❡ a✳ ❆ss✐♠✱

n.a=     

e s❡ n= 0,

(n−1).a+a s❡ n >0,

(−n).(−a) s❡ n <0.

❙❡❥❛G ✉♠ ❣r✉♣♦ q✉❛❧q✉❡r✱ a∈G✳ ❙❡❥❛hai={ai_{;i∈Z}✱ ❛ss✐♠ é ❢á❝✐❧ ✈❡r q✉❡ hai} é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ ❞❡♥♦♠✐♥❛❞♦ s✉❜❣r✉♣♦ ❝í❝❧✐❝♦ ❣❡r❛❞♦ ♣♦r a✳

✸✵ ●r✉♣♦s

❉❡✜♥✐çã♦ ✸✳✻✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✱ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ G✳ ❙❡❥❛ H ♦ ❝♦♥❥✉♥t♦ ❞❡

t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ G q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ❡❧❡♠❡♥t♦s

❞❡ S✱ ❡❧❡✈❛❞♦s ❛ ❡①♣♦❡♥t❡s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ♥❡❣❛t✐✈♦s ♦✉ ♥✉❧♦s✳ ❆ss✐♠ hSi s❡rá ✉♠

s✉❜❣r✉♣♦ ❞❡ G ♦ q✉❛❧ ❞✐r❡♠♦s q✉❡ é ♦ s✉❜❣r✉♣♦ ❞❡ G ❣❡r❛❞♦ ♣♦r S ❡ ❞❡♥♦t❛♠♦s ♣♦r H =hSi✳

❊①❡♠♣❧♦ ✸✳✸✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✭Z✱✰✮ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ 1 ♦✉−1✱ ❧♦❣♦Z=h1i✱ ♦❜s❡r✈❡ q✉❡ Z=h−1i✳

P♦❞❡♠♦s ♣❡♥s❛r ❡♠ ❣❡r❛❞♦r❡s ❝♦♠♦ s❡♥❞♦ ♦ ✏♥ú❝❧❡♦✑ ❞♦ ❣r✉♣♦❀ ✉♠❛ ✈❡③ q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ t❡r♠♦s ❞♦s ❣❡r❛❞♦r❡s✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦s ❣❡r❛❞♦r❡s ♣♦❞❡♠ ♠✉✐t❛s ✈❡③❡s s❡r tr❛❞✉③✐❞❛s ♣❛r❛ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ t♦❞♦ ♦ ❣r✉♣♦✳ ▲❡♠❛ ✸✳✶✳ ❙❡❥❛G✉♠ ❣r✉♣♦ ✜♥✐t♦✱ ♦✉ s❡❥❛ ♦ ❝♦♥❥✉♥t♦ ● é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ ❡g ∈G✳

❊♥tã♦ g−1 _=gn _{♣❛r❛ ❛❧❣✉♠ n∈N✳}

❉❡♠♦♥str❛çã♦✿ ❙❡Gé ✜♥✐t♦✱ ❞✐❣❛♠♦sG=a1, a2, ..., ak✱ ❡♥tã♦ t♦❞♦ ❡❧❡♠❡♥t♦a∈G t❡♠ ♦r❞❡♠ ✜♥✐t❛✳ ❋❛③❡♥❞♦ O(ai) = ni ♣❛r❛ i = 1, ..., k ❡ ❝♦♥s✐❞❡r❛♥❞♦ s ♦ ♣r♦❞✉t♦ ❞❡ss❛s ♦r❞❡♥s s=n1.n2...nk✱ t❡♠♦s q✉❡✿

as

i = (a

ni

i )r =e,∀a∈G✱

❡♠ q✉❡ r =n1.n2...ni−1.ni+1...nk✳

▲❡♠❛ ✸✳✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ G✳ ❊♥tã♦ G = hSi s❡✱ ❡

s♦♠❡♥t❡ s❡✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ G ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s

❞❡ S✳✭◆❡ss❡ ❝❛s♦ ♦s ✐♥✈❡rs♦s ❞❡ S ♥ã♦ sã♦ ♥❡❝❡ssár✐♦s✳✮

❉❡♠♦♥str❛çã♦✿ ❙❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ G ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡

❡❧❡♠❡♥t♦s ❞❡ S✱ ❡♥tã♦ t❡♠♦s q✉❡G=hSi✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ G=hSi✳ ▲♦❣♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡G ♣♦❞❡ s❡r ❡s❝r✐t♦

❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ s1.s2...sn✱ ♦♥❞❡ ❝❛❞❛ si ❡stá ❡♠ S ♦✉ é ✉♠ ✐♥✈❡rs♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ S✳ Pr♦✈❛r❡♠♦s ✐ss♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳

❙❡ n = 1✳ ❚❡♠♦s q✉❡ s1 ∈ S ♦✉ s1−1 ∈ S✳ ❙❡ s1 ∈ S✱ ❡♥tã♦ s1 é ❡s❝r✐t♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡ S✳ ❙❡ s−1

1 ∈ S✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✸✳✶✱ s

−1

1 ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ S✳

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❡♥♦r q✉❡ n❀ q✉❡r❡♠♦s ♠♦str❛r q✉❡ s1.s2...sn ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ S✳ P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s1.s2...sn−1 ❡ sn ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ♦ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡S✱ ❛ss✐♠✱ s1.s2...sn é ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡

S✳

●r✉♣♦s ❞❡ ❙✐♠❡tr✐❛s ✸✶

Pr♦♣♦s✐çã♦ ✸✳✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ G✳ ❙✉♣♦♥❤❛ q✉❡ ❛s

❞✉❛s ❝♦♥❞✐çõ❡s s❡❣✉✐♥t❡s sã♦ s❛t✐s❢❡✐t❛s✿

✶✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ S s❛t✐s❢❛③ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ P✳

✷✳ ❙❡ g ∈G ❡ h∈G s❛t✐s❢❛③❡♠ P✱ ❡♥tã♦ gh t❛♠❜é♠ s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ P✳

❊♥tã♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ hSi s❛t✐s❢❛③ P✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ▲❡♠❛ ✸✳✷✱ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ hSi ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

s1.s2...sn ♦♥❞❡n ∈N❡ ❝❛❞❛ si ∈S✳ Pr♦✈❛r❡♠♦s ❛ ♣r♦♣♦s✐çã♦ ✉s❛♥❞♦ ✐♥❞✉çã♦ s♦❜r❡n✳ ❙❡n= 1 ❡♥tã♦✱ ♣♦r ❤✐♣ót❡s❡✱s1 ∈S s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ P✳

❙✉♣♦♥❤❛✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ s1.s2...sn−1 s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡P✳ ❊♥tã♦✱ ♦ ♣r♦❞✉t♦

(s1.s2...sn−1)sn é ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s s❛t✐s❢❛③❡♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ P✱ ❧♦❣♦✱ ♣♦r

❤✐♣ót❡s❡✱ s❛t✐s❢❛③❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ P✳

✸✳✸ ●r✉♣♦s ❞❡ ❙✐♠❡tr✐❛s

P❡r♠✉t❛çã♦ é ♦ t❡r♠♦ ❡s♣❡❝í✜❝♦ ✉s❛❞♦ ♥❛ t❡♦r✐❛ ❞♦s ❣r✉♣♦s ♣❛r❛ ❞❡s✐❣♥❛r ✉♠❛ ❜✐❥❡çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♥❡❧❡ ♠❡s♠♦✳ ❙❡ A✐♥❞✐❝❛ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ ❞❡♥♦t❛r❡♠♦s

♣♦rS(A)♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡r♠✉t❛çõ❡s ❞♦s ❡❧❡♠❡♥t♦s ❞❡A✳ ❆ ❝♦♠♣♦s✐çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s

é✱ ♥❡st❡ ❝❛s♦✱ ✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡ S(A)✱ ♣♦✐s s❡f ❡ g sã♦ ♣❡r♠✉t❛çõ❡s ❞❡ A✱ ♦✉ s❡❥❛✱

s❡ f :A−→A✱ ❡g :A−→A sã♦ ❜✐❥❡çõ❡s✱ ❡♥tã♦ ❛ ❝♦♠♣♦st❛ ❣◦❢ ✿ ❆ −→ ❆ t❛♠❜é♠

é ✉♠❛ ❜✐❥❡çã♦✳ ❈❤❛♠❛✲s❡ (SA,◦)❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s s♦❜r❡ A✳

◗✉❛♥❞♦ A t❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s✱ A ={x1, x2, ..., xn}✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ {1,2, ..., n} ♣❛r❛ r❡♣r❡s❡♥t❛r ❛s ♣❡r♠✉t❛çõ❡s ❞♦s ❡❧❡♠❡♥t♦s ❞❡ A ❡ ❞❡♥♦t❛r❡✲

♠♦s S(A) ♣♦r Sn✳

➱ ❝♦♠✉♠ r❡♣r❡s❡♥t❛r ✉♠❛ ♣❡r♠✉t❛çã♦α∈Sn ♣♦r

α= 1 2 ... n

α(1) α(2) ... α(n)

!

❊①❡♠♣❧♦ ✸✳✹✳ ❙❡❥❛ A={1,2,3}✳ ❆s ♣❡r♠✉t❛çõ❡s ❞❡ A sã♦

α1 =

1 2 3 1 2 3

!

, α2 =

1 2 3 2 1 3

!

, α3 =

1 2 3 1 3 2

!

α4 =

1 2 3 3 2 1

!

, α5 =

1 2 3 3 1 2

!

, α6 =

1 2 3 2 3 1

!

♦✉ s❡❥❛✱ S3 ={α1, α2, α3, α4, α5, α6}✳ ❋❛③❡♥❞♦ α=α6 ❡β =α2✱ t❡♠♦s

α2 ₌ 1 2 3

2 3 1

!

1 2 3 2 3 1

!

= 1 2 3 3 1 2

!

✸✷ ●r✉♣♦s

❆♥❛❧♦❣❛♠❡♥t❡✱ α3 _{=e✱ β}2 _=e✱βα=α

3 ❡ αβ =α4✳

❉❡✜♥✐çã♦ ✸✳✼✳ ❯♠❛ ♣❡r♠✉t❛çã♦ α ∈ Sn ❝❤❛♠❛✲s❡ ❝✐❝❧♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ r ♦✉

r✲❝✐❝❧♦ q✉❛♥❞♦ ❡①✐st❡♠ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s a1✱ a2✱ ✳✳✳✱ ar∈ {1,2..., n} t❛✐s q✉❡

α(a1) = a2, α(a2) = a3, ... α(ar−1) =ar, α(ar) = a1 ❡

α(i) =i, ∀i∈ {1,2..., n} − {a1, a2, ..., ar}. ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ ✷✲❝✐❝❧♦ ❝❤❛♠❛✲s❡ tr❛♥s♣♦s✐çã♦✳

❊♠ ❣❡r❛❧✱ ❞❡♥♦t❛✲s❡ ✉♠ r✲❝✐❝❧♦ α ♣♦r α= (a1 a2 ... ar)✳

❊①❡♠♣❧♦ ✸✳✺✳ ◆♦ ❣✉♣♦ S5✱ ❛ ♣❡r♠✉t❛çã♦α = 1 2 3 4 5 3 2 5 1 4

!

é t❛❧ q✉❡α(1) = 3✱

α(3) = 5✱ α(5) = 4✱ α(4) = 1 ❡α(2) = 2✳ ▲♦❣♦✱ α= (1 3 5 4)✱ ♦✉ s❡❥❛✱α é ✉♠ ✹✲❝✐❝❧♦✳

❊①❡♠♣❧♦ ✸✳✻✳ ❊♠ S4

τ = 1 2 3 4 2 1 3 4

!

= (1 2) ❡ σ= 1 2 3 4 1 4 3 2

!

= (2 4)

sã♦ tr❛♥s♣♦s✐çõ❡s✳

❖❜s❡r✈❛çã♦ ✸✳✶✳ ❖❜s❡r✈❡ q✉❡ s❡µ= (a1 a2 ... ar)∈Sn✱ ❡♥tã♦µ−1 = (ar ar−1 ... a1)∈

Sn

❉❡✜♥✐çã♦ ✸✳✽✳ ❉♦✐s ❝✐❝❧♦s α, β ∈ Sn✱ ❞✐❣❛♠♦s α = (a1 a2 ... ar) ❡ β = (b1 b2 ... bk) sã♦ ❞✐t♦s ❝✐❝❧♦s ❞✐s❥✉♥t♦s q✉❛♥❞♦ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡{1,2, ..., n} é ♠♦✈✐❞♦ ♣♦r ❛♠❜♦s✳

❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ q✉❛♥❞♦

{a1, a2, ..., ar} ∩ {b1, b2, ..., bk}=∅

❊①❡♠♣❧♦ ✸✳✼✳ ❙❡❥❛ α ∈ S5 ❞❛❞♦ ♣♦r α =

1 2 3 4 5 4 3 2 5 1

!

✱ ❛ss✐♠ t❡♠♦s q✉❡ α = (2 3)(1 4 5)✳

Pr♦♣♦s✐çã♦ ✸✳✷✳ ❙❡ α, β ∈Sn sã♦ ❝✐❝❧♦s ❞✐s❥✉♥t♦s✱ ❡♥tã♦ αβ =βα✳

❉❡♠♦♥str❛çã♦✿ ❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ αβ(i) = βα(i) ♣❛r❛ t♦❞♦ i ∈ In✳ ❙❡ i ∈ In é ✜①❛❞♦ ♣♦r α❡ β✱ ❡♥tã♦ α✭β(i)✮ ❂α(i)❀ ❞❛ ♠❡s♠❛ ❢♦r♠❛ (βα)(i) ❂β(α(i))❂ β(i) ❂i✳

P♦rt❛♥t♦✱ ♣❛r❛ ❡ss❡ ❝❛s♦ t❡♠✲s❡ q✉❡ αβ(i) = βα(i)✳

❆❣♦r❛✱ s❡ α ♠♦✈❡ ♦ ❡❧❡♠❡♥t♦ i✱ ❞✐❣❛♠♦s α(i) = j 6= i✱ ❡♥tã♦ β(i) = i✱ ♣♦✐s α ❡ β sã♦

❞✐s❥✉♥t♦s✳ ❉❡ss❡ ♠♦❞♦✱

●r✉♣♦s ❞❡ ❙✐♠❡tr✐❛s ✸✸

P♦✐s α ♠♦✈❡ ♦ ❡❧❡♠❡♥t♦ j✱ ✉♠❛ ✈❡③ q✉❡ s❡ α(j) = j✱ ❡♥tã♦ α(i) = α(j)✱ ❝♦♠

i 6= j ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ α s❡r ✐♥❥❡t♦r❛✳ P♦rt❛♥t♦ αβ(i) = βα(i)✳ ❉❛ ♠❡s♠❛

❢♦r♠❛ ♠♦str❛✲s❡ ❡st❛ ✐❣✉❛❧❞❛❞❡ q✉❛♥❞♦ ♦ ❡❧❡♠❡♥t♦ ié ♠♦✈✐❞♦ ♣♦r β✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱

αβ =βα✳

❚❡♦r❡♠❛ ✸✳✸✳ ❚♦❞❛ ♣❡r♠✉t❛çã♦ α ∈ Sn− {e} ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ❝✐❝❧♦s ❞✐s❥✉♥t♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡st❛ ❢❛t♦r❛çã♦ é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✳

❉❡♠♦♥str❛çã♦✿ ❙❡ α ∈ Sn ❢♦r ✉♠ ❝✐❝❧♦✱ ❡♥tã♦ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ✐♠❡❞✐❛t♦✳ ❈❛s♦ ❝♦♥trár✐♦ ❝♦♥s✐❞❡r❡♠♦s O1, O2, ..., Ok✱ ❛s ❞✐st✐♥t❛s α✲ór❜✐t❛s ♥ã♦ tr✐✈✐❛✐s✱ ♦✉ s❡❥❛✿

❛sα✲ór❜✐t❛s ❝♦♠ ♠❛✐s ❞❡ ✉♠ ❡❧❡♠❡♥t♦✳ ❚❡♠♦s ❡♥tã♦ q✉❡α(Oi) =O✱ ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ i= 1, ..., n✱ ❞❡✜♥❛♠♦s✿

µi(j) =

(

α(j) s❡ j ∈Oi,

j s❡ j /∈Oi,

❈❧❛r❛♠❡♥t❡✱ µ é ✉♠ ❝✐❝❧♦✱ ♣♦✐s s❡ j /∈ Oi✱ ❡♥tã♦ µ(j) = j ❡✱ ♣♦rt❛♥t♦✱ ❛ µi ✲ór❜✐t❛ ❞❡

j é ✉♥✐tár✐❛✱ ✐st♦ é✱ é ✐❣✉❛❧ ❛ j✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ Oi é ✉♠❛ ór❜✐t❛ ❞❡ µi✱ ♣♦✐s µi ❡ α ❝♦✐♥❝✐❞❡♠ ❡♠ Oi✱ ❡ ❝♦♠♦ α(Oi) = µ(Oi) = Oi, αm ❝♦✐♥❝✐❞❡ ❝♦♠ µi ❡♠ Oi✱ ♣❛r❛ t♦❞♦ m ∈ Z✳ ❆❧é♠ ❞❡ µ1, µ2, ..., µk s❡r❡♠ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ♣❛r❡s ✈ê✲s❡ ❝❧❛r❛♠❡♥t❡ q✉❡

α =µ1µ2...µk✳

▼♦str❡♠♦s ❛❣♦r❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ ❢❛t♦r❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡✿ α =β1β2...βl✱ s❡♥❞♦ ♦s βi ❝✐❝❧♦s ♥ã♦ tr✐✈✐❛✐s ❞✐s❥✉♥t♦s✳ P❛r❛ ❝❛❞❛i = 1, ..., l✱ ❝❤❛♠❡♠♦s ❞❡ Ci ❛ ór❜✐t❛ ♥ã♦ tr✐✈✐❛❧ ❞❡ βi✳ ❉❡ss❡ ♠♦❞♦ C1, C2, ..., Cl sã♦ ór❜✐t❛s ♥ã♦ tr✐✈✐❛✐s ❞❡ α =β1β2...βl✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ l =k ❡✱ r❡♦r❞❡♥❛♥❞♦ s❡ ♥❡❝❡ssár✐♦✱ t❡♠♦s C1 =O1, C2 =O2, ..., Ck =Ok✳ ▲♦❣♦ µi =βi ❝♦♠ i= 1, ..., k ♣♦✐s µ(j) = α(j) = β(j)♣❛r❛ t♦❞♦ j ∈Oi✳ ❈♦r♦❧ár✐♦ ✸✳✶✳ ❚♦❞❛ ♣❡r♠✉t❛çã♦ α∈ Sn ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ tr❛♥s♣♦s✐✲ çõ❡s✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❜❛st❛ ♠♦str❛r q✉❡ t♦❞♦ ❝✐❝❧♦ ❡♠ Sn é ✉♠ ♣r♦❞✉t♦ ❞❡ tr❛♥s♣♦s✐çõ❡s✳ ❆ss✐♠✱ ❞❛❞♦ µ= (a1 a2 ... ar)✱ t❡♠♦s q✉❡

µ= (a1 ar)(a1 ar−1)...(a1 a2).

✸✹ ●r✉♣♦s

❈♦r♦❧ár✐♦ ✸✳✷✳ ✭✐✮ ❖ ❝♦♥❥✉♥t♦ ❞❡ tr❛♥s♣♦s✐çõ❡s {(1 2),(1 3), ...,(1n),} ❣❡r❛ Sn✳ ✭✐✐✮ ❆s tr❛♥s♣♦s✐çõ❡s {(1 2),(2 3), ...,(n−1n)} ❣❡r❛ Sn✳

❉❡♠♦♥str❛çã♦✿ (i) ❖❜s❡r✈❡ q✉❡ (a b) = (1 a)(1 b)(1 a)✱ ❛ss✐♠ ❜❛st❛ ✉t✐❧✐③❛r♠♦s ♦

❝♦r♦❧ár✐♦ ❛♥t❡r✐♦r✳

(ii) ❖❜s❡r✈❡ q✉❡ (1k) = (k−1 k)...(3 4)(2 3)(1 2)(2 3)(3 4)...(k−1k) ❡ ❛♣❧✐❝❛♠♦s ❛

♣❛rt❡ (i)✳

❊①❡♠♣❧♦ ✸✳✽✳ ◆♦t❡♠♦s q✉❡ ❛ ♣❡r♠✉t❛çã♦σ ∈S6❞❛❞❛ ♣♦rσ =

1 2 3 4 5 6 6 4 3 5 2 1

!

✱ é t❛❧ q✉❡ σ= (1 6)(2 4 5)✳ P♦rt❛♥t♦✱ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ tr❛♥s♣♦s✐çõ❡s✱

σ= (1 6)(2 5)(2 4).

❚❡♦r❡♠❛ ✸✳✹✳ ❙❡❥❛♠ µ1, µ2, ..., µk ∈ Sn ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❛♦s ♣❛r❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦s

r1, r2, ..., rk✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ❛ ♦r❞❡♠ ❞❛ ♣❡r♠✉t❛çã♦ α =µ1µ2...µk é ✐❣✉❛❧ ❛

mmc(r1, r2, ..., rk).

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ µ1, µ2, ..., µk sã♦ ❝✐❝❧♦s ❞✐s❥✉♥t♦s✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✷ µiµj =

µjµi q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ i, j ∈ {1,2, ..., k}✳ ▲♦❣♦✱

αs _{= (µ}

1µ2...µk)s=µs1µs2...µsk,∀s∈Z

❙❡♥❞♦ m = mmc(r1, r2, ..., rk)✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ i ∈ {1,2, ..., k}✱ ❡①✐st❡ λi ∈ Z t❛❧ q✉❡

m=λiri✳ ❆ss✐♠

µmi =µ λiri

i = (µ

ri

i ) λi

=e,

♣♦✐s ❛ ♦r❞❡♠ ❞❡µi éri✳ ▲♦❣♦✱

αm = (µ1µ2...µk)m =µm₁ µm₂ ...µmk =e. P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ αt _{=e✱ ♦✉ s❡❥❛✱(µ}

1µ2...µk)t=e✱ ❡♥tã♦

µt

1µt2...µtk=e.

▼❛s ❝♦♠♦ ♦s ❝✐❝❧♦s µ1, µ2, ..., µk sã♦ ❞✐s❥✉♥t♦s✱ ♦❜t❡♠♦s q✉❡

µt

i =e,∀i∈ {1,2, ..., k}

❍♦♠♦♠♦r✜s♠♦s ❞❡ ●r✉♣♦s ✸✺

❊①❡♠♣❧♦ ✸✳✾✳ ❈♦♥s✐❞❡r❡τ = 1 2 3 4 5 6 7 8 3 5 8 7 1 4 6 2

!

✳ ❈♦♠♦τ = (1 3 8 2 5)(4 7 6)✱

❡♠ q✉❡ µ1 = (1 3 8 2 5) ❡ µ2 = (4 7 6) sã♦ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✺ ❡ ✸✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s q✉❡ ❛ ♦r❞❡♠ ❞❡ τ é mmc(3,5) = 15✳

❚❡♦r❡♠❛ ✸✳✺✳ ❙❡ α, σ ∈ Sn✱ ❡♥tã♦ ασα−1 é ❛ ♣❡r♠✉t❛çã♦ ♦❜t✐❞❛ ❛♣❧✐❝❛♥❞♦ α ❛♦s ❡❧❡♠❡♥t♦s ❞♦s ❝✐❝❧♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❢❛t♦r❛çã♦ ❞❡ σ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ασα−1 _{❡ σ tê♠} ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❡ ❝✐❝❧♦s✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s τ =ασα−1_{✱ ❛ss✐♠✱ s❡ σ(i) =j✱ ❡♥tã♦}

(τ α)(i) = (ασα−1_{)α(i) = (ασ)(i) =α(j).}

❉❡ss❡ ♠♦❞♦✱ ❞❡s❞❡ q✉❡ (a1 a2 ... ar) s❡❥❛ ✉♠ ❝✐❝❧♦ ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ σ✱ t❡♠♦s q✉❡

(α(a1)α(a2)...α(ar))é ✉♠ ❝✐❝❧♦ ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ασα−1✳ P♦rt❛♥t♦✱ ❛ ❢❛t♦r❛çã♦ ❡♠ ❝✐❝❧♦s ❞❡ τ é ♦❜t✐❞❛ s✉❜st✐t✉✐♥❞♦✲s❡ x ♣♦r α(x) ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ σ✳ P♦r ✐ss♦✱τ ❡ σ

tê♠ ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❡ ❝✐❝❧♦s✳

❊①❡♠♣❧♦ ✸✳✶✵✳ ❉❛❞❛s ❛s ♣❡r♠✉t❛çõ❡s ❡♠ S6

α = 1 2 3 4 5 6 2 3 5 6 1 4

!

❡ σ = 1 2 3 4 5 6 1 2 3 5 6 4

!

✈❛♠♦s ❞❡t❡r♠✐♥❛r ασα−1 _{✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✸✳✺ ❡ ♣❡❧♦ ♠ét♦❞♦ tr❛❞✐❝✐♦♥❛❧✳ ❈♦♠♦}

σ = (4 5 6)✱ ❡♥tã♦

ασα−1 _{= (α(4)α(5)α(6)) = (6 1 4) = (1 4 6)} P❡❧♦ ♠ét♦❞♦ tr❛❞✐❝✐♦♥❛❧ t❡♠♦s✱

α−1 ₌ 1 2 3 4 5 6

5 1 2 6 3 4

!

❡

ασα−1 ₌ 1 2 3 4 5 6

2 3 5 6 1 4

!

1 2 3 4 5 6 1 2 3 5 6 4

!

1 2 3 4 5 6 5 1 2 6 3 4

!

= 1 2 3 4 5 6 4 2 3 6 5 1

!

= (1 4 6).

✸✳✹ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ●r✉♣♦s

✸✻ ●r✉♣♦s

❉❡✜♥✐çã♦ ✸✳✾✳ ❙❡❥❛♠ (G1, ⋆)❡ (G2,∗) ❞♦✐s ❣r✉♣♦s✳ ❯♠❛ ❢✉♥çã♦ f :G1 →G2 ❝❤❛♠❛✲ s❡ ❤♦♠♦♠♦r✜s♠♦ ❞❡ G1 ❡♠ G2 q✉❛♥❞♦ f(a ⋆ b) =f(a)∗f(b)✱ ♣❛r❛ t♦❞♦ ❛✱ ❜ ∈ G1 Pr♦♣♦s✐çã♦ ✸✳✸✳ ❙❡❥❛ f :G1 →G2 ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳ ❊♥tã♦✿

✭✶✮ f(e1) =e2✱ s❡♥❞♦ ei ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ Gi ✭✷✮ f(a−1_{) = f(a)}−1_{✱ ♣❛r❛ t♦❞♦ ❛ ∈ G1✳}

✭✸✮ Im(f) ={f(a) :a∈G1}é ✉♠ s✉❜❣r✉♣♦ ❞❡ G2✳

✭✹✮ ❙❡Hé ✉♠ s✉❜❣r✉♣♦ ❞❡G2✱ ❡♥tã♦ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛f−1_{(H)❞❡H♣♦rf✱f}−1_{(H) =}

{x∈G1 :f(x)∈H}✱ é ✉♠ s✉❜❣r✉♣♦ ❞❡G1✳ ❉❡♠♦♥str❛çã♦✿ ✭✶✮ ❈♦♠♦ e1 =e1.e1✱ ❡♥tã♦✿

f(e1) = f(e1.e1) = f(e1).f(e1)

▲♦❣♦✱ f(e1) é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡G2✱ ♦✉ s❡❥❛✱ f(e1) = e2

✭✷✮ P❛r❛ t♦❞♦a ∈G1, a.a−1 =e1✳ ❆ss✐♠✱

f(a1.a−1) = f(e1) = e2✱

♦✉ s❡❥❛✱ f(a).f(a−1_{) = e2✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ f(a}−1_{) =f(a)}−1

✭✸✮ ❙❡♥❞♦ f(e1) = e2✱ ❡♥tã♦Im(f) =6 ∅✳ ❆❣♦r❛✱ ❞❛❞♦s x, y ∈ Im(f) ❡①✐st❡♠ a, b∈G1 t❛✐s q✉❡ f(a) =x ❡f(b) = y✳ P♦r ✐ss♦✱

x.y−1 _=f(a).f(b)−1 _=f(a).f(b−1_{) =f(a.b}−1₎ ❞❡ ♠❛♥❡✐r❛ q✉❡✱ x.y−1 _{∈Im(f) ❡Im(f)< G}

2

✭✹✮ ❈♦♠♦ e2 ∈ H ❡ f(e1) = e2✱ ❡♥tã♦ f−1(H) =6= ∅✳ ❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ a, b ∈

f−1_{(H)✳ ❆ss✐♠✱ ♣♦r ❞❡✜♥✐çã♦✱ f(a) ∈ H ❡ f(b) ∈ H✳ ❈♦♠♦ H é s✉❜❣r✉♣♦ ❞❡ G2✱}

f(b)−1 _=f(b−1_{) t❛♠❜é♠ ❡stá ❡♠ H✱ ❞❡ ♠♦❞♦ q✉❡✿}

f(a.b−1_{) =f(a).f(b}−1_{) = f(a).f(b)}−1 _∈H

P♦rt❛♥t♦✱ a.b−1 _∈f−1_{(H)✱ ✐♠♣❧✐❝❛♥❞♦ q✉❡ f}−1_{(H)< G✳} ❉❡✜♥✐çã♦ ✸✳✶✵✳ ❖ ♥ú❝❧❡♦ ❞❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❢✿ G1 → G2 é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦

ker(f) ❂ ④❣ ∈ G1✿❢✭❣✮❂e2⑥✱ ♦✉ s❡❥❛✱ ker(f) é ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❞❡ e2✳

❚❡♦r❡♠❛ ✸✳✻✳ ❙❡❥❛ f :G1 →G2 ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳ ❊♥tã♦✿ ✭✶✮ ker(f) =e1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f é ✐♥❥❡t♦r❛✳