Probabilidade de Comutar em Grupos Finitos

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❈♦♠✉t❛r ❡♠ ●r✉♣♦s ❋✐♥✐t♦s

♣♦r

❆❧❡①❛♥❞r❡ ▼❛t♦s ❞❛ ❙✐❧✈❛ P✐r❡s ❞❡ ▼♦r❛❡s

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ▼❛rt✐♥♦ ●❛r♦♥③✐

❇r❛sí❧✐❛ ✲ ❉❋ ✷✵✶✼

(2)

(3)

(4)

✹

❘❊❙❯▼❖

(5)

✺

❆❇❙❚❘❆❈❚

(6)

✻

❈♦♥t❡ú❞♦

❘❊❙❯▼❖ ✹

❆❇❙❚❘❆❈❚ ✺

✶✳ ■◆❚❘❖❉❯➬➹❖ ✼

✷✳ ❈❖◆❈❊■❚❖❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✶✵

✷✳✶✳ ❈❖◆❏❯●❆➬➹❖ ✶✵

✷✳✷✳ ❘❊▲❆➬➹❖ ❊◆❚❘❊ ❈❖▼❯❚❆❚■❱■❉❆❉❊ ❊ ❈❖◆❏❯●❆➬➹❖ ✶✵

✷✳✸✳ ❆➬➹❖ ❉❊ ●❘❯P❖❙ ✶✶

✷✳✹✳ ❆➬➹❖ P❖❘ ❆❯❚❖▼❖❘❋■❙▼❖❙ ✶✸

✷✳✺✳ ❆➬➹❖ P❖❘ ❈❖◆❏❯●❆➬➹❖ ✶✹

✷✳✻✳ ❈▲❆❙❙❊❙ ❉❊ ❈❖◆❏❯●❆➬➹❖ ❊▼ ❯▼ ❙❯❇●❘❯P❖ ✶✽

✷✳✼✳ ❙❯❇●❘❯P❖ ❈❖▼❯❚❆❉❖❘ ❊ ❙❯❇●❘❯P❖ ❉❊❘■❱❆❉❖ ✷✸

✸✳ ❆ ❈❆▼■◆❍❖ ❉❖ ❚❊❖❘❊▼❆ ❉❊ ❇❊◆❉❊❘ ✷✺

✸✳✶✳ ●❘❯P❖❙ ◆■▲P❖❚❊◆❚❊❙ ❊ ❙❖▲Ú❱❊■❙ ✷✺

✸✳✷✳ ❖ ❙❯❇●❘❯P❖ ❉❊ ❋■❚❚■◆● ●❊◆❊❘❆▲■❩❆❉❖ ✸✶

✹✳ ❉❊▼❖◆❙❚❘❆➬➹❖ ❉❖ P❘■▼❊■❘❖ ❚❊❖❘❊▼❆ ✹✵

✺✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ▲■◆❊❆❘❊❙ ❉❊ ●❘❯P❖❙ ✹✺

✺✳✶✳ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ✹✺

✺✳✷✳ ❋❯◆➬Õ❊❙ ❉❊ ❈▲❆❙❙❊ ✺✻

✺✳✸✳ ❈❆❘➪❚❊❘ ❉❊ ❯▼❆ ❘❊P❘❊❙❊◆❚❆➬➹❖ ✺✼

✺✳✹✳ ❖ ❚❊❖❘❊▼❆ ❉❊ ❋❘❖❇❊◆■❯❙✲❙❈❍❯❘ ✻✾

✻✳ ❉❊▼❖◆❙❚❘❆➬➹❖ ❉❖ ❙❊●❯◆❉❖ ❚❊❖❘❊▼❆ ✼✺

✼✳ ❆P✃◆❉■❈❊ ✼✾

✼✳✶✳ P❘❖P❘■❊❉❆❉❊❙ ❉❖ ❚❘❆➬❖ ❉❊ ❯▼❆ ▼❆❚❘■❩ ✼✾

✼✳✷✳ ❈❆❘➪❚❊❘ ❉❊ P❊❘▼❯❚❆➬➹❖ ✽✶

✼✳✸✳ ❖ ❚❊❖❘❊▼❆ ❉❊ ❙❈❍❯❘✲❩❆❙❙❊◆❍❆❯❙ ✽✷

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✼

✶✳ ■◆❚❘❖❉❯➬➹❖

◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛✲s❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❝♦♠✉t❛r❡♠ q✉❛♥❞♦ sã♦ ❛❧❡❛t♦r✐❛♠❡♥t❡ s❡❧❡❝✐♦♥❛❞♦s ❡♠ ✉♠ ❣r✉♣♦ ✜♥✐t♦✳ ❆♥t❡s ❞❡ ❡♠❜❛r❝❛r ❡♠ ✉♠❛ ❜✉s❝❛ ❝❡♥tr❛❞❛ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ❛s♣❡❝t♦s q✉❛♥t✐t❛t✐✈♦s ✭q✉❡ ♥❛t✉r❛❧♠❡♥t❡ ❛❝♦♠♣❛♥❤❛♠ ❛ ♥♦çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✮✱ é ✐♥t❡r❡ss❛♥t❡ r❡✢❡t✐r s♦❜r❡ ♣♦t❡♥❝✐❛✐s ❢♦♥t❡s ❞❡ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ❝♦♠✉t❛t✐✈✐❞❛❞❡✳

❆♠♣❛r❛♥❞♦✲♥♦s ♥❛ ❞❡✜♥✐çã♦ ❜ás✐❝❛ ❞❡ q✉❡✱ ✧❞♦✐s ❡❧❡♠❡♥t♦s x, g ❞❡ ✉♠ ❣r✉♣♦ ❝♦♠✉t❛♠ s❡ xg ✏ gx✧✱ ❡ r❡❝♦rr❡♥❞♦ ❛ ❛❧❣✉♠❛s ❡q✉✐✈❛❧ê♥❝✐❛s ♠✉✐t♦ ❝♦♥❤❡❝✐❞❛s✱ ♣♦❞❡♠♦s ♦❜t❡r ❞✐❢❡r❡♥t❡s ♣❡rs♣❡❝t✐✈❛s s♦❜r❡ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ✧❝♦♠✉t❛r✧ ✿

xg ✏gx ô g✁1xg ✏x ô x✁1g✁1xg ✏1.

❈♦♥✈é♠ ♥ã♦ ♠❡♥♦s♣r❡③❛r ♦ ❝♦♥t❡ú❞♦ s✉❜❥❛❝❡♥t❡ ❛ r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s tã♦ ❞❡s✲ ♣r❡t❡♥s✐♦s❛s✳ ❉❡ ❢♦r♠❛ s✐♠✐❧❛r ❛♦ q✉❡ ♦❝♦rr❡ ❡♠ ♦✉tr♦s ❝❛♠♣♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✉♠❛♥♦✱ ❛✈❛❧✐❛r ✐♥❞✐r❡t❛♠❡♥t❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ❡♠ ❝❛❞❛ ❝❡♥ár✐♦ ❡s♣❡❝í✜❝♦ ❞❡ ❡st✉❞♦✱ ♣♦❞❡ s❡r t❛r❡❢❛ ♠❛✐s s✐♠♣❧❡s ❞♦ q✉❡ ♦ ✉s♦ ❞✐r❡t♦ ❞❛ ❞❡✜♥✐çã♦✳ ■ss♦ ♥ã♦ só ♠♦t✐✈❛ ❛ ❡s❝♦❧❤❛ ❞❛ ♣❡rs♣❡❝t✐✈❛ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ❡♠ ❝❛❞❛ ❝♦♥t❡①t♦ ♠❛s t❛♠❜é♠ ✐♥s♣✐r❛ ❛ ❝r✐❛çã♦ ❞❡ ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s q✉❡ ♣❡r♠✐t❛♠ ✐♥❢❡r✐r s♦❜r❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝♦♠✉t❛r✳ ❉❡♥tr❡ ♦s ♦❜❥❡t♦s ♠❛✐s r❡❧❡✈❛♥t❡s✱ ♠❡r❡❝❡♠ ❞❡st❛q✉❡ ♦s ❝♦♠✉t❛❞♦r❡s ❡ ♦s ❛✉t♦♠♦r✜s♠♦s ✐♥t❡r♥♦s ✐♥❞✉③✐❞♦s ♣♦r ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦✳ ❉❛❞❛ ❛ ✐♠♣♦rtâ♥✲ ❝✐❛ ❞❡ss❡s ú❧t✐♠♦s✱ tr❛t❛r❡♠♦s ❢r❡qü❡♥t❡♠❡♥t❡ ❞❛ ❛♥á❧✐s❡ ❞❛ ❛çã♦ ♣♦r ❝♦♥❥✉❣❛çã♦✳ ❯♠ ❝❛s♦ ❧✐♠✐t❡ ✐♠♣♦rt❛♥t❡✱ ♥♦ q✉❡ ❝♦♥❝❡r♥❡ à ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ é ♦ ❞♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s✳ ❆❧é♠ ❞❡st❡s s❡r❡♠✱ ♣r❡❝✐s❛♠❡♥t❡✱ ♦s ❣r✉♣♦s ❝♦♠✉t❛t✐✈♦s✱ ♣♦ss✉❡♠ ❡str✉t✉r❛ ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞❛✳ ■ss♦ ❝♦♥❞✉③ ❛ ♣❡♥s❛r q✉❡✱ s❡ ❞✐s♣✉s❡r♠♦s ❞❡ ♠ét♦❞♦s q✉❡ ♣❡r♠✐t❛♠ ❞❡✜♥✐r q✉ã♦ ❧♦♥❣❡ ❞❡ s❡r ❛❜❡❧✐❛♥♦ ✉♠ ❣r✉♣♦ ❡stá✱ s❡rá ♣♦ssí✈❡❧ ♦❜t❡r ❞❛❞♦s s♦❜r❡ ❛ ❝♦♠✉t❛çã♦ ❡♠ ✉♠ ❣r✉♣♦ ♥ã♦✲❛❜❡❧✐❛♥♦✳

◆❡ss❛ ❧✐♥❤❛ ❞❡ ❛♥á❧✐s❡✱ ❞❡✈♦t❛r❡♠♦s ❡s♣❡❝✐❛❧ ❛t❡♥çã♦ ❛♦s q✉♦❝✐❡♥t❡s ❞❡ ✉♠ ❣r✉♣♦ G ♣♦r ❞♦✐s ❞❡ s❡✉s s✉❜❣r✉♣♦s ❝❛r❛❝t❡ríst✐❝♦s ♠❛✐s ❛♠♣❧❛♠❡♥t❡ ❝♦♥❤❡❝✐❞♦s✿

✌ ♦ ❝❡♥tr♦✱Z♣Gq ❀

✌ ♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦✱ G✶✳

P♦r s❡r ✐s♦♠♦r❢♦ ❛ Inn♣Gq ✱ ❣r✉♣♦ ❞♦s ❛✉t♦♠♦r✜s♠♦s ✐♥t❡r♥♦s ❞❡ G✱ ♦ q✉♦❝✐✲ ❡♥t❡G④Z♣Gq ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çõ❡s ❞✐r❡t❛s s♦❜r❡ ❝♦♥❥✉❣❛çã♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱G④G✶

❢♦r♥❡❝❡ ❛ r❡❢❡rê♥❝✐❛ ❞❡ s❡r ♦ ♠❛✐♦r q✉♦❝✐❡♥t❡ ❛❜❡❧✐❛♥♦ ❛ss♦❝✐❛❞♦ ❛ G✳

❊ss❡s rá♣✐❞♦s ❡①❡♠♣❧♦s ✐❧✉str❛♠ ♦ ✉s♦ ❞❡ ❛tr✐❜✉t♦s ✐♥❞✐r❡t♦s ♣❛r❛ ❝♦♥❝❧✉✐r s♦❜r❡ ❛ ❝♦♠✉t❛çã♦ ❡ t❡rã♦ ♣r❡s❡♥ç❛ ♠❛r❝❛♥t❡ ♥♦ tr❛❜❛❧❤♦ ❛ s❡r ❞❡s❡♥✈♦❧✈✐❞♦✱ ♦ q✉❛❧ ❝✉❧♠✐♥❛rá ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❞♦✐s t❡♦r❡♠❛s q✉❡ ❡st❛❜❡❧❡❝❡♠ ❧✐♠✐t❡s ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r✳

(8)

✽

❙❡❥❛G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠ n✳ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ G s❡❧❡❝✐♦♥❛❞♦s ❛❧❡❛t♦r✐❛♠❡♥t❡ ✭❡ ❝♦♠ r❡♣♦s✐çã♦✮ ❝♦♠✉t❡♠ é ❞❡✜♥✐❞❛ ❛tr❛✈és ❞❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿

cp♣Gq ✏ ⑤t♣x, yq PG✂G:xy✏yx✉⑤

⑤G✂G⑤ .

❈✉♠♣r❡ s❛❧✐❡♥t❛r q✉❡✱ ❡♠ ✉♠❛ t❛❧ ❞❡✜♥✐çã♦✱ ❛ss✉♠❡✲s❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✉♥✐❢♦r♠❡ ♥♦ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ G✂G✳

▲❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥t❛ q✉❡✱ ❡♠ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ♣♦r s✉❛ ♣ró♣r✐❛ ❡ssê♥❝✐❛✱ ✈❛❧❡ cp♣Gq ✏ 1✱ ♦ ❝❛s♦ ❞❡ ✐♥t❡r❡ss❡ ♣rát✐❝♦ é ❥✉st❛♠❡♥t❡ ♦ ❞♦s ❣r✉♣♦s ♥ã♦✲❛❜❡❧✐❛♥♦s

✭cp♣Gq ➔1✮✳ ❈♦♥✈é♠ ❛✐♥❞❛ ♦❜s❡r✈❛r q✉❡ ♥ã♦ ❡①✐st❡ ♦ ♦✉tr♦ ❡①tr❡♠♦ ❞♦ ❡s♣❡❝tr♦✱

✐st♦ é✱ q✉❡ ♥ã♦ ♦❝♦rr❡ ♦ ❝❛s♦ cp♣Gq ✏ 0✳ ■ss♦ ♣♦rq✉❡✱ ❡♠ q✉❛❧q✉❡r ❣r✉♣♦ G✱ ♦

❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡✱ 1G✱ ❝♦♠✉t❛ ❝♦♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ G✳

❯♠ ❢❛t♦ q✉❡ ❝❤❛♠❛ ❜❛st❛♥t❡ ❛ ❛t❡♥çã♦ é q✉❡ ♦ ✈❛❧♦r ♠á①✐♠♦ q✉❡ ♣♦❞❡ s❡r ❛ss✉♠✐❞♦ ♣♦r cp♣Gq ❡♠ ❣r✉♣♦s ♥ã♦✲❛❜❡❧✐❛♥♦s é 5④8✱ ❝♦♥❢♦r♠❡ s❡rá ❞❡♠♦♥str❛❞♦

♥♦ t❡①t♦✳ ❊ss❡ ❝✉r✐♦s♦ r❡s✉❧t❛❞♦ ❢♦✐ ♦r✐❣✐♥❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✶✾✼✸ ✭❬✻❪✮ ❡ ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ♥♦ ❡st✉❞♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r✳

▼❡s♠♦ s❡♥❞♦ t❡❝♥✐❝❛♠❡♥t❡ s✐♠♣❧❡s ❡✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ ✐♥t✉✐t✐✈❛✱ ❡ss❛ ❞❡✜♥✐çã♦ ❞❡ cp♣Gq ♥ã♦ é ♠✉✐t♦ ❝♦♥✈❡♥✐❡♥t❡ s♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ♠❛♥✐♣✉❧❛çã♦ ♠❛t❡✲ ♠át✐❝❛✳ ■ss♦ ♥❛t✉r❛❧♠❡♥t❡ ♠♦t✐✈❛ ❛ ❜✉s❝❛ ❞❡ ❢♦r♠❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❝❛❧❝✉❧❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ s❡♥❞♦ ❛ ♠❛✐s ♥♦tá✈❡❧ ❛ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r✳

❙❡ ❞❡♥♦t❛r♠♦s ♣♦rk♣Gq ♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ♥♦ ❣r✉♣♦G✱ ✈❡r❡♠♦s q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r é ❞❛❞❛ ♣♦r✿

cp♣Gq ✏ k♣Gq ⑤G⑤ .

❯♠ ❛s♣❡❝t♦ ✐♥t❡r❡ss❛♥t❡ ❛ss♦❝✐❛❞♦ ❛ ❡ss❛ ♥♦✈❛ ❢♦r♠✉❧❛çã♦ é q✉❡ ❡❧❛ ❡✈✐❞❡♥❝✐❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ q✉❡ ❡st❛♠♦s ❡st✉❞❛♥❞♦ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❡ ✐♥❢♦r♠❛çõ❡s ♣✉r❛♠❡♥t❡ ❛❧❣é❜r✐❝❛s✿ ❛ ♦r❞❡♠ ❞♦ ❣r✉♣♦ ❡ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦✳ P♦r ❡st❛r♠♦s tr❛t❛♥❞♦ ❝♦♠ ❣r✉♣♦s ✜♥✐t♦s✱ ♦ ❞❡s❛✜♦ ♠❛✐♦r r❡s✐❞❡ ❡♠ ♦❜t❡r ♦ ✈❛❧♦r ❞❡ k♣Gq ✱ r❛③ã♦ ♣❡❧❛ q✉❛❧ ♠✉✐t♦s tr❛❜❛❧❤♦s s❡ ❞❡❞✐❝❛r❛♠ ❛ ♣❡sq✉✐s❛r t❛❧ ✐♥✈❛r✐❛♥t❡✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❬✸❪✱ ❬✶✸❪✱ ❬✷❪ ❡✱ ♠❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ❬✹❪✳

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✾

❆♣❡s❛r ❞❡ ❡st❛r❡♠ ❞✐s♣♦♥í✈❡✐s ✈ár✐❛s té❝♥✐❝❛s q✉❡ s❡ ♣r♦♣õ❡♠ ❛ ❝❛❧❝✉❧❛r cp♣Gq✱ ❝♦♥❝❡♥tr❛r❡♠♦s ♦s ❡s❢♦rç♦s ♥❛ ❛♥á❧✐s❡ ❞❡ ❞✉❛s ❞❡❧❛s✳ ❉❡ ♠♦❞♦ ❛ ❞❡♠♦♥str❛r ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s✱ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❚❡♦r❡♠❛s 9 ❡ 14 ❝♦♥st❛♥t❡s ❡♠ ❬✺❪✱

s❡rã♦ ❛❞♦t❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s s❡❣✉✐♥t❡s ❛❜♦r❞❛❣❡♥s✿

✶✮ ❆♥á❧✐s❡ ❞❛ ❝♦♥❥✉❣❛çã♦ ❡♠ ✉♠ ❣r✉♣♦ ✭❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s ❛ss♦❝✐❛❞♦s✮❀ ✷✮ ❚❡♦r✐❛ ❞❡ ❘❡♣r❡s❡♥t❛çõ❡s ▲✐♥❡❛r❡s ❡ ❈❛r❛❝t❡r❡s✳

P❛ss❡♠♦s✱ ❛❣♦r❛✱ ❛♦s ❡♥✉♥❝✐❛❞♦s ❞♦s ❞♦✐s t❡♦r❡♠❛s✳ ❚❡♦r❡♠❛ ✹✳✶ ✭❬✺❪ ✲ ❚❡♦r❡♠❛ ✾✮✳

❙❡♥❞♦ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ sol♣Gq ♦ s❡✉ r❛❞✐❝❛❧ s♦❧ú✈❡❧✱ ✈❛❧❡✿

cp♣Gq ↕ rG:sol♣Gqs✁1④2✳

❚❡♦r❡♠❛ ✻✳✶ ✭❬✺❪ ✲ ❚❡♦r❡♠❛ ✶✹✮✳

✭❛✮ P❛r❛ q✉❛❧q✉❡r ❣r✉♣♦ ✜♥✐t♦ G é ✈á❧✐❞❛ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡✿

cp♣Gq ➙ ✁ ₁

⑤G⑤☎ ➳

χPIrr♣Gq χ♣1q

✠2 ✳

✭❜✮ ❙❡ G é ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠ ♣❛r ❡ ❝♦♠ ❝❡♥tr♦ tr✐✈✐❛❧✱ ❡♥tã♦✿ ⑤G⑤ ➔ ⑤CG♣xq⑤

3

✱ ♣❛r❛ ❛❧❣✉♠xPG✁ t1✉✳

❖ ❚❡♦r❡♠❛ ✹✳✶ ❢♦r♥❡❝❡ ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r ❡✱ ♥❛ ♣❛✈✐♠❡♥t❛çã♦ ❞♦ ❝❛♠✐♥❤♦ ♣❛r❛ ❞❡♠♦♥strá✲❧♦✱ ❡st✉❞❛r❡♠♦s ✈ár✐♦s ❝♦♥❝❡✐t♦s ❛✉①✐❧✐❛r❡s✳ ❉❡♥tr❡ ❡❧❡s ♣♦❞❡♠♦s ❧✐st❛r ♦s ❞❡ ❣r✉♣♦s q✉❛s✐s✐♠♣❧❡✱ s❡♠✐✲ s✐♠♣❧❡s✱ ❛❧♠♦st s✐♠♣❧❡ ❡ ♦ ❞❡ ✉♠ s✉❜❣r✉♣♦ ❝❛r❛❝t❡ríst✐❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❋✐tt✐♥❣ ●❡♥❡r❛❧✐③❛❞♦✱F ✝♣Gq✳ Pr♦✈❛r❡♠♦s t❛♠❜é♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❇❡♥❞❡r✱ ♦ q✉❛❧ r❡❧❛❝✐♦♥❛ F ✝♣Gq ❝♦♠ ♦ s✉❜❣r✉♣♦ ❞❡ ❋✐tt✐♥❣✱ F♣Gq ✱ ❡ ❝♦♠ ♦✉tr♦ s✉❜❣r✉♣♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ♦ ❧❛②❡r ❞❡ G ❡ ❞❡♥♦t❛❞♦ ♣♦rE♣Gq ✳

❏á ♦ ❚❡♦r❡♠❛ ✻✳✶ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ♣❛r❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r✱ ✉s❛♥❞♦ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ❣r❛✉ ❞♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡ ✉♠ ❣r✉♣♦✳ ❚♦❞♦ ♦ s✉♣♦rt❡ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ s❡ ♣♦ss❛ ❞❡♠♦♥strá✲❧♦ é ♣r♦✈✐❞♦ ♥♦ t❡①t♦✱ ♥✉♠❛ s❡çã♦ q✉❡ tr❛t❛ ❞❡ ❘❡♣r❡s❡♥t❛çõ❡s ▲✐♥❡❛r❡s✳

➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡✱ ❛♣❡s❛r ❞❡ ✉t✐❧✐③❛r ❝♦♥❝❡✐t♦s r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❡♠ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ s❡ ❛♣ó✐❛ ❡♠ ❞♦✐s r❡s✉❧t❛❞♦s ✐♥t❡r♠❡❞✐ár✐♦s ✭❬✷❪ ❡ ❬✶✸❪✮✱ q✉❡ r❡❝♦rr❡♠ à ❈❧❛ss✐✜❝❛çã♦ ❞♦s ●r✉♣♦s ❙✐♠♣❧❡s ❋✐♥✐t♦s ✭❈❋❙●✮✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❞❡s♣❡✐t♦ ❞❡ s❡✉ ❡♥✉♥❝✐❛❞♦ ♠❛✐s ❝♦♠♣❛❝t♦✱ ♦ ♣r✐♠❡✐r♦ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✧♠❡♥♦s ❡❧❡♠❡♥t❛r✧ q✉❡ ♦ s❡❣✉♥❞♦✳

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✶✵

✷✳ ❈❖◆❈❊■❚❖❙ ❋❯◆❉❆▼❊◆❚❆■❙

✷✳✶✳ ❈❖◆❏❯●❆➬➹❖✳

❉✐③❡♠♦s q✉❡ ❞♦✐s ❡❧❡♠❡♥t♦s x ❡y ❞❡ ✉♠ ❣r✉♣♦ Gsã♦ ❝♦♥❥✉❣❛❞♦s s❡ ❡①✐st❡ g PG t❛❧ q✉❡ y ✏ g✁1xg✳ ◗✉❛♥❞♦ ❡①✐st❡ ✉♠ t❛❧ g✱ é ❝♦♠✉♠ ❞✐③❡r q✉❡ y é ❝♦♥❥✉❣❛❞♦ ❛ x ♣♦r g ❡ ✉s❛r ❛ ♥♦t❛çã♦ y ✏ xg✳ ❆ r❡❧❛çã♦ ❜✐♥ár✐❛ ❡♠ q✉❡ ✧x ✒ y✧ s✐❣♥✐✜❝❛ q✉❡ ✧❡①✐st❡ g P G t❛❧ q✉❡ y ✏ g✁1xg✧ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❆ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ α P G ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ t❛❧ r❡❧❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ α ❡ ❞❡♥♦t❛❞❛ ♣♦rαG ♦✉ Cl♣αq✳

❆ ❝♦♥❥✉❣❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ✜①♦ g P G é t❛♠❜é♠ ❝♦♠✉♠❡♥t❡ r❡❢❡r❡♥❝✐❛❞❛ ❝♦♠♦ ♦ ❛✉t♦♠♦r✜s♠♦ ✐♥t❡r♥♦ ✐♥❞✉③✐❞♦ ♣♦r g✿

θg :GÑG

xÞÑθg♣xq ✏ xg ✏g✁ 1

xg✳

❯♠❛ ♦❜s❡r✈❛çã♦ ❜ás✐❝❛ é q✉❡ s❡x❡g ❝♦♠✉t❛♠✱ ❡♥tã♦x✏xg✳ ❉❡ss❛ ❢♦r♠❛✱ ❡♠ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ♦ ú♥✐❝♦ ❛✉t♦♠♦r✜s♠♦ ✐♥t❡r♥♦ é ❛ ✐❞❡♥t✐❞❛❞❡ ✭Inn♣Gq ❂ ✶✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡Gé ♥ã♦✲❛❜❡❧✐❛♥♦✱ ❞❡✈❡♠ ❡①✐st✐r ❛✉t♦♠♦r✜♠♦s ✐♥t❡r♥♦s ❞✐st✐♥t♦s ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡✱ ♥❡ss❡ ❝❛s♦✱ ♦ ❡st✉❞♦ ❞❛ ❝♦♥❥✉❣❛çã♦ ♣♦❞❡ s❡ ♠♦str❛r ❢r✉tí❢❡r♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❝♦♠✉t❛t✐✈✐❞❛❞❡✳

✷✳✷✳ ❘❊▲❆➬➹❖ ❊◆❚❘❊ ❈❖▼❯❚❆❚■❱■❉❆❉❊ ❊ ❈❖◆❏❯●❆➬➹❖✳ ❙❡❥❛g ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❡ ✉♠ ❣r✉♣♦ G✳ ❉❡✜♥✐♠♦s ♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡ g ❡♠ G❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s x ❞♦ ❣r✉♣♦Gq✉❡ ❝♦♠✉t❛♠ ❝♦♠ ✉♠ ❞❛❞♦ g PG✳

CG♣gq ✏ txPG⑤xg ✏gx✉ ✏ txPG⑤g✁ 1

xg ✏x✉ ✏ txP G⑤xg ✏x✉ ✳

❖s ❡❧❡♠❡♥t♦sxq✉❡ ❝❡♥tr❛❧✐③❛♠ ✉♠ ❞❛❞♦g P G❣❡r❛♠ ✉♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ ✉♥✐tár✐❛ ♦✉ s✐♥❣❧❡t♦♥ ✭xg _✏_{x✮✳ ❉❡ss❛ ❢♦r♠❛✱ t♦❞♦}_x_P _G_{t❛❧ q✉❡}_xg _✏_gx_{♣❡r♠❛♥❡❝❡} ✜①♦ q✉❛♥❞♦ s❡ ♣r♦♠♦✈❡ ❛ ❝♦♥❥✉❣❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❞❡G✳ ➱ ✐♠♣♦rt❛♥t❡ ❛✐♥❞❛ ♥♦t❛r q✉❡ ♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡ ✉♠ ❡❧❡♠❡♥t♦g PGé ✉♠ s✉❜❣r✉♣♦ ❞❡G✳ ✭CG♣gq ↕G✮ ❊♠ r❡s✉♠♦✱ s❡ ❡st✐✈❡r♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❞❡t❡r♠✐♥❛r ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦G q✉❡ ❝♦♠✉t❛♠ ❝♦♠ ✉♠ ❝❡rt♦g PG✱ ♣♦❞❡♠♦s ❛❞♦t❛r ❛s s❡❣✉✐♥t❡s ❛❜♦r❞❛❣❡♥s✿

✭✶✮ t❡st❛r ❝❛❞❛ xPG ❡ ✈❡r✐✜❝❛r s❡ ♦❝♦rr❡ xg ✏gx❀

✭✷✮ ❞❡✐①❛r g P ● ❛❣✐r ♣♦r ❝♦♥❥✉❣❛çã♦ s♦❜r❡ ❝❛❞❛ x P G ❡ ✐❞❡♥t✐✜❝❛r t♦❞♦s ❛q✉❡❧❡s ❡❧❡♠❡♥t♦s q✉❡ ♣❡r♠❛♥❡❝❡♠ ✜①♦s✳

(11)

✶✶

❉❛❞♦ ✉♠ ❣r✉♣♦ G✱ s❡❥❛X ❸G ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛r❜✐trár✐♦✳ ❉❡✜♥✐♠♦s✿

✌ ♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡X ❡♠ G é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡Gq✉❡ ❝♦♠✉t❛♠ ❝♦♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡X✱ ✐st♦ é✱CG♣Xq ❂ tg PG⑤xg ✏gx, ❅xP X ✉✳

❖❜s❡r✈❛♥❞♦ q✉❡ CG♣Xq ❂ ↔

x_PX

CG♣xq✱ s❡❣✉❡CG♣Xq ↕ G✳

✌ ♦ ❝❡♥tr♦ ❞♦ ❣r✉♣♦ G é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s yP G q✉❡ ❝♦♠✉t❛♠ ❝♦♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡G✳ ❆ss✐♠✱Z♣Gq ✏CG♣Gq ✏ tyPG⑤xy✏yx, ❅xP G✉ ❡ ❞❡❝♦rr❡ q✉❡G é ❛❜❡❧✐❛♥♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Z♣Gq ✏G✳

✌ ♦ ♥♦r♠❛❧✐③❛❞♦r ❞❡ X ❡♠ G é ♦ s✉❜❣r✉♣♦ NG♣Xq ❂ t g P G ⑤ Xg ✏X ✉✳ ❉❛❞♦ ✉♠ s✉❜❣r✉♣♦H ↕G✱ t❡♠✲s❡H ➐NG♣Hq✳

❈♦♠♣✐❧❛♠♦s✱ ♥❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r✱ ✉♠❛ sér✐❡ ❞❡ ❢❛t♦s s♦❜r❡ ❝❡♥tr❛❧✐③❛❞♦r❡s ❡ ♥♦r♠❛❧✐③❛❞♦r❡s✱ q✉❡ s❡rã♦ ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ❚❛✐s r❡s✉❧t❛✲ ❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ✈ár✐❛s r❡❢❡rê♥❝✐❛s ❝❧áss✐❝❛s ❞❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❡ s❡rã♦ ❛ss✉♠✐❞♦s s❡♠ ❞❡♠♦♥str❛çã♦✳

Pr♦♣♦s✐çã♦ ✷✳✶ ✭Pr♦♣r✐❡❞❛❞❡s ❇ás✐❝❛s ❞♦ ❈❡♥tr❛❧✐③❛❞♦r ❡ ❞♦ ◆♦r♠❛❧✐③❛❞♦r✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭❛✮ ❙❡ X, Y sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ G t❛✐s q✉❡ X ❸Y✱ ❡♥tã♦ CG♣Yq ❸CG♣Xq ❀ ✭❜✮ ❙❡ H ↕G ❡ g PG✱ ❡♥tã♦ CH♣gq ✏ CG♣gq ❳H❀

✭❝✮ ❙❡♥❞♦ H ↕J ↕G✱ ✈❛❧❡ NJ♣Hq ✏ NG♣Hq ❳J❀

✭❞✮ P❛r❛ q✉❛❧q✉❡r s✉❜❣r✉♣♦ H ↕G✱ ✈❛❧❡ CG♣Hq ➐ NG♣Hq ❀

✭❡✮ ❖ q✉♦❝✐❡♥t❡ NG♣Hq④CG♣Hq é ✐s♦♠♦r❢♦ ❛ ✉♠ s✉❜❣r✉♣♦ ❞❡ ❆✉t✭H✮✳

✷✳✸✳ ❆➬➹❖ ❉❊ ●❘❯P❖❙✳

❖✉tr❛ ♠❛♥❡✐r❛ ❜❛st❛♥t❡ ❢r❡qü❡♥t❡ ❞❡ tr❛t❛r ❛ ❝♦♥❥✉❣❛çã♦ ❢❛③ ✉s♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ G ❛❣❡ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ Ω ✭♦✉ q✉❡ G ♣❡r♠✉t❛ ♦s ♣♦♥t♦s ❞❡ Ω✮ s❡✱ ❛ ❝❛❞❛ g P G ❡ ❛ ❝❛❞❛ α PΩ✱ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦αg ❞❡Ω ✭✐♠❛❣❡♠ ❞❛ ❛çã♦ ❞❡ g s♦❜r❡ α✮ t❛❧ q✉❡✿

✭✐✮ α1 ✏α,❅αPΩ❀

✭✐✐✮ ♣αg_qh _✏

αgh,❅α PΩ❡ ❅g, hPG✳

❉❡ss❛ ❢♦r♠❛✱ ✉♠❛ ❛çã♦ ❞❡ G s♦❜r❡ Ωé ✉♠❛ ❢✉♥çã♦✿

µ: Ω✂GÑΩ

(12)

✶✷

◆♦t❛çã♦✿ ◆❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ♦♣t❛♠♦s ♣♦r ❡s♣❡❝✐✜❝❛r ✉♠❛ ❛çã♦ à ❞✐r❡✐t❛ ✳ ❚❛❧ ❝♦♥✈❡♥çã♦ s❡rá ♠❛♥t✐❞❛ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ t❡①t♦✳

❈♦♠❡ç❛r❡♠♦s ❛ ❛♥á❧✐s❡✱ r❡✈✐s❛♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ór❜✐t❛s ❡ ❡st❛❜✐❧✐③❛❞♦r❡s ❞❡ ✉♠❛ ❛çã♦ ❡ ❡st❛❜❡❧❡❝❡♥❞♦ ❛❧❣✉♠❛s r❡❧❛çõ❡s ✐♠♣♦rt❛♥t❡s ❡♥tr❡ ❡❧❡s✿

✌ ór❜✐t❛✿ s✉❜❝♦♥❥✉♥t♦ ❞❡ Ω q✉❡ ❝♦♥té♠ ♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ❞❛ ❛çã♦ ❞❡

❝❛❞❛ g P Gs♦❜r❡ ♦ ❡❧❡♠❡♥t♦ α PΩ✳

O♣αq ✏ αG ✏ tαg ⑤g P G✉ ❸ Ω✳

✌ ❡st❛❜✐❧✐③❛❞♦r ❞❡ ✉♠ ♣♦♥t♦✿ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ G q✉❡ ✜①❛♠ ✉♠ ❞❛❞♦ α PΩ✳ ✭❚❛❧ ❝♦♥❥✉♥t♦ é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✮✳

Gα ✏StabG♣αq ✏ tg PG⑤αg ✏α✉ ↕G✳

❆♥t❡s ❞❡ ✐♥✐❝✐❛r ❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✷✱ r❡✈✐s✐t❛r❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ Ór❜✐t❛✲❡st❛❜✐❧✐③❛❞♦r✳

❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❛❣✐♥❞♦ s♦❜r❡ Ω ❡ O ✉♠❛ ór❜✐t❛ ❞❡ss❛ ❛çã♦✳ ❙❡❥❛ α P O ❡

❞❡♥♦t❡♠♦s H ✏Gα✱ ♦ ❡st❛❜✐❧✐③❛❞♦r ❞❡ α ❡♠ G✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦✿

OØ t Hx⑤xP G✉✳

❙❡❥❛♠ f :O Ñ tHx⑤xPG✉ ✉♠❛ ❢✉♥çã♦ ❡ β ✉♠ ❡❧❡♠❡♥t♦ ❞❡ O✳

❚♦♠❡♠♦sxPG t❛❧ q✉❡ β ✏αx ❡ ❢❛ç❛♠♦sf♣βq ✏Hx✳ ✭■✮f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s✿

αx ✏αy ñ ♣αxqy✁1 ✏ ♣αyqy✁1 ✏α1 ✏α ✻ xy✁1 PH ✏Gα ✻ xy✁ 1 _✏

hPH

✻ x✏hy, hP H ✻ Hx✏Hy✳

✭■■✮f é s♦❜r❡❥❡t✐✈❛✱ ♣♦✐s✱ ❞❛❞❛ ✉♠❛ ❝❧❛ss❡Hx✱ ❡①✐st❡xPGt❛❧ q✉❡f♣αxq ✏Hx✳ ✭■■■✮f é t❛♠❜é♠ ✐♥❥❡t✐✈❛✳

❆❞♠✐t❛♠♦sf♣βq ✏f♣γq✳ ❊♥tã♦ β✏αx ❡ γ ✏αy✱ ❝♦♠ Hx✏Hy✳ ▼❛s ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ y✏hx♣❛r❛ ❛❧❣✉♠ hPH ✏Gα ❡✱ ♣♦rt❛♥t♦✿

γ ✏αy ✏αhx ✏ ♣αhqx ✏αx ✏β✳

❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛ ❞✐r❡t❛ ❞❡ f s❡r ❜✐❥❡t✐✈❛✱ ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❛ ór❜✐t❛ ❞❡ α ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❧❛t❡r❛✐s ❞♦ ❡st❛❜✐❧✐③❛❞♦rGα✱ ♦✉ s❡❥❛✿

|O|✏ rG:Gαs ✏

|G| |Gα|

✭❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡ ✲ ❋❈P✮✳

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✶✸

Pr♦♣♦s✐çã♦ ✷✳✷✳ ❉❛❞♦s ✉♠ ❣r✉♣♦ G ❡ ✉♠ ❝♦♥❥✉♥t♦ Ω✱ ❞❡♥♦t❡♠♦s ♣♦r µ ✉♠❛ ❛çã♦ ❞❡ G s♦❜r❡ Ω✿

µ: Ω✂GÑΩ

♣ω, xq ÞÑ ωx✳

❙❡ α, β P Ω ❡stã♦ ♥❛ ♠❡s♠❛ ór❜✐t❛✱ ❡♥tã♦✿

✭❛✮ rG:Gαs ✏ rG:Gβs ❀

✭❜✮ ❊①✐st❡xPGt❛❧ q✉❡ Gβ ✏ ♣Gαqx✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡Gα ❡Gβ sã♦ s✉❜❣r✉♣♦s ❝♦♥❥✉❣❛❞♦s ❞❡ G❀

✭❝✮ ❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠✲s❡ Gαx ✏ ♣G_αqx✳

❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❉❛❞♦ q✉❡ α, β ❡stã♦ ♥❛ ♠❡s♠❛ ór❜✐t❛✱ ✈❛❧❡ O♣αq ✏ O♣βq✳ ❯s❛♥❞♦ ♦ ❢❛t♦ q✉❡|O♣αq| ❂ ❬G:Gα❪ ❡|O♣βq| ❂ ❬G:Gβ❪✱ s❡❣✉❡ ❛ t❡s❡✳

✭❜✮ ▲❡♠❜r❛♥❞♦ q✉❡ β ❡stá ❡♠ O♣αq✱ ❡①✐st❡ x P G t❛❧ q✉❡ β ✏ αx✳ ❚♦♠❛♥❞♦ g PGα✱ t❡♠✲s❡ αg ✏α✳ ❆❧é♠ ❞✐ss♦✱ gx P ♣Gαqx✱ ♦✉ s❡❥❛✱ x✁

1

gxP ♣Gαqx✳ ❘❡s✉❧t❛✿

βx✁1gx✏ ♣αxqx✁1gx ❂αxx✁1gx ✏αgx✏αx ✏β✳

▼❛s ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ x✁1gxP Gβ✱ ✐st♦ é✱ ♣Gαqx ❸Gβ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ ❛ ✐♥❝❧✉sã♦ r❡✈❡rs❛ ❡ ♦❜té♠✲s❡ ❛ t❡s❡✳

✭❝✮ P♦r ❞❡✜♥✐çã♦ ❞❡ ór❜✐t❛✱ t❡♠✲s❡αx PO♣αq,❅xP G✳ ❉❡ ✭❜✮✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

✷✳✹✳ ❆➬➹❖ P❖❘ ❆❯❚❖▼❖❘❋■❙▼❖❙✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛♠ ❞♦✐s ❣r✉♣♦s N ❡ H t❛✐s q✉❡ H ❛❣❡ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ N✳ ❉✐③❡♠♦s q✉❡ H ❛❣❡ ♣♦r ❛✉t♦♠♦r✜s♠♦s s♦❜r❡ N s❡✱ ❅x, y P N ❡ ❅hPH✱ ✈❛❧❡ ♣xyqh _✏_xh_yh_✳

❆ss✐♠✱ ❛ ❢✉♥çã♦ θh✱ ✐♥❞✉③✐❞❛ ♣♦r ❝❛❞❛ hPH✱ é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡N✳

θh :N ÑN

nÞÑθh♣nq ✏nh ✱

P❛r❛ ✈❡r✐✜❝❛r t❛❧ ❛✜r♠❛çã♦✱ ❞❡✈❡✲s❡ ♦❜s❡r✈❛r ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ ❛çã♦ ❞❡ H s♦❜r❡ N é✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ ❛çã♦ ❞♦ ❣r✉♣♦ H s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ N ❡✱ ♣♦rt❛♥t♦✱ θh é ✉♠❛ ♣❡r♠✉t❛çã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ N✳ ❆ss✐♠✱ ❞❛❞♦s x, y PN✱ t❡♠✲s❡✿

θh♣xyq ✏ ♣xyqh ✏xh☎yh ✏θh♣xq ☎θh♣yq✳

(14)

✶✹

✷✳✺✳ ❆➬➹❖ P❖❘ ❈❖◆❏❯●❆➬➹❖✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❉❛❞♦ ✉♠ ❣r✉♣♦ G✱ ❞❡✜♥✐♠♦s ❛ ❛çã♦ ♣♦r ❝♦♥❥✉❣❛çã♦ ✭❞❡ G s♦❜r❡ G✮ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

µ:G✂GÑG

♣α, gq ÞÑαg ✏g✁1αg✳

❘❡✉♥✐♠♦s✱ ❛ s❡❣✉✐r✱ ♥ã♦ só ❛ t❡r♠✐♥♦❧♦❣✐❛ tí♣✐❝❛ ♠❛s t❛♠❜é♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❛ss♦❝✐❛❞❛s à ❛çã♦ ♣♦r ❝♦♥❥✉❣❛çã♦✿

✌ ❉♦✐s ❡❧❡♠❡♥t♦s ❝♦♥❥✉❣❛❞♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♦r❞❡♠❀

✌ ❆ ❛çã♦ ♣♦r ❝♦♥❥✉❣❛çã♦ ✭❞❡G s♦❜r❡G✮ é ✉♠❛ ❛çã♦ ♣♦r ❛✉t♦♠♦r✜♠♦s✳ ■ss♦ t❛♠❜é♠ ❛❝♦♥t❡❝❡ ❝♦♠ ✉♠❛ ❛çã♦ ❞❡ Gs♦❜r❡ ✉♠ s✉❜❣r✉♣♦ N ➐G❀

✌ O♣αq ✏ αG ✏ tαg ⑤g P G✉ ✏ t g✁1αg⑤g PG✉ ✏Cl♣αq ❀ ✌ Gα ✏StabG♣αq ✏ t g PG⑤αg ✏α✉ ✏CG♣αq ❀

✌ G é ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞❡ s✉❛s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦❀

✌ ❖ ♥ú♠❡r♦ ❞❡ ❝♦♥❥✉❣❛❞♦s ❞❡ ✉♠ ❞❛❞♦ ❡❧❡♠❡♥t♦ α ❡♠ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G é ♦ í♥❞✐❝❡ ❞♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡ α ❡♠ G✱ ♦✉ s❡❥❛✱ ⑤Cl♣αq⑤ ✏ rG:CG♣αqs ❀ ✌ ❉❛❞♦ q✉❡ CG♣αq é ✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ t❡♠✲s❡|G| ✏|CG♣αq|☎ ❬G : CG♣αq❪

❡✱ ♣♦rt❛♥t♦✱|G| ❂|CG♣αq|☎ ⑤Cl♣αq⑤✳ ❉❛í r❡s✉❧t❛ q✉❡ t♦❞♦s ♦s t❛♠❛♥❤♦s ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞✐✈✐❞❡♠|G|❀

✌|G| ✏|Z♣Gq|

k ➳

i✏l 1

❬G : CG♣xiq❪ ✱ ❡♠ q✉❡ xl 1, . . . , xk sã♦ r❡♣r❡s❡♥t❛♥t❡s ❞❛s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ é ♠❛✐♦r q✉❡ 1✳

❉❛❞♦ α P G✱ ♦ ❝♦♥❥✉♥t♦ tα✉ é ✉♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ α P Z♣Gq✱ ♣♦✐s✿ α P Z♣Gq ô αg ✏ gα,❅g P G ô g✁1αg ✏ α,❅g P G ô αg ✏ α,❅g PGôCl♣αq ✏ tα✉ ✳

❱❛❧❡ ♦❜s❡r✈❛r q✉❡✱ ♣❛r❛ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ t❡♠✲s❡ Z♣Gq ✏ G✱ ❞♦♥❞❡ s❡❣✉❡ k♣Gq ✏|G|✳ ■ss♦ ❡✈✐❞❡♥❝✐❛ q✉❡ ♦ ❡st✉❞♦ ❞❛s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦✱ ♥❡ss❡ ❝❛s♦✱ ♥ã♦

tr❛③ q✉❛❧q✉❡r ✐♥❢♦r♠❛çã♦ r❡❧❡✈❛♥t❡✳

❉❡♥tr❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❧✐st❛❞❛s ❛❝✐♠❛✱ ♠❡r❡❝❡ ❡s♣❡❝✐❛❧ ❛t❡♥çã♦ ❛ q✉❡ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ♥✉♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦✳

❙❡♥❞♦Cl♣xq ✏ txg ⑤xPG✉ ❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡xPG✱ ♠♦str❛r❡♠♦s q✉❡✿ ✌ ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♠❡s♠❛ ❝❧❛ss❡ ❧❛t❡r❛❧ ❞❡CG♣xq ❣❡r❛♠ ♦ ♠❡s♠♦ ❝♦♥❥✉❣❛❞♦ ✌ ❡❧❡♠❡♥t♦s ❡♠ ❞✐st✐♥t❛s ❝❧❛ss❡s ❧❛t❡r❛✐s ❞❡ CG♣xq ❞ã♦ ♦r✐❣❡♠ ❛ ❝♦♥❥✉❣❛❞♦s

❞✐st✐♥t♦s✳

(15)

✶✺

Pr♦♣♦s✐çã♦ ✷✳✸ ✭❈á❧❝✉❧♦ ❞♦ t❛♠❛♥❤♦ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦✮✳ ❖ ♥ú♠❡r♦ ❞❡ ❝♦♥❥✉❣❛❞♦s ❞❡ x ❡♠ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G é ♦ í♥❞✐❝❡ ❞♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡ x ❡♠ G✳ ❉❡♠♦♥str❛çã♦✳ ❆♥t❡s ❞❡ ✐♥✐❝✐❛r ❛ ❞❡♠♦♥str❛çã♦ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛✱ ❝♦♥✈é♠ ♥♦t❛r q✉❡ t❛❧ Pr♦♣♦s✐çã♦ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❚❡♦r❡♠❛ Ór❜✐t❛✲❊st❛❜✐❧✐③❛❞♦r✳ ◆♦ ❡♥t❛♥t♦✱ s❡♥❞♦ ❛ ❝♦♥❥✉❣❛çã♦ ✉♠ ❝♦♥❝❡✐t♦ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ♦ r❛❝✐♦❝í♥✐♦ ✉s❛❞♦ ♥✉♠❛ ♣r♦✈❛ ❞✐r❡t❛ ♠❡r❡❝❡ r❡❣✐str♦ ❡s♣❡❝í✜❝♦✳

✭✶✮ ❙✉♣♦♥❤❛ q✉❡ a✱ b P G ❡st❡❥❛♠ ♥❛ ♠❡s♠❛ ❝❧❛ss❡ ❧❛t❡r❛❧ à ❞✐r❡✐t❛ ❞❡ CG♣xq ❡♠ G✳ ❊♥tã♦ b ✏ha✱ ❝♦♠ hPCG♣xq✳ ✭✻b✁

1

❂ ♣haq✁1 ❂a✁1h✁1✮✳ ▼❛s hPCG♣xq ñhx ❂ xh ❡✱ ♣♦rt❛♥t♦✿

xb ❂ b✁1xb ❂ a✁1h✁1xha ❂ a✁1h✁1hxa ❂ a✁1xa ❂xa✳

✭✷✮ ❙❡a✱b PG❡stã♦ ❡♠ ❞✐st✐♥t❛s ❝❧❛ss❡s ❧❛t❡r❛✐s à ❞✐r❡✐t❛ ❞❡CG♣xq ❡♠G✱ s❡❣✉❡ b✁1xb ✘a✁1xa✳ ❙❡ t❛❧ ♥ã♦ ❢♦ss❡ ♦ ❝❛s♦✱ t❡rí❛♠♦s b✁1xb ❂a✁1xa ❡ ✈✐r✐❛ ab✁1xb ❂ xa✻ab✁1x❂xab✁1✱ ♦ q✉❡ s✐❣♥✐✜❝❛ab✁1 PCG♣xq✱ ♦✉ s❡❥❛✱ ✐ss♦ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡ra, b ♣❡rt❡♥❝❡♠ à ♠❡s♠❛ ❝❧❛ss❡ ❧❛t❡r❛❧ ❞❡ CG♣xq ❡♠ G✳ ✭▼❛s ✐ss♦ ❝♦♥tr❛r✐❛ ❛ ❡s❝♦❧❤❛

✐♥✐❝✐❛❧ ❞❡ a ❡ b✮✳

Pr♦♣♦s✐çã♦ ✷✳✹ ✭◆ú♠❡r♦ ❞❡ ❈❧❛ss❡s ❞❡ ❈♦♥❥✉❣❛çã♦ ❡♠ ✉♠ ●r✉♣♦ ❋✐♥✐t♦✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠ n✳ ❉❡♥♦t❛♥❞♦ ♣♦r k♣Gq ♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❡♠ G✱ ✈❛❧❡♠ ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿

✭❛✮ k♣Gq ✏ 1

|G| ☎

➳

x_PG

|CG♣xq|✏

1

n ☎ ➳

x_PG

|CG♣xq| ;

✭❜✮ cp♣Gq ✏ k♣Gq

|G| .

❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ C ✏ t ♣x, yq P G✂G : xy ✏ yx ✉✱ ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❝✉❥❛s ❡♥tr❛❞❛s sã♦ ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠✉t❛♠ ♥♦ ❣r✉♣♦G✳ P❛r❛ ❝❛❞❛ x P G ✭✜①♦✮✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞♦ t✐♣♦ ✭x✱y✮ ❡♠ C é ❞❛❞♦ ♣♦r

|CG♣xq|✳ ❙❡♥❞♦|G| ❂ rG:CG♣xqs ☎|CG♣xq|✱ s❡❣✉❡✿

|C| ❂ ➳

xPG

|CG♣xq| ❂ ➳

xPG

|G|

rG:CG♣xqs ❂

|G|☎ ➳

xPG

1

rG:CG♣xqs ✭✯✮

P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✱ s❡ ❞♦✐s ♣♦♥t♦s x❡ y❡♠ G ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ ór❜✐t❛ ✭❝❧❛ss❡✮✱ s❡✉s ❡st❛❜✐❧✐③❛❞♦r❡s ✭❝❡♥tr❛❧✐③❛❞♦r❡s✮ sã♦ s✉❜❣r✉♣♦s ❝♦♥❥✉❣❛❞♦s ❡ ♣♦ss✉❡♠ ♠❡s♠♦ í♥❞✐❝❡✳

❚♦♠❡♠♦s ✉♠ r❡♣r❡s❡♥t❛♥t❡xi ❞❛i✲és✐♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦✱Ci ✱ ❡ ❞❡♥♦t❡♠♦s

(16)

✶✻

rG:CG♣yqs ✏ rG:CG♣xiqs ✏|Ci|✏mi

❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ❛i✲és✐♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦✱ ✈❛❧❡✿ ➳

yPCi

1

rG:CG♣yqs ❂ ➳

yPCi

1

mi ✏ mi☎

1

mi ✏

1

❋❛③❡♥❞♦k ✏k♣Gq✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✭✯✮✿

|C|❂|G|☎➳

xPG

1

rG:CG♣xqs ❂

|G|☎

k ➳

l✏1

1❂|G|☎k♣Gq

❯s❛♥❞♦ t❛❧ ❢❛t♦ ❡♠ ✭✯✮ ✱ ❝♦♥❝❧✉í♠♦s ❛ ♣r♦✈❛ ❞♦ ✐t❡♠✳

✭❜✮ ❘❡❝♦r❞❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛çã♦✱ cp♣Gq ✿

cp♣Gq ✏ ⑤t♣x, yq PG✂G:xy✏yx✉⑤

⑤G✂G⑤ ✏

|C| |G|2

❘❡❝♦rr❡♥❞♦ ❛♦ ✐t❡♠ ✭❛✮✱ ♦❜t❡♠♦s ❛ ❡①♣r❡ssã♦ ❜ás✐❝❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡cp♣Gq✿

cp♣Gq ✏ k♣Gq

|G| .

Pr♦♣♦s✐çã♦ ✷✳✺ ✭❈♦t❛ ❙✉♣❡r✐♦r ♣❛r❛cp♣Gq ❡♠ ❣r✉♣♦s ♥ã♦✲❛❜❡❧✐❛♥♦s✮✳ ❙❡G é ✉♠ ❣r✉♣♦ ♥ã♦✲❛❜❡❧✐❛♥♦✱ ❡♥tã♦ cp♣Gq ↕ 5④8✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❝❛s♦ cp♣Gq ✏ 5④8 ♦❝♦rr❡

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ❝❡♥tr♦ t❡♠ ✐♥❞í❝❡ 4✳

❉❡♠♦♥str❛çã♦✳ ❆♥t❡s ❞❡ ✐♥✐❝✐❛r ❛ ❞❡♠♦♥str❛çã♦✱ ✈❛❧❡ r❡❣✐str❛r q✉❡ t❛❧ r❡s✉❧t❛❞♦ é ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✻❪ ✭✶✾✼✸✮✳ ❆♣❡s❛r ❞❡ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠ tr❛❜❛❧❤♦ q✉❡ ❞❡s♣❡rt♦✉ ♦ ✐♥t❡r❡ss❡ ♥♦ ❡st✉❞♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r✳

✭■✮ ❆ ❡q✉❛çã♦ ❞❛s ❝❧❛ss❡s s❡ ❡s❝r❡✈❡|G| ✏|Z♣Gq| |C1| ☎ ☎ ☎ |Ct|✱ ❡♠ q✉❡ Ci r❡♣r❡s❡♥t❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ ♥ã♦✲❝❡♥tr❛❧ ✭❡✱ ♣♦rt❛♥t♦✱ ♥ã♦✲✉♥✐tár✐❛✮✳ ❆ss✐♠✱|Ci|➙2✱ ♣❛r❛ ❝❛❞❛ i✏1,2, . . . , t ❡ r❡s✉❧t❛ ✭|G|✁|Z♣Gq|q ➙2☎t✳ ❙❡♥❞♦ k ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❝❧❛ss❡s✱ s❡❣✉❡k ✏t |Z♣Gq| ❡ ❞❛í k ↕ ♣|G| |Z♣Gq|q④2✳

✭■■✮ ❙❡♥❞♦ G ✉♠ ❣r✉♣♦ ♥ã♦✲❛❜❡❧✐❛♥♦✱ t❡♠♦s q✉❡ ♦ q✉♦❝✐❡♥t❡ G④Z♣Gq é ♥ã♦✲ ❝í❝❧✐❝♦ ❡✱ ♣♦rt❛♥t♦✱|G④Z♣Gq| ➙ 4✳ ❆ss✐♠✱|Z♣Gq| ↕|G|④4 ❡✱ ❛♣❧✐❝❛♥❞♦ ✭■✮✱ ✈❡♠

k↕5④8☎|G|✳

✭■■■✮ ❯s❛♥❞♦ ✭■■✮ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ ❡①♣r❡ssã♦ ❜ás✐❝❛ ♣❛r❛ cp♣Gq ♦❜t✐❞❛ ♥❛ Pr♦♣♦s✐çã♦ ✷✳✹✱ s❡❣✉❡ ❛ t❡s❡✳

✭■❱✮ ❱❡❥❛♠♦s ❛❣♦r❛ ❛ s✐t✉❛çã♦ ❡♠ q✉❡ ♦❝♦rr❡ ❛ ✐❣✉❛❧❞❛❞❡✳

▼♦str❛♠♦s✱ ♥❛ ❛♥á❧✐s❡ ❛♥t❡r✐♦r✱ q✉❡✿ k ↕ ♣1④2q ☎ ♣|G| |Z♣Gq|q ↕ 5④8☎|G|✳

(17)

✶✼

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❛❞♠✐t❛♠♦s q✉❡|Z♣Gq| ✏|G|④4✳ Z ✏ Z♣Gq ↕CG♣xq,❅x P G✳ ❙❡xé ♥ã♦✲❝❡♥tr❛❧✱ s❡❥❛C♦ ❝❡♥tr❛❧✐③❛❞♦r ❞❡ x❡♠G✳ ❆s ✐♥❝❧✉sõ❡sZ ➔C ❡C ➔G sã♦ ❡str✐t❛s✱ ♣♦✐s✿

✌ ❙❡ ♦❝♦rr❡ss❡ Z ✏ C✱ ❡♥tã♦ x ❡st❛r✐❛ ❡♠ Z✳ ▼❛s ✐ss♦ é ❛❜s✉r❞♦✱ ✈✐st♦ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❡s❝♦❧❤❛ ❞❡ x ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ ♥ã♦✲❝❡♥tr❛❧✳

✌ ❙❡ ✈❛❧❡ss❡C ✏ G✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❝♦♠✉t❛r✐❛♠ ❝♦♠ x✳ ▼❛s✱ ♣♦r ❞❡✜♥✐çã♦ ❞♦ ❝❡♥tr♦ Z ✏ Z♣Gq✱ ✐ss♦ s✐❣♥✐✜❝❛r✐❛ x P Z✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

❙❡♥❞♦ Z ✉♠ s✉❜❣r✉♣♦ ❞❡C✱ ❞❡✈❡♠♦s t❡r|C|✏ ♣1④2q ☎|G|✳ ❘❡s✉❧t❛✿

|Cl♣xq|✏ |G| |CG♣xq| ✏

|G| |C| ✏2.

❘❡❝♦rr❡♥❞♦ à ❡q✉❛çã♦ ❞❛s ❝❧❛ss❡s ❡ ✉s❛♥❞♦ ❛ ✐♥❢♦r♠❛çã♦|Z|✏ ♣1④4q ☎|G|✱ s❡❣✉❡✿

|G|✏|Z| ♣k♣Gq✁|Z|q☎2 ✻ ♣3④8q☎|G|✏k♣Gq✁♣1④4q☎|G|✻ k♣Gq ✏ ♣5④8q ☎|G|

❊①❡♠♣❧♦ ✷✳✸✳ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♠✉t❛r ❡♠ ●r✉♣♦s ❉✐❡❞r❛✐s

❆ tít✉❧♦ ❞❡ ✐❧✉str❛çã♦✱ r❡❣✐str❛♠♦s✱ ♥❛ t❛❜❡❧❛ ❛ s❡❣✉✐r✱ ♦ ✈❛❧♦r ❞❡ cp♣Gq ♣❛r❛ ❣r✉♣♦s ❞✐❡❞r❛✐sDn✳ ❖s ❝á❧❝✉❧♦s ❢♦r❛♠ ❢❡✐t♦s ❝♦♠ ♦ s♦❢t✇❛r❡ ●❆P ❡n❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❡♠ ❛♥á❧✐s❡✳ ➱ ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r q✉❡ ♦ ✈❛❧♦r ❞❡ cp♣Gq ♣❛r❛ D4 é ♦ ♠á①✐♠♦ ♣♦ssí✈❡❧ ✭❝♦♥❢♦r♠❡ Pr♦♣♦s✐çã♦ ✷✳✺✮✳

❞✐❡❞r❛❧ cp♣Gq ❞✐❡❞r❛❧ cp♣Gq ❞✐❡❞r❛❧ cp♣Gq

D3 1④2 D20 13④40 D120 21④80

D4 5④8 D30 3④10 D130 17④65

D5 2④5 D40 23④80 D140 73④280

D6 5④14 D50 7④25 D150 13④50

D7 7④16 D60 11④40 D160 83④320

D8 1④3 D70 19④70 D170 22④85

D9 2④5 D80 43④160 D180 31④120

D10 7④22 D90 4④15 D190 49④190

D11 3④8 D100 53④200 D196 101④392

(18)

✶✽

✷✳✻✳ ❈▲❆❙❙❊❙ ❉❊ ❈❖◆❏❯●❆➬➹❖ ❊▼ ❯▼ ❙❯❇●❘❯P❖✳

❊st✉❞❛r❡♠♦s✱ ❛❣♦r❛✱ ❛❧❣✉♥s ❧❡♠❛s q✉❡ ❡st❛❜❡❧❡❝❡♠ r❡❧❛çõ❡s ❡♥tr❡ ♦s ♥ú♠❡r♦s ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ ❡ ❞❡ ✉♠ s❡✉ s✉❜❣r✉♣♦✳ ❚❛✐s r❡s✉❧t❛❞♦s s❡rã♦ ❡♠♣r❡❣❛❞♦s✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ Pr✐♠❡✐r♦ ❚❡♦r❡♠❛✳

▲❡♠❛ ✷✳✻✳ ❙❡♥❞♦ H ✉♠ s✉❜❣r✉♣♦ ❞❡ G✱ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ H ❡ ❞❡ G✿

✭❛✮ k♣Hq ➔ rG:Hs ☎k♣Gq, se H ✘G❀ ✭❜✮ k♣Gq ↕ rG:Hs ☎k♣Hq ✳

❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❈♦♥❢♦r♠❡ ✈✐st♦ ♥❛ Pr♦♣♦s✐çã♦ ✷✳✹✱ t❡♠♦s✿

k♣Hq ✏ 1

|H| ☎

➳

xPH

|CH♣xq|✏ r G:Hs

|G| ☎

➳

xPH

|CH♣xq| ✭■✮

❙❡♥❞♦ H ❡ G s✉❜❝♦♥❥✉♥t♦s ❞❡ G ❡♠ q✉❡ H ❸ G✱ ❛ Pr♦♣♦s✐çã♦ ✷✳✶ ♣❡r♠✐t❡ ❡s❝r❡✈❡r✿

rG:Hs

|G| ☎

➳

xPH

|CH♣xq|↕ r G:Hs

|G| ☎

➳

xPH

|CG♣xq| ✭■■✮

❙❡♥❞♦H✱ ♣♦r ❤✐♣ót❡s❡✱ ✉♠ s✉❜❣r✉♣♦ ♣ró♣r✐♦ ❞❡ G✱ t❡♠♦s✿ rG:Hs

|G| ☎

➳

xPH

|CG♣xq|➔

rG:Hs

|G| ☎

➳

xPG

|CG♣xq| ✭■■■✮

❯♥✐♥❞♦ ♦s ❢❛t♦s ✭■✮✱ ✭■■✮✱ ❡ ✭■■■✮✱ r❡s✉❧t❛✿

k♣Hq ✏ rG:Hs ☎ 1

|G|

➳

xPH

|CH♣xq|➔ rG:Hs ☎

1

|G|

➳

xPG

|CG♣xq|✏ rG:Hs ☎k♣Gq

✭❜✮ ❙❛❜❡♠♦s q✉❡✿

✻ |CG♣xq|✏ rG:Hs ☎|CH♣xq|☎r

H :CH♣xqs rG:CG♣xqs ✭■❱✮

❆ Pr♦♣♦s✐çã♦ ✷✳✸ ❡st❛❜❡❧❡❝❡ q✉❡✱ ♣❛r❛ ✉♠ ❣r✉♣♦ q✉❛❧q✉❡r✿|Cl♣xq|✏ rG:CG♣xqs

(19)

✶✾

rH :CH♣xqs ↕ rG:CG♣xqs

❯s❛♥❞♦ t❛❧ ✐♥❢♦r♠❛çã♦ ❡♠ ✭■❱✮✱ ✈❡♠✿ |CG♣xq|↕ rG:Hs ☎|CH♣xq| ✭❱✮ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡♠ ✭❱✮ ✈❛❧❡ ♣❛r❛ t♦❞♦xP G✱ s❡❣✉❡✿

➳

xPG

|CG♣xq| ↕ rG:Hs ☎ ➳

xPG

|CH♣xq|✏ rG:Hs ☎ ➳

yPH

|CG♣yq| ✭❱■✮

❯s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ✭❱✮ ❡♠ ✭❱■✮✱ r❡s✉❧t❛✿ ➳

x_PG

|CG♣xq| ↕ rG:Hs ☎ ➳

y_PH

|CG♣yq|

↕ rG:Hs ☎ ➳ yPH

rG:Hs ☎|CH♣yq|✏

|G|

|H| ☎ rG:Hs ☎

➳

yPH

|CH♣yq|

✻ 1

|G| ☎

➳

xPG

|CG♣xq| ↕ rG:Hs ☎

1

|H| ☎

➳

yPH

|CH♣yq|

❆ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡✿ k♣Gq ↕ rG:Hs ☎k♣Hq ✳

▲❡♠❛ ✷✳✼✳ ❙❡♥❞♦ N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G✱ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿

✭❛✮ CG♣xqN

N ❸CG④N♣xNq ❀

✭❜✮ CG♣xq CN♣xq ✕

CG♣xqN

N ❀

✭❝✮ k♣Gq ↕ k♣G④Nq ☎k♣Nq ✳

❉❡♠♦♥str❛çã♦✳ ✭❛✮ ◆♦t❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡✱ s❡♥❞♦ N ➐G ❡CG♣xq ↕ G✱ t❡♠✲s❡ q✉❡ CG♣xqN é s✉❜❣r✉♣♦ ❞❡ G✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ q✉❡ N ➐ CG♣xqN✱ s❡❣✉❡ q✉❡ ♣CG♣xqNq④N ↕ G④N✳ ❚♦♠❡♠♦s ❛❣♦r❛ ✉♠❛ ❝❧❛ss❡ ❧❛t❡r❛❧ σN ❡♠ ♣CG♣xqNq④N✳ ❚❡♠♦s σ ✏y☎n✱ ❝♦♠ yPCG♣xq ❡ nPN✳

yP CG♣xq ñy☎x✏x☎y ✻ yN ☎xN ✏xN ☎yN ✻ yN PCG④N♣xNq

❙❡♥❞♦ σ✏y☎n✱ s❡❣✉❡σN ✏ ♣y☎nqN ✏yN ❡✱ ♣♦rt❛♥t♦✱ σN PCG_④N♣xNq ✭❜✮ ❈♦♥❢♦r♠❡ ✈✐st♦ ♥♦ ✐t❡♠ ✭❛✮✱ N ➐CG♣xqN ✭■✮

❙❡♥❞♦ CG♣xq ↕CG♣xqN✱ t❡♠♦s ✭N ❳CG♣xqq ➐CG♣xq✳

(20)

✷✵

CN♣xq ➐ CG♣xq ✭■■✮

❘❡❝♦rr❡♠♦s ❛♦ ❙❡❣✉♥❞♦ ❚❡♦r❡♠❛ ❞♦ ■s♦♠♦r✜s♠♦ ♣❛r❛ ♠♦str❛r q✉❡✿

CG♣xqN

N ✕

CG♣xq N ❳CG♣xqq

❡✱ ❛♣❧✐❝❛♥❞♦ ✭■■✮✱ ♦❜t❡♠♦s CG♣xqN

N ✕

CG♣xq CN♣xq

✭❝✮ ❈♦♠❜✐♥❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ✭❛✮ ❡ ✭❜✮✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

CG♣xq CN♣xq ✕

CG♣xqN

N ❸CG④N♣xNq ✳

P♦r ❡st❛r♠♦s tr❛t❛♥❞♦ ❝♦♠ ❣r✉♣♦s ✜♥✐t♦s✱ ♣♦❞❡♠♦s ♣❛ss❛r ❞❡ t❛✐s ✐♥❝❧✉sõ❡s ❛ ❞❡s✐❣✉❛❧❞❛❞❡s q✉❡ ❡♥✈♦❧✈❡♠ ❛s ♦r❞❡♥s ❞♦s ❣r✉♣♦s✿

|CG♣xq|↕

CG④N♣xNq

☎|CN♣xq| ❡ ➳

xPG

|CG♣xq|↕ ➳

xPG

CG④N♣xNq

☎|CN♣xq| ✭■✮

❙❡♥❞♦ ρ ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❝❧❛ss❡ ❧❛t❡r❛❧ xN✱ t❡♠✲s❡ ρN ✏xN✳ ❆ss✐♠✱ ♦ ❝❡♥tr❛✲ ❧✐③❛❞♦rCG④N♣ρNq ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❝❡♥tr❛❧✐③❛❞♦rCG④N♣xNq,❅ρPxN✳ ❉❡ss❛ ❢♦r♠❛✱ ❢❛③❡♥❞♦ S✏xN✱ ❛ s♦♠❛ ❡♠ ✭■✮ ♣♦❞❡ s❡r r❡❡s❝r✐t❛✿

➳

xPG

CG④N♣xNq

☎|CN♣xq|✏ ➳

S_PG_④N ✁

CG④N♣Sq ☎

➳

xPS

|CN♣xq| ✠

✭■■✮

❉❛❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s A, B ❸G✱ ❝♦♥s✐❞❡r❡♠♦s ♦s ❝♦♥❥✉♥t♦s✿ Q✏ t ♣a, bq PA✂B :ab✏ba✉ ❡ CA♣xq ✏ ta PA : ax✏xa✉ ✳

❈♦♠ ❡ss❛s ♥♦t❛çõ❡s✱ ✈❡♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛ a P A ✭✜①♦✮✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ♣a, bq ❡♠Q é ❞❛❞♦ ♣♦r ⑤CB♣aq⑤✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❝❛❞❛ bP B ✭✜①♦✮✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r❡s ♣a, bq ❡♠ Q é ❞❛❞♦ ♣♦r ⑤CA♣bq⑤✳ ❯♥✐♥❞♦ ❛s ❞✉❛s ♣❡rs♣❡❝t✐✈❛s✿

➳

aPA

⑤CB♣aq⑤ ✏ ⑤Q⑤ ✏ ➳

bPB

⑤CA♣bq⑤ ✭■■■✮

❆♣❧✐❝❛♥❞♦ ✭■■■✮ ❛♦s s✉❜❝♦♥❥✉♥t♦s N ↕G ❡ xN ✏S ❸G✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ ➳

xPS

|CN♣xq|✏ ➳

τPN

|CS♣τq|✳

❊ss❡ ú❧t✐♠♦ ❢❛t♦ ♥♦s ♣❡r♠✐t❡ ✈♦❧t❛r ❛ ✭■■✮ ❡ tr❛♥s❢♦r♠❛r ✉♠❛ s♦♠❛ s♦❜r❡ ❛ ❝❧❛ss❡ ❧❛t❡r❛❧S✱ ❡♠ ♦✉tr❛✱ s♦❜r❡ ♦ s✉❜❣r✉♣♦ ♥♦r♠❛❧ N✿

➳

SPG④N ✁

CG④N♣Sq ☎

➳

x_PS

|CN♣xq| ✠

✏ ➳

SPG④N ✁

CG④N♣Sq ☎

➳

τ_PN

|CS♣τq| ✠

(21)

✷✶

❉❛❞♦ ♦ s✉❜❣r✉♣♦ N ➐ G✱ s❡❥❛♠ t P N✱ x P G✱ S :✏ xN✳ ❈♦♠ ❛s ♠❡s♠❛s ♥♦t❛çõ❡s ❥á ✉t✐❧✐③❛❞❛s✱ ✈❛♠♦s ♠♦str❛r q✉❡✿ ⑤CS♣tq⑤ ↕ ⑤CN♣tq⑤ ✳ ❉❡ ❢❛t♦✿

◗✉❛♥❞♦ CS♣tq é ✈❛③✐♦✱ t❡♠✲s❡ 0 ✏ ⑤CS♣tq⑤ ↕ ⑤CN♣tq⑤✳ ❈♦♠ ✐ss♦ ❡♠ ♠❡♥t❡✱ ♣♦❞❡♠♦s s✉♣♦r CS♣tq ♥ã♦✲✈❛③✐♦✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❡ y P CS♣tq✳ ❉❛❞♦ ✉♠ t❛❧ y✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ CS♣tq ❸yCN♣tq✱ ♣♦✐s ⑤yCN♣tq⑤ ✏ ⑤CN♣tq⑤✳

❚♦♠❡♠♦s ✉♠z PCS♣tq✳ ❊♥tã♦ ❡①✐st❡n PN t❛❧ q✉❡z ✏xn❡ ✈❛❧❡xnt♣xnq✁ 1

✏t✳ ▲❡♠❜r❛♥❞♦ q✉❡ yt✏ty✱ s❡❣✉❡✿

t✏y✁1ty ✏y✁1xntn✁1x✁1y ✱ ♠♦str❛♥❞♦ q✉❡y✁1xn PCG♣tq ✳

P♦r ♦✉tr♦ ❧❛❞♦ y✁1x P N ✭♣♦✐s y P S ✏ xN✮✳ ❆ss✐♠ y✁1xn P N ❡✱ ♣♦rt❛♥t♦✱ y✁1xn PCN♣tq✳ ❋✐♥❛❧♠❡♥t❡✱ z ✏xn✏y♣y✁

1

xnq P yCN♣tq ❡ ❞❛íCS♣tq ❸yCN♣tq ✳

❈♦♠❜✐♥❛♥❞♦ ❡ss❡ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❝♦♠ ❛ ❡①♣r❡ssã♦ ❡♠ ✭■❱✮✱ s❡❣✉❡✿

1

|G|

➳

xPG

|CG♣xq|↕

1

|G|

➳

SPG④N

CG④N♣Sq ☎

➳

τPN

|CS♣τq|

↕ 1

|G④N|

➳

SPG④N

CG④N♣Sq

☎

✁ ₁

|N|

➳

τPN

|CN♣τq| ✠

❘❡❝♦rr❡♥❞♦ à ❡①♣r❡ssã♦ ♣❛r❛ k♣Gq ❞❛❞❛ ♥❛ Pr♦♣♦s✐çã♦ ✷✳✹✱ ♦❜t❡♠♦s ❛ t❡s❡✳

❖ ✐t❡♠ ✭❝✮ ❞♦ ▲❡♠❛ ✷✳✼ é ❞❡ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡✱ ♣♦✐s ♣❡r♠✐t❡ ❞✐✈✐❞✐r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❛❧❝✉❧❛r ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ♣❛r❛ k♣Gq✳ ❈❛s♦ s❡❥❛ ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ N ❡ ✉♠ q✉♦❝✐❡♥t❡ G④N ♣❛r❛ ♦s q✉❛✐s s❡ ❝♦♥s✐❣❛ ❧✐♠✐t❛r ♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦✱ t❡r❡♠♦s ✉♠ ❧✐♠✐t❡ ♣❛r❛ k♣Gq✳ ❱❡r❡♠♦s t❛♠❜é♠✱ ♥♦ ❧❡♠❛ ❛ s❡❣✉✐r✱ ❝♦♠♦ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ cp♣Gq✳ ❚❛❧ ❡str❛té❣✐❛ t❡rá ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳

▲❡♠❛ ✷✳✽✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦✳

✭❛✮ P❛r❛ t♦❞♦ s✉❜❣r✉♣♦ ♣ró♣r✐♦ H ➔G✱ ✈❛❧❡✿

rG:Hs✁2☎cp♣Hq ➔ cp♣Gq ↕cp♣Hq ❀

✭❜✮ ◗✉❛♥❞♦ N ➌G✱ cp♣Gq ↕cp♣Nq ☎cp♣G④Nq ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✈❛❧❡ cp♣Gq ↕cp♣G④Nq ❀

✭❝✮ ❯♠❛ s❡çã♦ ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G é q✉❛❧q✉❡r ✐♠❛❣❡♠ ❤♦♠♦♠ór✜❝❛ ❞❡ ✉♠ s✉❜❣r✉♣♦ ❞❡ G✳ P❛r❛ t♦❞❛ s❡çã♦ X ❞❡ G✱ t❡♠✲s❡ cp♣Gq ↕cp♣Xq ❀

(22)

✷✷

✭❡✮ ❙❡ ♦ ❣r✉♣♦ G é ♥ã♦✲❛❜❡❧✐❛♥♦✱ cp♣Gq ➔ 3④2☎ rG : CG♣xqs✁ 1

✱ ♣❛r❛ ❛❧❣✉♠ x P ♣G✁Z♣Gqq✳ ❙❡ ♦ ❝❡♥tr♦ ❩✭G✮ é tr✐✈✐❛❧✱ ✈❛❧❡ cp♣Gq ↕ rG : CG♣xqs✁

1

♣❛r❛ ❛❧❣✉♠ xPG✁ t1✉ ✳

❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❯s❛♥❞♦ ♦s ít❡♥s ✭❛✮ ❡ ✭❜✮ ❞♦ ▲❡♠❛ ✷✳✻✱ ♦❜t❡♠♦s✿

rG:Hs✁1☎k♣Hq ➔ k♣Gq ↕ rG:Hs ☎k♣Hq ✱ q✉❛♥❞♦H é s✉❜❣r✉♣♦ ♣ró♣r✐♦ ❞❡G✳ ▲❡♠❜r❛♥❞♦ q✉❡ k♣Gq ✏|G|☎cp♣Gq ❡ k♣Hq ✏|H|☎cp♣Hq✱ ✈❡♠✿

rG:Hs✁1☎|H|☎cp♣Hq ➔|G|☎cp♣Gq ↕ rG:Hs ☎|H|☎cp♣Hq ✳

❉✐✈✐❞✐♥❞♦ t♦❞♦s ♦s ♠❡♠❜r♦s ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♣♦r |G|✱ ♦❜t❡♠♦s ❛ t❡s❡✳

✭❜✮ ❯s❛♠♦s ♦ ❢❛t♦ q✉❡ k♣G④Nq ✏|G|

|N| ☎cp♣G④Nq ❡ ❛♣❧✐❝❛♠♦s ♦ ▲❡♠❛ ✷✳✼ ✭❝✮✳

❉❛❞♦ q✉❡ cp♣Nq ↕1✱ t❡♠♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ cp♣Gq ↕cp♣G④Nq✳

✭❝✮ ◆♦t❡♠♦s q✉❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ ✉♠❛ s❡çã♦ ❞❡ G é ✉♠ ❣r✉♣♦ ❞❛ ❢♦r♠❛ Y④N✱ ❡♠ q✉❡ N ➐ Y ❡ Y ↕ G✳ ❙❡❥❛ X ✏ Y④N ✉♠❛ s❡çã♦ ❞❡ G✳ P❡❧♦ ✐t❡♠ ✭❜✮✱ t❡♠♦s cp♣Yq ↕cp♣Y④Nq✳ ▼❛s cp♣Gq ↕ cp♣Yq✱ ♣♦r ✭❛✮✳ ❈♦♠❜✐♥❛♥❞♦ ✭❛✮ ❡ ✭❜✮✱ s❡❣✉❡✿

cp♣Gq ↕ cp♣Yq ↕ cp♣Y④Nq ✻ cp♣Gq ↕ cp♣Xq ✳ ✭❞✮ ❙❡❣✉❡ ❞♦ ❢❛t♦ q✉❡k♣G✂Hq ✏ k♣Gq ☎k♣Hq ✳

✭❡✮ ❋❛③❡♥❞♦Z ✏Z♣Gq ❡ k✏k♣Gq ✱ ❛ ❡q✉❛çã♦ ❞❛s ❝❧❛ss❡s s❡ ❡s❝r❡✈❡✿

|G|✏|Z| ➳

i

rG:CG♣xiqs ✱

❡♠ q✉❡ ♦sxi sã♦ r❡♣r❡s❡♥t❛♥t❡s ❞❛s ❝❧❛ss❡s ❞❡ t❛♠❛♥❤♦ s✉♣❡r✐♦r ❛ ✶✳ ✭■✮ P❛r❛ ♣r♦✈❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣r♦♣♦st❛✱ ❜❛st❛ ♠♦str❛r q✉❡✿

k ➔ 3④2 ☎|G| ☎ rG:CG♣xqs✁ 1

✱ ♣❛r❛ ❛❧❣✉♠ xP ♣G✁Zq ✱

♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ q✉❡✿ k ➔ 3④2 ☎|CG♣xq|✱ ♣❛r❛ ❛❧❣✉♠x ♥ã♦✲❝❡♥tr❛❧✳ ❉❡♥♦t❛♥❞♦ ♣♦rc ❛ ♠á①✐♠❛ ♦r❞❡♠ ❞❡CG♣xq ♣❛r❛xPG✁Z✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡k ➔3④2 ☎c ♣❛r❛ ✉♠ t❛❧ x ✳ ❘❡t♦r♥❛♥❞♦ à ❡q✉❛çã♦ ❞❛s ❝❧❛ss❡s✱ ♦❜t❡♠♦s✿

|G|✏|Z| ➳

i

❬G:CG♣xiq❪ ➙|Z| ♣k✁|Z|q☎

|G|

c ✻ |G|→|G|✁|Z|➙ ♣k✁|Z|q☎

|G|

c ✳

(23)

✷✸

▲❡♠❜r❛♥❞♦ q✉❡ Z ↕ CG♣xq ❡ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❡s❝♦❧❤❛ ❞❡ x ✭♥ã♦✲❝❡♥tr❛❧✮✱ s❡❣✉❡ q✉❡ Z ❞❡✈❡ s❡r ✉♠ s✉❜❣r✉♣♦ ♣ró♣r✐♦ ❞❡ CG♣xq ✳ ❆ss✐♠✿

rCG♣xq:Zs ➙2 ✻|Z|↕1④2☎|CG♣xq| ✻ |Z|↕1④2☎c

❯s❛♥❞♦ ❡st❡ ú❧t✐♠♦ ❢❛t♦ ❡♠ k➔c |Z|✱ s❡❣✉❡k ➔3④2 ☎c

✭■■✮ ❉❛❞♦ xP ♣G✁Zq✱ ❝♦♠ Z ✏1✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡✿

k ↕|G| ☎ rG:CG♣xqs✁ 1

✭♦✉✱ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱ k ↕|CG♣xq|↕c✮✳

▼❛s✱ ♥♦ ❝❛s♦ ❡s♣❡❝í✜❝♦Z ✏1✱ ❛ ✐♥❡q✉❛çã♦ k ↕c |Z|✁1s❡ r❡❞✉③ ❛k ↕c✳

✷✳✼✳ ❙❯❇●❘❯P❖ ❈❖▼❯❚❆❉❖❘ ❊ ❙❯❇●❘❯P❖ ❉❊❘■❱❆❉❖✳

❖ ❝♦♠✉t❛❞♦r ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s x, y P G é ♦ ❡❧❡♠❡♥t♦ rx, ys ✏ x✁1y✁1xy✳ ❉❛❞♦ q✉❡ xy ✏ yxrx, ys✱ ♦ ❝♦♠✉t❛❞♦r ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ✧❝♦rr❡çã♦✧ ❛♣❧✐❝❛❞❛ ❛♦ t❡r♠♦ yx ♣❛r❛ q✉❡ s❡ t♦r♥❡ ✐❣✉❛❧ ❛ xy✳ P❡❧❛ ❞❡✜♥✐çã♦✱ ♣♦❞❡✲s❡ ❛✐♥❞❛ ✈❡r q✉❡ rx, ys ✏1ôx ❡ y ❝♦♠✉t❛♠✳

❙❡❥❛♠ H ↕G ❡K ↕G s✉❜❣r✉♣♦s ❞❡ G✿

✌ rH, Ks ❂ ① rh, ks ⑤ h P H e k P K ② ❞❡♥♦t❛ ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s ❝♦♠✉t❛❞♦r❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ H ❝♦♠ ❡❧❡♠❡♥t♦s ❞❡ K ❡ é ❝❤❛♠❛❞♦ ♦ s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r ❞❡ H ❡K✳

✌ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❡✜♥✐♠♦s G✶ ✏ rG, Gs ✏ ① rx, ys ⑤ x, y P G ②✱ ♦ ❝❤❛♠❛❞♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ G ✭s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s ❝♦♠✉t❛❞♦r❡s ❡♠ G✮✳ ➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡G✶ ✏1ôGé ❛❜❡❧✐❛♥♦✳

✌ ❉✐③❡♠♦s q✉❡ H ❝❡♥tr❛❧✐③❛ K s❡ H↕CG♣Kq✳ ✌ ❉✐③❡♠♦s q✉❡ H ♥♦r♠❛❧✐③❛ K s❡ H ↕NG♣Kq✳

Pr♦♣♦s✐çã♦ ✷✳✾ ✭❙✉❜❣r✉♣♦ ❉❡r✐✈❛❞♦ ❡ ◗✉♦❝✐❡♥t❡s ❆❜❡❧✐❛♥♦s ✮✳ ❉❛❞♦s ✉♠ ❣r✉♣♦ G ❡ s❡✉ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ G✶✱ t❡♠✲s❡✿

✭❛✮ ❉❛❞♦ N ➐G✱ ♦ q✉♦❝✐❡♥t❡ G④N é ❛❜❡❧✐❛♥♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G✶ ❸N ❀ ✭❜✮ ❙❡❥❛ ϕ : G Ñ A ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ❡♠ q✉❡ A é ✉♠ ❣r✉♣♦

❛❜❡❧✐❛♥♦✳ ❊♥tã♦ G✶ ❸Ker♣ϕq ❀

✭❝✮ ❚♦❞♦ s✉❜❣r✉♣♦ H ❞❡ G q✉❡ ❝♦♥té♠ G✶ é ♥♦r♠❛❧ ❡♠ G✳

❉❡♠♦♥str❛çã♦✳ ❱✐❞❡ ✭❬✶✻❪✱ ✐t❡♠ ✷✳✷✸✱ ♣❣ ✸✸✮ ❡ ✭❬✶✺❪✱ ✐t❡♠ ✸✳✺✷✱ ♣❣ ✺✾✮✳

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✷✹

❆ Pr♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❛♣r❡s❡♥t❛ ✉♠❛ ❝♦❧❡tâ♥❡❛ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❛ss♦❝✐❛❞❛s ❛♦ s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r✳ ❘❡❝♦rr❡r❡♠♦s ♠✉✐t❛s ✈❡③❡s ❛ t❛✐s ❢❛t♦s ♣❛r❛ ♣r♦✈❛r r❡s✉❧t❛✲ ❞♦s q✉❡ ❡♥✈♦❧✈❡♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ✉♠ s✉❜❣r✉♣♦ ❝❡♥tr❛❧✐③❛r ♦✉ ♥♦r♠❛❧✐③❛r ♦✉tr♦ s✉❜❣r✉♣♦✳ ■ss♦ s❡rá ♣❛rt✐❝✉❧❛r♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞♦ ❙✉❜❣r✉♣♦ ❞❡ ❋✐tt✐♥❣ ●❡♥❡r❛❧✐③❛❞♦ ✭❙❡çã♦ ✸✳✷✮✳

Pr♦♣♦s✐çã♦ ✷✳✶✵ ✭Pr♦♣r✐❡❞❛❞❡s ❞♦ ❙✉❜❣r✉♣♦ ❈♦♠✉t❛❞♦r✮✳ ❙❡❥❛♠ H ↕G ❡ K ↕ G s✉❜❣r✉♣♦s ❞❡ G✳ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭❛✮ rH, Ks ✏ rK, Hs ❀

✭❜✮ H ❝❡♥tr❛❧✐③❛ K s❡✱ ❡ s♦♠❡♥t❡ s❡✱ rH, Ks ✏1❀

✭❝✮ H ♥♦r♠❛❧✐③❛K s❡✱ ❡ s♦♠❡♥t❡ s❡✱ rH, Ks ↕K ❀ ✭❞✮ ❙❡ H↕G ❡ K ➐G✱ ❡♥tã♦ rH, Ks ↕K❀

✭❡✮ ❙❡ H, K ➐G✱ ❡♥tã♦ rH, Ks ↕ H❳K✳ ❆❧é♠ ❞✐ss♦✱ s❡ H❳K ✏1✱ ❡♥tã♦

H ↕CG♣Kq ❀

✭❢✮ H ❸NG♣rH, Ksq✱ ♦✉ s❡❥❛✱ H ♥♦r♠❛❧✐③❛ ♦ s✉❜❣r✉♣♦ rH, Ks ❀

✭❣✮ ❙❡❥❛♠ K ↕ H ↕ G ❝♦♠ K ➐ G✳ ❊♥tã♦ rH, Gs ↕ K s❡✱ ❡ s♦♠❡♥t❡ s❡✱ H④K ↕Z♣G④Kq ❀

✭❤✮ ❙❡H, K ↕G❡ϕ :GÑLé ❤♦♠♦♠♦r✜s♠♦✱ ❡♥tã♦ϕ♣rH, Ksq ✏ rϕ♣Hq, ϕ♣Kqs❀ ✭✐✮ ✭▲❡♠❛ ❞♦s ✸ ❙✉❜❣r✉♣♦s✮ ✲ ❙❡❥❛♠ ❞❛❞♦s três s✉❜❣r✉♣♦s H, K, L ↕ G✱ q✉❡

s❛t✐s❢❛③❡♠ rH, K, Ls ✏1 ❡ rK, L, Hs ✏1✳ ❊♥tã♦ ✈❛❧❡ rL, H, Ks ✏1✳

❉❡♠♦♥str❛çã♦✳ ❱✐❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ ✭❬✶✻❪✱ ♣❣ ✶✶✷✮ ❡ ✭❬✾❪✱ ✐t❡♠ ✹✳✾✱ ♣❣ ✶✷✻✮✳

❱❛❧❡ ❛t❡♥t❛r ♣❛r❛ ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ✐♥t❡r❡ss❛♥t❡s ❞♦ ✐t❡♠ ✭❣✮✿

✭■✮ rG, Gs ↕K s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G④K ↕Z♣G④Kq✳ ❉❛❞♦ q✉❡ ❛ ✐♥❝❧✉sã♦ r❡✈❡rs❛ é s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛✱ s❡❣✉❡ G✶ ↕K ôG④K é ❛❜❡❧✐❛♥♦✳ ✭✈✐❞❡ Pr♦♣♦s✐çã♦ ✷✳✾✮

✭■■✮ ❋❛③❡♥❞♦ ❛❣♦r❛K ✏Z♣Gq✱ s❡❣✉❡ G✶ ↕Z♣Gq ôG④Z♣Gq é ❛❜❡❧✐❛♥♦✳ ❱❡♠♦s ❛✐♥❞❛ q✉❡ s❡ G④Z♣Gq é ♥ã♦✲❛❜❡❧✐❛♥♦✱ ❡♥tã♦ G✶ ❺Z♣Gq✳

Pr♦♣♦s✐çã♦ ✷✳✶✶✳ ❉❛❞♦s ♦s ❣r✉♣♦sG❡K✱ s❡❥❛H ✉♠ s✉❜❣r✉♣♦ ❞❡G✳ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭❛✮ H ↕GñH✶ ↕G✶❀

✭❜✮ ϕ :GÑK é ❤♦♠♦♠♦r✜s♠♦ ñ ϕ♣G✶q ↕ K✶ ❀

✭❝✮ ϕ :GÑK é ❤♦♠♦♠♦r✜s♠♦ s♦❜r❡❥❡t✐✈♦ ñ ϕ♣G✶q ✏K✶✳

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✷✺

✸✳ ❆ ❈❆▼■◆❍❖ ❉❖ ❚❊❖❘❊▼❆ ❉❊ ❇❊◆❉❊❘

✸✳✶✳ ●❘❯P❖❙ ◆■▲P❖❚❊◆❚❊❙ ❊ ❙❖▲Ú❱❊■❙✳

❯♠❛ sér✐❡ ♥♦r♠❛❧ ❞❡ ✉♠ ❣r✉♣♦ G é ✉♠❛ s❡qüê♥❝✐❛ ✜♥✐t❛ ❞❡ s✉❜❣r✉♣♦s ♥♦r♠❛✐s ✭Gi ➐ G✮ t❛✐s q✉❡ 1 ✏ G0 ↕ G1 ↕ ☎ ☎ ☎ ↕ Gn ✏ G✳ ❈❛❞❛ q✉♦❝✐❡♥t❡ Gi 1④Gi é ❞❡♥♦♠✐♥❛❞♦ ✉♠ ❣r✉♣♦ ❢❛t♦r ❞❛ sér✐❡✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❛ sér✐❡ ♥♦r♠❛❧ é ♦ ♥ú♠❡r♦ ❞❡ ✐♥❝❧✉sõ❡s ❡str✐t❛s✱ ♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦ ❞❡ ❢❛t♦r❡s ♥ã♦✲tr✐✈✐❛✐s✳

❯♠❛ sér✐❡ ❛❜❡❧✐❛♥❛ ♦✉ sér✐❡ s♦❧ú✈❡❧ ♣❛r❛ ✉♠ ❣r✉♣♦G é ✉♠❛ sér✐❡ ♥♦r♠❛❧✱ ❝✉❥♦s ❣r✉♣♦s ❢❛t♦r❡s ✭Gi 1④Gi✮ sã♦ ❛❜❡❧✐❛♥♦s✳

❯♠❛ sér✐❡ ❝❡♥tr❛❧ ♣❛r❛ ✉♠ ❣r✉♣♦Gé ✉♠❛ sér✐❡ ♥♦r♠❛❧✱ ❝✉❥♦s ❢❛t♦r❡s s❛t✐s❢❛③❡♠✿

Gi 1 Gi

↕Z ✁_G

Gi ✠

, ❅iP t0,1, . . . , n✁1✉ ✳

❯♠ ❣r✉♣♦ é ❞✐t♦ s♦❧ú✈❡❧ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠❛ sér✐❡ s♦❧ú✈❡❧✳ ❯♠ ❣r✉♣♦ é ❞✐t♦ ♥✐❧♣♦t❡♥t❡ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠❛ sér✐❡ ❝❡♥tr❛❧✳

❈♦♠♣❧❡♠❡♥t❛♠♦s ❡ss❛ r❡✈✐sã♦ ✐♥✐❝✐❛❧ s♦❜r❡ ❣r✉♣♦s ♥✐❧♣♦t❡♥t❡s ❡ s♦❧ú✈❡✐s✱ ❛♣r❡✲ s❡♥t❛♥❞♦ ✉♠❛ ❝♦❧❡tâ♥❡❛ ❞❡ t❡r♠♦s q✉❡ s❡rã♦ r❡❧❡✈❛♥t❡s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❞❡♠❛✐s tó♣✐❝♦s tr❛t❛❞♦s ♥❡ss❡ t❡①t♦✿

✌ ❯♠ s✉❜❣r✉♣♦S ↕G é ❞✐t♦ s✉❜♥♦r♠❛❧ ❡♠ G s❡ ❡①✐st✐r ✉♠❛ sér✐❡ ✜♥✐t❛ ❞❡ s✉❜❣r✉♣♦sHi ↕Gt❛✐s q✉❡ S ✏H0➌H1 ➌ ☎ ☎ ☎ ➌Hr ✏G✳

✌ ❯♠ s✉❜❣r✉♣♦ ♣ró♣r✐♦ M ➔ G é ❞✐t♦ ✉♠ s✉❜❣r✉♣♦ ♠❛①✐♠❛❧ ❞❡ G s❡ ♥ã♦ ❡①✐st❡ s✉❜❣r✉♣♦Lt❛❧ q✉❡M ➔L➔G✳ ✭M ➔Gé ❞✐t♦ ♠❛①✐♠❛❧ s❡✱ s❡♠♣r❡ q✉❡M ↕H ↕G✱ ❡♥tã♦ H ✏M ♦✉H ✏G✮✳

✌ P❛r❛ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G✱ ❞❡✜♥❡✲s❡ ♦ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐✱ Φ♣Gq✱ ❝♦♠♦ ❛

✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s s✉❜❣r✉♣♦s ♠❛①✐♠❛✐s ❞❡G✳

Φ♣Gq ✏ ↔

M maxG M ✳

❯♠ ❡❧❡♠❡♥t♦ xPGé ❞✐t♦ ✉♠ ♥ã♦✲❣❡r❛❞♦r s❡ ♣♦❞❡ s❡r ♦♠✐t✐❞♦ ❞❡ q✉❛❧✲ q✉❡r ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ G✳ ❉❡ss❛ ❢♦r♠❛✱ s❡ x é ✉♠ ♥ã♦ ❣❡r❛❞♦r ❞❡ G✱ ✈❛❧❡✿ G✏ ①x, Y② ñG✏ ①Y② ✳

❖ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ ❞❡ ✉♠ ❣r✉♣♦ Gé ❝❛r❛❝t❡ríst✐❝♦ ❡♠G ❡ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦✱ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡Gq✉❡ sã♦ ♥ã♦✲❣❡r❛❞♦r❡s✳

(26)

✷✻

F♣Gq ✏ ①M ↕G⑤M ➐G ❡M é ♥✐❧♣♦t❡♥t❡ ② ✳

◗✉❛♥❞♦Gé ✉♠ ❣r✉♣♦ ✜♥✐t♦✱ ❞❡♠♦♥str❛✲s❡ q✉❡F♣Gq é ♦ ♠❛✐♦r s✉❜❣r✉♣♦ ♥♦r♠❛❧ ♥✐❧♣♦t❡♥t❡ ❞❡ G ✭✈✐❞❡ ❬✾❪✱ ✐t❡♠ ✶✳✷✽✱ ♣❣ ✷✻✮✳

✌ ❉❛❞♦ ✉♠ s✉❜❣r✉♣♦H ↕G✱ ♦ ♥♦r♠❛❧ ❝♦r❡ ❞❡H❡♠Gé ♦ s❡❣✉✐♥t❡ s✉❜❣r✉♣♦ ♥♦r♠❛❧✿

coreG♣Hq ✏ ↔

x_PG

Hx ➐G✳

❖ ♥♦r♠❛❧ ❝♦r❡ ❞❡ H ❡♠ G é ♦ ♠❛✐♦r s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G ❝♦♥t✐❞♦ ♥♦ s✉❜❣r✉♣♦ H✳

✌ ❆ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s p✲s✉❜❣r✉♣♦s ❞❡ ❙②❧♦✇ ❞❡ ✉♠ ❣r✉♣♦ G é ❞❡♥♦♠✐✲ ♥❛❞❛ ♦ p✲❝♦r❡ ❞❡ G✳ ❙❡ p é ✉♠ ♣r✐♠♦ ❡ P PSylp♣Gq✱ ❡♥tã♦✿

Op♣Gq ✏ ↔

Sylp♣Gq ✏ ↔

xPG

Px ✏coreG♣Pq ✳

❖ p✲❝♦r❡ ❞❡ G é ♦ ♠❛✐♦r p✲s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G✳

✌ ❙❡♥❞♦ p ✉♠ ♣r✐♠♦ q✉❡ ❞✐✈✐❞❡ ❛ ♦r❞❡♠ ❞❡ ✉♠ ❣r✉♣♦ G✱ ♦ s✉❜❣r✉♣♦ ❞❡ ❋✐tt✐♥❣ ♣♦❞❡ t❛♠❜é♠ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

F♣Gq ✏ ➵

p divide|G|

Op♣Gq ✳

✌ ❙❡❥❛ K ✉♠ s✉❜❣r✉♣♦ t❛❧ q✉❡ 1 ➔ K ➐ G✳ ❊♥tã♦ K é ❞✐t♦ ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ♠✐♥✐♠❛❧ ❞❡ G s❡ ♥ã♦ ❡①✐st✐r s✉❜❣r✉♣♦ ♥♦r♠❛❧ L ❞❡ G t❛❧ q✉❡

1 ➔ L ➔ K✳ ✭K t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡ ❞❡♥tr❡ t♦❞♦s ♦s s✉❜❣r✉♣♦s ♥♦r♠❛✐s ♥ã♦✲tr✐✈✐❛✐s✮✳ ❱❛❧❡ ♥♦t❛r q✉❡✱ à ❡①❝❡çã♦ ❞♦ ❣r✉♣♦ tr✐✈✐❛❧✱ t♦❞♦s ♦s ❣r✉♣♦s ✜♥✐t♦s ♣♦ss✉❡♠ s✉❜❣r✉♣♦s ♥♦r♠❛✐s ♠✐♥✐♠❛✐s✳ ❆❧é♠ ❞✐ss♦✱ s❡ G é s✐♠♣❧❡s✱ ❡♥tã♦G é ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ♠✐♥✐♠❛❧ ❞❡ G✳

✌ ❖ s♦❝❧❡ ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦G✱ ❞❡♥♦t❛❞♦Soc♣Gq✱ é ♦ s✉❜❣r✉♣♦ ❝❛r❛❝t❡ríst✐❝♦ ❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s s✉❜❣r✉♣♦s ♥♦r♠❛✐s ♠✐♥✐♠❛✐s ❞❡ G✿

Soc♣Gq ✏ ①M ↕G⑤M é ♥♦r♠❛❧ ♠✐♥✐♠❛❧ ② ✳