Pr rtt - - PowerPoint PPT Presentation
Pr rtt rs t s Pr s Prt t t
❇♦❛③ ❚s❛❜❛♥ P❛rt✐❛❧❧② ❥♦✐♥t ✇✐t❤ ❉❛✈✐❞ ●❛r❜❡r✱ ❆r❦❛❞✐✉s ❑❛❧❦❛✱ ▼✐♥❛ ❚❡✐❝❤❡r✱ ●❛r② ❱✐♥♦❦✉r ❇❛r✲■❧❛♥ ❯♥✐✈❡rs✐t②
❆❧✐❝❡ ❛♥❞ ❇♦❜ ✇✐s❤ t♦ ❝♦♠♠✉♥✐❝❛t❡ ♦✈❡r ❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧✳ ∃ ❊✣❝✐❡♥t ✫ s❡❝✉r❡ ♠❡t❤♦❞s ✐❢ t❤❡② s❤❛r❡ ❛ s❡❝r❡t ✭✏❦❡②✑✮✿ ❙②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥ ✭❆❊❙✱✳ ✳ ✳ ✮✳ ❍♦✇ t♦ ❞❡❝✐❞❡ ❛ s❤❛r❡❞ s❡❝r❡t ❦❡② ♦✈❡r ❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧❄ ❉✐✣❡✕❍❡❧❧♠❛♥ ✶✾✼✻✳ ❑❡② ❊①❝❤❛♥❣❡ Pr♦t♦❝♦❧✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❜r❡❛❦t❤r♦✉❣❤ ✐♥ ❝r②♣t♦❣r❛♣❤②✳ ■♥ t❤✐s ♠✐♥✐❝♦✉rs❡✿ ❖♥❧② ♣❛ss✐✈❡ ❛❞✈❡rs❛r✐❡s✳ ❚❤❡ ❦❡r♥❡❧ ♦♥ ✇❤✐❝❤ ♠♦r❡ ✐♥✈♦❧✈❡❞ P❑❈ ✐s ❜✉✐❧t✳
❈♦✉rt❡s② ♦❢ ❲✐❦✐♣❡❞✐❛
❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ {✵, ✶, . . . , p − ✶} G = g, |G| = p b ∈ {✵, ✶, . . . , p − ✶} ga
= gab K = ga b = gab ❊①♣♦♥❡♥t✐❛t✐♦♥✳ x → gx ✈✐❛ sq✉❛r❡ ❛♥❞ ♠✉❧t✐♣❧②✱ O(❧♦❣✷ p)✳
❉✐✣❡✕❍❡❧❧♠❛♥ Pr♦❜❧❡♠✳ (ga, gb) → gab✳ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠✳ gx → x✳ ❉▲P ≥ ❉❍P✳ ❇♦t❤ ❛r❡ ǫ✲❤❛r❞✳ ❚s ✷✵✵✻✳ ◆♦♥❡ ❞❡♣❡♥❞s ♦♥ ❣❡♥❡r❛t♦r ❝❤♦✐❝❡✳
❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠✳ gx → x✳ ❉❡♣❡♥❞s ♦♥ t❤❡ ❣r♦✉♣✦ G = (Zp, +)✳ g = ✶✳ ✏gx✑ = x · g = x · ✶ = x✳ G ≤ (Z∗
p, ·)✳ ◗✉✐t❡✱ ❜✉t ♥♦t ❡♥♦✉❣❤✱ ❤❛r❞✿
◆❋❙✳ n := ❧♦❣✷(p)✿ ✷ (✶.✸✸ + o(✶))n✶/✸(❧♦❣✷ n)✷/✸✳ n ◆❋❙ ❲♦r❦ Pr❡❞✐❝t✐♦♥ ❨❡❛r ❇r♦❦❡♥ ✺✷✺ ✷✹✼ ✷✵✵✷ ✺✼✽ ✷✹✾ ✷✵✵✸ ✻✻✹ ✷✺✷ ✷✵✵✺ ✼✻✽ ✷✺✺ ✷✵✵✾ ✶✵✷✹ ✷✻✷ ✷✵✶✻❄ ✶✵, ✵✵✵ ❜✐ts ♣r✐♠❡ ❢♦r ✏❡t❡r♥❛❧✑ s❡❝✉r✐t②❄ ■♠♣r❛❝t✐❝❛❧✳
G ≤ ❊❧❧✐♣t✐❝ ❈✉r✈❡✳ ◆♦t❤✐♥❣ ❜❡tt❡r t❤❛♥ ✷n/✷✳ ❨❡t✳ ❊❈❈✳ ❘✐❝❤ ♠❛t❤❡♠❛t✐❝s → · · · → ❛❧❣♦r✐t❤♠✐❝ ❜r❡❛❦t❤r♦✉❣❤s❄ ◗✉❛♥t✉♠ ❈♦♠♣✉t❡rs✳ ❇r❡❛❦ ❛❧❧ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊Ps✳ ❚❤❡♦r❡t✐❝✳ ❇✉t ✇❤❛t ✐s ②♦✉r ❛❧t❡r♥❛t✐✈❡❄ ❘✐✈❡st✲❙❤❛♠✐r✲❆❞❧❡♠❛♥ ✭❘❙❆✱ ✶✾✼✽✮✳ ❆s ❡❛s② ❛s ❉▲P ✐♥ Z∗
p✳
▲❛tt✐❝❡✲❜❛s❡❞❄ ▼❛②❜❡✳ ❍♦✇ ❛❜♦✉t ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❣r♦✉♣s❄ ❲■◆✴❲■◆✿ ◆❡✇ ❑❊P ✴ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s✳
❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ✶✾✼✻✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ {✵, ✶, . . . , p − ✶} G = g, |G| = p b ∈ {✵, ✶, . . . , p − ✶} ga
= gab K = ga b = gab
❑♦✕▲❡❡✕❈❤❡♦♥✕❍❛♥✕❑❛♥❣✕P❛r❦ ✷✵✵✵✳ G ♥♦♥❝♦♠♠✉t❛t✐✈❡✳ gx := x−✶gx✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ A A, B ≤ G, g ∈ G, [A, B] = ✶ b ∈ B ga
= gba K = ga b = gab
G = X | R✳ ❲♦r❞ Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r g = ✶✳ ❈♦♥❥✉❣❛❝② Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r g, h ❛r❡ ❝♦♥❥✉❣❛t❡✳ ✭❆❑❆ ●❡♥❡r❛❧✐③❡❞ ❲♦r❞ Pr♦❜❧❡♠✳✮ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r G, H ❛r❡ ✐s♦♠♦r♣❤✐❝✳ ❖r✐❣✐♥❛❧❧②✱ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✳ ❈r②♣t♦ ✉s❡s t❤❡ s❡❛r❝❤ ✈❡rs✐♦♥s✳ ❯♥❧✐❦❡ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✱ t❤❡ s❡❛r❝❤ ♣r♦❜❧❡♠s ❛r❡ ❞❡❝✐❞❛❜❧❡✱ ❜✉t ✇❡ ❛s❦ ❢♦r ❡✣❝✐❡♥t s♦❧✉t✐♦♥s✳ Pr♦♣♦s❡❞ ♣❧❛t❢♦r♠✳ ❆rt✐♥✬s ❜r❛✐❞ ❣r♦✉♣✳ ✭❚❇❉✮ ▼♦t✐✈❛t❡❞ ❛ ♥❡✇ ❧✐♥❡ ♦❢ r❡s❡❛r❝❤ ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✳
■❞❡♥t✐t② ❜r❛✐❞✿
✶✿
σ−✶
✶ ✿
σ−✶
✶ σ✷✿
σ−✶
✶ σ✷σ−✶ ✶ ✿
σ−✶
✶ σ✷σ−✶ ✶ σ✷✿
σ−✶
✶ σ✷σ−✶ ✶ σ✷σ−✶ ✶ ✿
σ−✶
✶ σ✷σ−✶ ✶ σ✷σ−✶ ✶ σ✷✿
❆ ❈❤❛❧❧❛❤✳
❇✿ ❇r❛✐❞s ✴ ✐s♦t♦♣②✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ❈♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❜r❛✐❞s✳ ■♥✈❡rs✐♦♥✿ ▼✐rr♦r ❜r❛✐❞✳
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✶
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✷
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✸
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✹
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✺
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✻
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✼
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✽
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵
σ✾
❋❛r ❈♦♠♠✉t❛t✐✈✐t②✿ σiσj = σjσi ❢♦r i + ✶ < j✳ = ❚r✐♣❧❡ r❡❧❛t✐♦♥✿ σiσi+✶σi = σi+✶σiσi+✶✳ =
❚❤✐♥❦ ❉❍ ❑❊P ✐♥ (Z/pZ)∗ ✐♥st❡❛❞ ♦❢ Z∗
p✿
✶✳ ▼❛② ♥♦t ❣❡t t❤❡ s❛♠❡ ❦❡② ✐❢ ❝❤♦✐❝❡ ♥♦t ❝❛♥♦♥✐❝❛❧✦ ✷✳ ❇r❡❛❦❛❜❧❡✦ ◆♦r♠❛❧ ❢♦r♠✿ n → (n ♠♦❞ p)✿ ✶✳ ❊♥s✉r❡s s❛♠❡ ❦❡②✳ ✷✳ ❍✐❞❡s t❤❡ ❣❡♥❡r❛t✐♦♥ ✐♥❢♦✳ ❇r❛✐❞ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ✉s❡s ❇ ❛s ♣❧❛t❢♦r♠ ❣r♦✉♣✳ ◆♦r♠❛❧ ❢♦r♠ ✐♥ ❇❄
❇+ = ▼♦♥
= σjσi (i + ✶ < j), σiσi+✶σi = σi+✶σiσi+✶
✶✳ ❊q✉✐✈❛❧❡♥t ♣♦s✐t✐✈❡ ❜r❛✐❞s ❛r❡ ♣♦s✐t✐✈❡✲❡q✉✐✈❛❧❡♥t✳ ✷✳ ∴ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ♦❢ ♣♦s✐t✐✈❡ ❜r❛✐❞s ❛r❡ ✜♥✐t❡✳ ✸✳ ▲❡①✲♠✐♥✐♠❛❧ r❡♣r❡s❡♥t❛t✐✈❡s ❛r❡ ♥♦r♠❛❧ ❢♦r♠s ✐♥ ❇+✳ ◆♦t ❡✣❝✐❡♥t✱ ❜✉t t❤❡ t❤❡♠❡ ✇✐❧❧ ❜❡❝♦♠❡ ✉s❡❢✉❧ ❧❛t❡r✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ❤❡♥❝❡❢♦rt❤ ✇♦r❦ ✐♥✿ ❇N✿ σ✶, . . . , σN−✶ ≤ ❇✱ s✉♣♣♦rt❡❞ ❜② t❤❡ ❧❡❢t♠♦st n str❛♥❞s✳
✶ ✷ ✸ ✹ ∆ = (σ✶σ✷σ✸)(σ✶σ✷)σ✶ σi∆ = ∆σN−i ∆✷ ∈ Z(❇N) σi∆−✶ = ∆−✶σN−i ∆ ∈ σi❇+ ∆ ∈ ❇+σi ∆σ−✶
i
∈ ❇+ ∀b ∈ ❇N∃ ♠✐♥✐♠❛❧ |p|, b = ∆i·
❇+
∈ p p❧❡①♠✐♥ := ❧❡① ♠✐♥✐♠✉♠ ♦❢ t❤❡s❡ p✬s
inf (b) := i ✭♠❛①✐♠❛❧✮
a ≤ b✿ ∃p ∈ ❇N+, ap = b✳ ❇N+ = {p ∈ ❇N : ✶ ≤ p}✳ p ∈ S✿ ✶ ≤ p ≤ ∆✳ P❡r♠✉t❛t✐♦♥ ❜r❛✐❞s✿ S ∼ =❡✛ SN✳ ❈❛♥♦♥✐❝❛❧ ❡①♣r❡ss✐♦♥ ❜② tr❛♥s♣♦s✐t✐♦♥s (i, i + ✶)✳ ❆❞②❛♥ ✶✾✽✹✕❚❤✉rst♦♥ ✶✾✾✷✕❊❧r✐❢❛✐✕▼♦rt♦♥ ✶✾✾✹ ◆♦r♠❛❧ ❋♦r♠✳ b = ∆✐♥❢(b)p✶p✷ · · · pℓ pi ∈ P ♦❢ ♠❛①✐♠❛❧ ❧❡♥❣t❤✱ i = ✶, ✷, . . . , ℓ ✭❧❡❢t✲✇❡✐❣❤t❡❞✮✳ ❈♦♠♣❧❡①✐t②✿ |b|✷N ❧♦❣ N✳
G = ❇N✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ A A, B ≤ G, g ∈ G, [A, B] = ✶ b ∈ B ga
= gba K = ga b = gab
A, B ≤ G, g ∈ G, [A, B] = ✶✳ ❇❉❍ Pr♦❜❧❡♠✳ (ga, gb) → gab ✭a ∈ A, b ∈ B✮✳ ❈♦♥❥✉❣❛❝② ❙❡❛r❝❤ Pr♦❜❧❡♠ ✭❈❙P✮✳ gx → ˜ x, gx = g ˜
x ✭g, x ∈ G✮✳
❈❙P✶✳ ga → ˜ a ∈ CG(B), ga = g ˜
a✳
❈❙P✷✳ ga → ˜ a ∈ A, ga = g ˜
a✳
❈❙P✷ ≥ ❈❙P✶ ≥ ❇❉❍ Pr♦❜❧❡♠✳
❇✉r❛✉ ✶✾✸✻✳ σi → Ii−✶ ⊕ ✶ − t t ✶ ✵
▼♦♦❞② ✾✶✱ ▲♦♥❣✕P❛t♦♥ ✾✸✱ ❇✐❣❡❧♦✇ ✾✾✳ ◆♦t ❢❛✐t❤❢✉❧ ❢♦r N ≥ ✺✳ ▲❛✇r❡♥❝❡✕❑r❛♠♠❡r✳ ▲❑: ❇N − → ●▲(N
✷)(Z[t±✶, q±✶])✳
❇✐❣❡❧♦✇ ✷✵✵✶ ✭❏❆▼❙✮✱ ❑r❛♠♠❡r ✷✵✵✷ ✭❆♥♥❛❧s✮✿ ▲❑ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❢❛✐t❤❢✉❧ ❢♦r ❛❧❧ N✳ ❈❤❡♦♥✕❏✉♥ ✷✵✵✸✳ ✶✳ ▲❑ ❊✈❛❧✉❛t✐♦♥✿ ❋❛st✳ ■♥✈❡rs✐♦♥✿ ❘♦✉❣❤❧② N✻ ✭❛❝❝❡♣t❛❜❧❡✮✳ ✷✳ ❙✉✣❝✐❡♥t t♦ ✜♥❞ t❤❡ ❦❡②✬s ✐♠❛❣❡ κ ✐♥ ❛ ✜❡❧❞ Z[t±✶, q±✶]/p, f (t), g(q) ✇✐t❤ κ ♠♦❞ p, f (t), g(q) = κ✳
❇❉❍ Pr♦❜❧❡♠✳ (ga, gb) → gab ✭a ∈ A, b ∈ B✮✳ ❈❤❡♦♥✕❏✉♥ ✷✵✵✸✳ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛tt❛❝❦✳ ❆ss✉♠❡ G ∼ =❡✛ ♠❛tr✐① ❣r♦✉♣✳ ❚❤✐♥❦ G ✐s ❛ ♠❛tr✐① ❣r♦✉♣✳ ga = a−✶ga ⇐ ⇒ a · ga = g · a ❙♦❧✈❡
= g · a a · B = B · a = ⇒ α s✳t✳
= g · α α · B = B · α ❚❤❡♥ gb α = gbα = gαb = (gα)b = ga b = gab = K ! P♦ss✐❜❧②✱ α / ∈ G✱ ❜✉t t❤✐s ✇♦r❦s ✦ ❈♦♠♣❧❡①✐t②✿ (n✷)✸ = N✶✷✳
Pr♦❜❧❡♠✳ ❋✐♥❞ G ✇✐t❤♦✉t ❛♥② r❡♣r❡s❡♥t❛t✐♦♥ t❤❛t ✐s✿ ✶✳ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧✱ ✷✳ ❢❛✐t❤❢✉❧✱ ❛♥❞ ✸✳ ❡✣❝✐❡♥t❧② ❝♦♠♣✉t❛❜❧❡ ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✳
❈❤❛✕❑♦✕▲❡❡✕❍❛♥✕❈❤❡♦♥ ✷✵✵✶✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a✶ ∈ A✶, a✷ ∈ A✷ A✶, A✷, B✶, B✷ ≤ G, g ∈ G b✶ ∈ B✶, b✷ ∈ B✷ a✶ga✷
K = b✶ a✶ga✷ b✷ ❈❤❡♦♥✕❏✉♥ ✷✵✵✸✳ ❙✐♠✐❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ❛tt❛❝❦✿ c = a✶ga✷ ⇐ ⇒ a−✶
✶
· c = g · a✷✳
Pr♦❜❧❡♠✳ ❋✐♥❞ ❛♥ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐① ✐♥ ❛ s✉❜s♣❛❝❡ ♦❢ Mn(F)✳ ❈❤❡♦♥✕❏✉♥ ❍❡✉r✐st✐❝✳ P✐❝❦ ✏r❛♥❞♦♠✑ ❡❧❡♠❡♥ts ✉♥t✐❧ ✐♥✈❡rt✐❜❧❡✳ ❚s✳ ❆ss✉♠❡ s♣❛♥{A✶, . . . , Am} ∩ ●▲n(F) = ✵✳ ❚❤❡♥ Pr(|α✶A✶ + · · · + αmAm| = ✵) ≥ ✶ − n |F|. Pr♦♦❢✿ f (x✶, . . . , xm) := |x✶A✶ + · · · + xmAm| ∈ F[x✶, . . . , xm]✱ ♥♦♥③❡r♦✱ ❞❡❣r❡❡ n✳ ❙❝❤✇❛rt③ ✶✾✽✵✕❩✐♣♣❡❧ ✶✾✽✾ ▲❡♠♠❛✳ f (x✶, . . . , xm) ∈ F[x✶, . . . , xm] ♥♦♥③❡r♦ ❞❡❣r❡❡ n✳ Pr(f (x✶, . . . , xm) = ✵) ≥ ✶ − n |F|.
❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a✶ ∈ G g ∈ G b✷ ∈ G B ≤ CG(a✶)
b✶ ∈ B a✶ga✷
K = b✶a✶ga✷b✷
❚s ✭❢r❡s❤✦✮✳ ❆ss✉♠❡ G ≤ M = Mn(F) ✭❡q✳✱ ❡✛✳ r❡♣r❡s❡♥t❛❜❧❡✮✳ ❑❡② ♦❜s❡r✈❛t✐♦♥s✳ ✶✳ ❈❛♥✬t ❝♦♥str❛✐♥t s♦❧✉t✐♦♥s ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s t♦ ❣r♦✉♣s✱ ❝❛♥ ❝♦♥str❛✐♥t s♦❧✉t✐♦♥s t♦ s✉❜s♣❛❝❡s✦ ✷✳ H = g✶, . . . , gk ≤ G ⇒ CG(H) ⊆ CM(H) = CM(g✶, . . . , gk)✳ ∴ CG(H) ❝♦♠♣✉t❛❜❧❡ ❜② s♦❧✈✐♥❣ xg✶ = g✶x ✳ ✳ ✳ xgk = gkx ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥ t❤❡ n✷ ❡♥tr✐❡s ♦❢ x✱ kn✻ ♦♣❡r❛t✐♦♥s✳ ✸✳ CM(g✶, . . . , gk) ✐s ❛ ✈❡❝t♦r s✉❜s♣❛❝❡ ♦❢ M✳ ✹✳ CM(CM(H)) ❝♦♠♣✉t❛❜❧❡✿ ❞✐♠(CM(H)) ≤ n✷ ❡q✉❛t✐♦♥s✳ ■♥ ✷✱✹✿ ▼❛② ✉s❡ ✐♥st❡❛❞ ❢❡✇ r❛♥❞♦♠ g ∈ H, CM(H)✳
g, a✶, b✷ ∈ G✱ B ≤ CG(a✶)✱ A ≤ CG(b✷)✱ a✷ ∈ A✱ b✶ ∈ B✳ ❙❤♣✐❧r❛✐♥✕❯s❤❛❦♦✈ Pr♦❜❧❡♠✳ (a✶ga✷, b✶gb✷) → a✶b✶ga✷b✷✳ a✷ ∈ A ⇒ a✷ ∈ CM(CM(A)) ⇐ ⇒ a−✶
✷
∈ CM(CM(A))✳ A ≤ CG(b✷) ⇒ b✷ ∈ CG(A) ⊆ CM(A) ⇒ [CM(CM(A)), b✷] = ✶. ❆tt❛❝❦ ✭❚s✮✳ ✶✳ ❈♦♠♣✉t❡ ❜❛s❡s ❢♦r t❤❡ s✉❜s♣❛❝❡s CM(B)✱ CM(CM(A))✳ ✷✳ ❙♦❧✈❡ a✶g = a✶ga✷ · a−✶
✷
✇✐t❤ a✶ ∈ CM(B), a−✶
✷
∈ CM(CM(A)) ✐♥✈❡rt✐❜❧❡✳ ✸✳ ∃ s♦❧✉t✐♦♥✿ (a✶, a−✶
✷ )✳
✹✳ ˜ a✶ b✶gb✷ ˜ a✷
!
= b✶˜ a✶g ˜ a✷b✷ = b✶a✶ga✷b✷ = K ! ✺✳ ❈♦♠♣❧❡①✐t② ≤ n✷ · (n✷)✸ = N✶✻✱ ❤❡✉r✐st✐❝❛❧❧② N✶✷✳ ◆♦t ♣r❛❝t✐❝❛❧✱ ❜✉t ✇♦rst✲❝❛s❡ ♣♦❧②t✐♠❡✳
❆♥s❤❡❧✕❆♥s❤❡❧✕●♦❧❞❢❡❧❞ ✶✾✾✾✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ v(x✶, . . . , xk) ∈ Fk a✶, . . . , ak ≤ G w(x✶, . . . , xk) ∈ Fk a = v(a✶, . . . , ak) b✶, . . . , bk ≤ G b = w(b✶, . . . , bk) b✶a, . . . , bka
K = w(b✶a, . . . , bka)−✶b a−✶v(a✶b, . . . , akb) = a−✶ab = a−✶b−✶ab = (ba)−✶b = w(b✶a, . . . , bka)−✶b
a ∈ a✶, . . . , ak, b ∈ b✶, . . . , bk ≤ G✳ ❈♦♠♠✉t❛t♦r ❑❊P Pr♦❜❧❡♠✳ (b✶a, . . . , bka, a✶b, . . . , akb) → a−✶b−✶ab. ❈♦♥❥✉❣❛❝② ❙❡❛r❝❤ Pr♦❜❧❡♠ ✭❈❙P✮✳ gx → ˜ x, gx = g ˜
x✳
▼✉❧t✐♣❧❡ ❈❙P✳ (g✶x, . . . , gkx) → ˜ x, (g✶x, . . . , gkx) = (g✶˜
x, . . . , gk ˜ x)✳
▼✉❧t✐♣❧❡ ❈❙P ✐s ❡❛s② ✐♥ ♠❛tr✐① ❣r♦✉♣s✳
a ∈ a✶, . . . , ak, b ∈ b✶, . . . , bk ≤ G✳ ❈♦♠♠✉t❛t♦r ❑❊P Pr♦❜❧❡♠✳ (b✶a, . . . , bka, a✶b, . . . , akb) → a−✶b−✶ab. ❚s✱ ▲✐♥❡❛r ❈❡♥tr❛❧✐③❡r ❆tt❛❝❦ ✭❢r❡s❤✦✮✳ ❲▲❖● G ✐s ❛ ♠❛tr✐① ❣r♦✉♣✳ ✶✳ ❈♦♠♣✉t❡ ❛ ❜❛s❡ ❢♦r CM(CM(b✶, . . . , bk))✳ ✷✳ ❙♦❧✈❡ b✶a = a · b✶a ✳ ✳ ✳ bka = a · bka ; a✶b = b · a✶b ✳ ✳ ✳ akb = b · akb ✇✐t❤ a ✐♥✈❡rt✐❜❧❡✱ b ∈ CM(CM(b✶, . . . , bk)) ✐♥✈❡rt✐❜❧❡✳ ✸✳ ∃ s♦❧✉t✐♦♥✿ (a, b)✳ ˜ a−✶˜ b−✶˜ a˜ b = ˜ a−✶˜ b−✶(˜ aa−✶a)˜ b = ˜ a−✶(˜ aa−✶)˜ b−✶a˜ b = a−✶a
˜ b = a−✶ab = K !
✳ ✳ ✳ ❛♥❞ ✇♦rs❡✿ ♦❢ ♠② ❧❡❝t✉r❡ s❡r✐❡s❄ ◆♦t q✉✐t❡✿ ✶✳ N✶✷ ✐s ✐♠♣r❛❝t✐❝❛❧✿ ✷✾✻ ✭t✐♠❡s ❝♦♥st❛♥ts✮ ❢♦r N = ✷✺✻✳ ✷✳ ❚❤❡r❡ ❛r❡ ❛❞❞✐t✐♦♥❛❧ ❜r❛✐❞✲P❑❈ ♣r♦♣♦s❛❧s ✭❉❡❤♦r♥♦②✱ ❑❛❧❦❛✱✳ ✳ ✳ ✮✳ ✸✳ ❚❤❡ ♦t❤❡r ♣r♦❜❧❡♠s ✭❈❙P✱ ▼✉❧t✐♣❧❡ ❈❙P✱✳ ✳ ✳ ✮ r❡♠❛✐♥ ♦♣❡♥✳ ▲✐♥❡❛r ❈❡♥tr❛❧✐③❡r ❆tt❛❝❦s s❡❡♠ ❛♣♣❧✐❝❛❜❧❡ t♦ s♦♠❡ ♦❢ t❤❡ ♦t❤❡r ❑❊Ps✳ Pr♦❜❛❜❧② ♥♦t ❛❧❧✿ ❋✐❛t✕❙❤❛♠✐r ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s❡❞ ♦♥ ❈❙P✱ ❡t❝✳ ❚❤❡ ♦♥❧② ✇❛② t♦ r✉❧❡ ♦✉t ✭♠♦st ♦❢✮ t❤✐s ❛♣♣r♦❛❝❤ ✐s t♦ s♦❧✈❡ t❤❡ ❈❙P✳
❆ss✉♠❡✿ ❋✐♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ❡✣❝✐❡♥t❧② s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠ ✭❜❡tt❡r✿ ♥♦r♠❛❧ ❢♦r♠✮✳ ❈♦♥❥✉❣❛❝② ❙❡❛r❝❤ Pr♦❜❧❡♠ ✭❈❙P✮✳ gx → ˜ x, gx = g ˜
x ✭g, x ∈ G✮✳
❘♦♦t ❙❡❛r❝❤ Pr♦❜❧❡♠✳ x✷ → ˜ x, x✷ = ˜ x✷✳ ❉♦✉❜❧❡ ❈♦s❡t Pr♦❜❧❡♠✳ agb ∈ AgB → ˜ a ∈ A, ˜ b ∈ B✱ agb = ˜ ag ˜ b✳ H✶, . . . , Hk ≤ G✱ w(t✶, . . . , tk+m) ∈ Fk+m✱ p✶, . . . , pm ∈ G✳ ❙♦❧✉t✐♦♥ ❙❡❛r❝❤ Pr♦❜❧❡♠✳ w(h✶, . . . , hk, p✶, . . . , pm) → ˜ h✶ ∈ H✶, . . . , ˜ hk ∈ Hk✱ w(h✶, . . . , hk, p✶, . . . , pm) = w(˜ h✶, . . . , ˜ hk, p✶, . . . , pm)✳
❙♦❧✉t✐♦♥ ❙❡❛r❝❤ Pr♦❜❧❡♠✳ w(h✶, . . . , hk, p✶, . . . , pm) → ˜ h✶ ∈ H✶, . . . , ˜ hk ∈ Hk✱ w(h✶, . . . , hk, p✶, . . . , pm) = w(˜ h✶, . . . , ˜ hk, p✶, . . . , pm)✳ ❖❜s❡r✈❛t✐♦♥s✳ ❙✉✣❝❡s t♦✿ ✶✳ ❋✐♥❞ t❤❡ ❧❡❛❞✐♥❣ ✈❛r✐❛❜❧❡✳ ✷✳ ❋✐♥❞ ❛ ✏s♠❛❧❧✑ ❧✐st ❝♦♥t❛✐♥✐♥❣ t❤❡ s♦❧✉t✐♦♥✳ ▲❡♥❣t❤✲❜❛s❡❞ ❛❧❣♦r✐t❤♠s✳ ❋✐♥❞ ❧❡❛❞✐♥❣ ✈❛r✐❛❜❧❡ ✰ ❡①♣r❡ss✐♦♥ ✐♥ ✐ts s✉❜❣r♦✉♣✳ ❚♦♦ ❛♠❜✐t✐♦✉s✱ ❜✉t t❤❡② ❛r❡ ❤❡✉r✐st✐❝✳ ❆ss✉♠♣t✐♦♥s✿ ✶✳ h✶, . . . , hk s❛♠♣❧❡❞ ✭s♦♠❡✇❤❛t✮ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ✷✳ ∃ ✏✇❡❧❧✲❜❡❤❛✈❡❞✑ ❧❡♥❣t❤ ❢✉♥❝t✐♦♥✿ ❯s✉❛❧❧② ℓ(hg) > ℓ(g)✳
G = g✶, . . . , gn ✭s②♠♠❡tr✐❝ ❣❡♥❡r❛t✐♥❣ s❡t✮✳
gx = g−✶
ik g−✶ ik−✶ · · · g−✶ i✶ ggi✶ · · · gik−✶gik
gxg−✶
j
= gjg−✶
ik g−✶ ik−✶ · · · g−✶ i✶ ggi✶ · · · gik−✶gikg−✶ j
gxg−✶
ik
= g−✶
ik−✶ · · · g−✶ i✶ ggi✶ · · · gik−✶
❍♦♣❡❢✉❧❧②✱ s❤♦rt❡st ❧❡♥❣t❤ ❢♦r gik✳ P❡❡❧ ♦✛ gik ❛♥❞ ❝♦♥t✐♥✉❡ t♦ gik−✶ ❡t❝✳ ▼❛② ✉s❡ {g✶, . . . , gn}m ❛s ❣❡♥❡r❛t♦rs✳ ❈♦♠♣❧❡①✐t②✿
k m · nm✳
■♥ ❇N✿ ❯s❡ ℓ(g) = ❧❡♥❣t❤ ♦❢ t❤❡ ♥♦r♠❛❧ ❢♦r♠ ♦❢ g✳ ◆♦ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❣✐✈❡♥✳
P❛t❡rs♦♥✕❘❛③❜♦r♦✈ ✶✾✾✶✳ ▼✐♥✐♠❛❧ ❧❡♥❣t❤ ✐♥ ❇ ✐s ◆P✲❤❛r❞✳ P❛t❡rs♦♥✕❘❛③❜♦r♦✈ ✶✾✾✶✳ ■s ▼✐♥✐♠❛❧ ❧❡♥❣t❤ ✐♥ ❇N ♣♦❧②✲t✐♠❡❄ ❇❡r❣❡r ✶✾✾✹✳ ❨❡s ✐♥ ❇✸✳ ❇✐r♠❛♥✳ ■s ▼✐♥✐♠❛❧ ❧❡♥❣t❤ ✐♥ ❇ ✐s ◆P✲❤❛r❞ ❢♦r ❇❑▲ ❣❡♥❡r❛t♦rs❄ ❍♦❝❦✕❚s ✷✵✶✵✳ ℓ(b) ≤ ℓR(b) ≤ (|∆| − ✶)ℓ(b) ✐♥ ❇N✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ℓR(b) = ℓ(b) ✐♥ ❇✸✳ ❍♦❝❦✕❚s ✷✵✶✵✳ ❆♣♣r♦①✐♠❛t❡ ❆rt✐♥ ❧❡♥❣t❤ ✉s✐♥❣ ❇❑▲ ℓR✳ ✭❆✳●✳✮ ▼②❛s♥✐❦♦✈✕❙❤♣✐❧r❛✐♥✕❯s❤❛❦♦✈ ✷✵✵✻✳ ❊①♣❡r✐♠❡♥t❛❧❧②✿ ❉❡❤♦r♥♦② ❤❛♥❞❧❡ r❡❞✉❝t✐♦♥ ✰ ∆✲❝♦♥❥✉❣❛t✐♦♥ ❣✐✈❡s ❡①❝❡❧❧❡♥t ❧❡♥❣t❤ ❢✉♥❝t✐♦♥✳
✶✳ ▲❡♥❣t❤ ♦❢ r❛t✐♦♥❛❧ ❢♦r♠ ❜❡tt❡r t❤❛♥ ♥♦r♠❛❧ ❢♦r♠✳ ✷✳ ❍✉❣❤❡s✕❚❛♥♥❡♥❜❛✉♠ ▲❇❆ s✉❝❝❡❡❞s ♦♥❧② ❢♦r t♦② ♣❛r❛♠❡t❡rs✱ ✇✐t❤ ❧♦♥❣ ❣❡♥❡r❛t♦rs✳
▼✉❝❤ ❜❡tt❡r✱ ❜✉t ❛❧s♦ ♥❡❡❞s s♦♠❡✇❤❛t ❧♦♥❣ ❣❡♥❡r❛t♦rs✳ ✭❆✳❉✳✮ ▼②❛s♥✐❦♦✈✕❯s❤❛❦♦✈ ✷✵✵✼✳ ❱❛r✐❛t✐♦♥ ♦❢ ▼❡♠♦r②✲❡♥❤❛♥❝❡❞ ▲❇❆✿ ❑❡❡♣ ❛❧❧ ✭❛♥❞ ♦♥❧②✮ t❤❡ st❡♣s r❡❞✉❝✐♥❣ ❧❡♥❣t❤✳ ❆❣❛✐♥st ❈♦♠♠✉t❛t♦r ❑❊P ✐♥ B✽✵✿ ✶✳ ❱❡r② s✉❝❝❡ss❢✉❧ ✇❤❡♥ |gi| ≥ ✷✵✳ ✷✳ ❋❛✐❧s ✇❤❡♥ |gi| ≤ ✶✵✳ ❚❤❡ ❈♦♠♠✉t❛t♦r ❑❊P ✇❛s ♥❡✈❡r ❛tt❛❝❦❡❞ ❢♦r |gi| ≈ ✶✵✳
❚❤❡ ❤❛r❞❡st ❝❛s❡ ❢♦r ▲❇❆✿ ✶✳ ♦♥❡ ✐♥st❛♥❝❡✱ ✷✳ s❤♦rt ❣❡♥❡r❛t♦rs✱ ✸✳ ♠❛♥② r❡❧❛t✐♦♥s✳ ❋♦r r❡❛s♦♥❛❜❧❡ ♣❛r❛♠❡t❡rs✿ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✿ ✵✪✳ ❋♦r ❛❧❧ ♠❡♥t✐♦♥❡❞ ❛❧❣♦r✐t❤♠s✳
❊①❛♠♣❧❡ ✶✳ g ❝♦♥❥✉❣❛t❡ t♦ h := gb ✭g, b ∈ ❇N ✐♥❞❡♣❡♥❞❡♥t✮✳ ❘❡❞✉❝✐♥❣ g ❧❡♥❣t❤ ✇♦♥✬t ❣❡t ✉s t♦ h✦ ❊①❛♠♣❧❡ ✷✳ g := uv ❝♦♥❥✉❣❛t❡ t♦ h := vu ✭u, v ∈ ❇N ✐♥❞❡♣❡♥❞❡♥t✮✳ ❚❤❡ ▲❇❆ ❤❡✉r✐st✐❝ ✐s ♠❡❛♥✐♥❣❧❡ss ❤❡r❡✳ ❑♦✈❛❧②♦✈❛✕❚s❛❜❛♥ ✷✵✶✵✳ ❙♦❧✉t✐♦♥✿ ▼❡❡t ✐♥ t❤❡ ▼✐❞❞❧❡ ✭♠❡♠♦r②✲❡♥❤❛♥❝❡❞✮ ▲❇❆✳
■❞❡❛ s✐♠✐❧❛r t♦ ❆✯ ❛❧❣♦r✐t❤♠ ❢♦r s❤♦rt❡st ♣❛t❤s ✐♥ ❛ ❣r❛♣❤✳
❆ss✉♠♣t✐♦♥✳ {h ∈ gG : ℓ(h) ≤ K} ✜♥✐t❡✳ ❈♦♠♣❧❡①✐t②✳ ❍❡✉r✐st✐❝❛❧❧②✱ √ M✱ M = |{h ∈ gG : ℓ(h) ✭♥❡❛r✮ ♠✐♥✐♠❛❧}|.
❆❧❣♦r✐t❤♠✳ ■♥♣✉t✿ ❈♦♥❥✉❣❛t❡ g, h✳ Sg := ∅, Sh := ∅✳ g✵ := g, h✵ := h✳ ▲♦♦♣ ✉♥t✐❧ ❛ ❝♦♠♣✉t❡❞ ❝♦♥❥✉❣❛t❡ ♦❢ h ✐s ✐♥ Sg✱ ♦r ✈✐❝❡ ✈❡rs❛✳ ✶✳ ❆❞❞ ❛❧❧ ❝♦♥❥✉❣❛t❡s ♦❢ g✵ ❜② ❣❡♥❡r❛t♦rs t♦ Sg✳ ✷✳ ❆❞❞ ❛❧❧ ❝♦♥❥✉❣❛t❡s ♦❢ h✵ ❜② ❣❡♥❡r❛t♦rs t♦ Sh✳ ✸✳ g✵ ∈r♥❞ ℓ✲♠✐♥✐♠❛❧ ❡❧❡♠❡♥ts ♦❢ Sg ♥♦t t❛❦❡♥ ❜❡❢♦r❡✳ ✹✳ h✵ ∈r♥❞ ℓ✲♠✐♥✐♠❛❧ ❡❧❡♠❡♥ts ♦❢ Sh ♥♦t t❛❦❡♥ ❜❡❢♦r❡✳ ❋✐♥✐t❡ t✐♠❡✳ ❊✈❡r② ❞♦❣ ❤❛s ✐ts ❞❛②✿ {h ∈ gG : ℓ(h) ≤ K} ✜♥✐t❡✳ ❊①❛♠♣❧❡✳ ❇✶✻✱ g, x ∈ {σ±✶
✶ , . . . , σ±✶ N−✶}✸✷✱ (g, gx)✳
❊①❈❆◆✶✻▲✸✷✳t①t
▼❡t❤♦❞♦❧♦❣②✳ ❊✣❝✐❡♥t❧② ❝♦♠♣✉t❛❜❧❡✿ ✶✳ g → ✜♥✐t❡ Ig ⊆ gG❀ ✷✳ g ∼ h ⇒ Ig = Ih❀ ✸✳ x ✇✐t❤ gx ∈ Ig❀ ✹✳ ❈♦♠♣✉t❡ Ig ❢r♦♠ ❛♥② s✐♥❣❧❡ ❡❧❡♠❡♥t✱ ❜② ❝♦♥❥✉❣❛t✐♦♥s✳ ❈❙P ❙♦❧✉t✐♦♥✳ ●✐✈❡♥ g ∼ h✿ ✶✳ ❈♦♥❥✉❣❛t❡ g ✐♥t♦ Ig✳ ✷✳ ❈♦♥❥✉❣❛t❡ h ✐♥t♦ Ih = Ig✳ ✸✳ ❇✉✐❧❞ Ig ❜② ❝♦♥❥✉❣❛t✐♦♥s ❢r♦♠ g✱ ✉♥t✐❧ h✬s ❝♦♥❥✉❣❛t❡ ✐s ❢♦✉♥❞✳ ❍❡✉r✐st✐❝✳ ▼♦r❡ ❡✣❝✐❡♥t❧②✱ ❜✉✐❧❞ Ig, Ih ✉♥t✐❧ t❤❡② ♠❡❡t✳ ❋♦r ❈♦♥❥✉❣❛❝② ❉❡❝✐s✐♦♥ Pr♦❜❧❡♠✿ Ih ∩ Ig ✐♥t❡rs❡❝t❄
❚❤✐♥❦ r✐♥❣✳ ❘❡❞✉❝❡ ❝②❝❧✐❝❛❧❧② ✭❡q✉✐✈❛❧❡♥t❧②✱ ❝②❝❧❡✮✳ y−✶x−✶x−✶xyyxxy−✶xxy x−✶x−✶xyyxxy−✶xx x−✶xyyxxy−✶x xyyxxy−✶ x−✶y−✶xxy−✶xyyyx y−✶xxy−✶xyyy xxy−✶xyy xy−✶xyyx y−✶xyyxx xyyxxy−✶ Ig := ❛❧❧ ❝②❝❧✐❝ r♦t❛t✐♦♥s ♦❢ t❤❡ ❝②❝❧✐❝❛❧❧② r❡❞✉❝❡❞ ❢♦r♠ ♦❢ g = ❈②❝❧❡ ♦❢ t❤❡ ❝②❝❧✐♥❣ ♦r❜✐t ♦❢ g✳
b ≤ c✿ bp = c✱ p ∈ ❇N+✳ ▲❡❢t ✐♥✈❛r✐❛♥t✿ b ≤ c ⇒ db ≤ dc✳ ∆i ≤ ∆ip✶ · · · pℓ
≤ ∆i+ℓ. ❈❛♥♦♥✐❝❛❧ ❧❡♥❣t❤ ♦❢ b✿ ℓ✳ ✐♥❢(b) := i s✉♣(b) := i + ℓ b ∈ [i, i + ℓ] = [✐♥❢(b), s✉♣(b)] b ∈ [i, ∞)✿ i ≤ ✐♥❢(b)✳
❡①♣s✉♠: ❇N → Z s✉♠ ♦❢ ❡①♣♦♥❡♥ts✳ ❲❡❧❧✲❞❡✜♥❡❞❀ ❝♦♥❥✲✐♥✈❛r✐❛♥t✳
❋✐♥✐t❡ ♥♦♥❡♠♣t② ❝♦♥❥✉❣❛❝② ✐♥✈❛r✐❛♥t✳ ❈❢✳ ▲❇❆✦ ❆❧❧ ❡❧❡♠❡♥ts ♦❢ ❙❙(b) ❤❛✈❡ t❤❡ s❛♠❡ ✐♥❢✱ ✐♥❢(b)✳ ❈❧❛ss✐❝❛❧❧②✱ ✐♥❢(b) = ♠❛①(✐♥❢(b❇N))✱ ❙❙(b) := b❇N ∩ [✐♥❢(b), ∞)✳ ❊❧r✐❢❛✐✕▼♦rt♦♥ ✶✾✾✹✳ ▼✐♥✐♠✐③❡ ❛❧s♦ t❤❡ ❝❛♥♦♥✐❝❛❧ ❧❡♥❣t❤ ♦❢ p✳ ❙✉♣❡r ❙✉♠♠✐t ❙❡t✿ ❙❙❙(b) := {∆ip ∈ b❇N : p ♠✐♥✐♠❛❧ ❧❡♥❣t❤ ❛♥❞ ❝❛♥♦♥✐❝❛❧ ❧❡♥❣t❤}✳ ❆❧❧ ❡❧❡♠❡♥ts ♦❢ ❙❙(b) ❤❛✈❡ t❤❡ s❛♠❡ s✉♣✱ s✉♣(b)✳ ❈❧❛ss✐❝❛❧❧②✱ s✉♣(b) = ♠✐♥(s✉♣(❙❙(b)))✱ ❙❙❙(b) = b❇N ∩ [✐♥❢(b), s✉♣(b)]✳
■♥ t❤❡ ❢r❡❡ ❣r♦✉♣✱ ❝②❝❧✐♥❣ ❜r✐♥❣s g t♦ t❤❡ ❝♦♥❥✉❣❛❝② ✐♥✈❛r✐❛♥t s❡t✳ ❈②❝❧✐♥❣ ✐♥ ❇N✿ ∆ip✶p✷ · · · pℓ = p✶∆ip✷ · · · pℓ − → ∆ip✷ · · · pℓp✶, ❛♥❞ ♠♦✈✐♥❣ t♦ ♥♦r♠❛❧ ❢♦r♠✳ ❈♦♥❥✉❣❛t✐♦♥ ❜② p✶ = p✶∆i✳ i ♠❛② ♦♥❧② ✐♥❝r❡❛s❡✱ ℓ, |p| ♠❛② ♦♥❧② ❞❡❝r❡❛s❡✳ ❊❧r✐❢❛✐✕▼♦rt♦♥ ✶✾✾✹✱ ❇✐r♠❛♥✕❑♦✕▲❡❡ ✷✵✵✶✳ ❈②❝❧✐♥❣ |∆| t✐♠❡s ✐♥❝r❡❛s❡s ✐♥❢(b) ✭✐❢ ♥♦t ♠❛①✐♠❛❧✮✳ ❉❡❈②❝❧✐♥❣✿ ∆ip✶ · · · pℓ−✶pℓ − → pℓ∆ip✶ · · · pℓ−✶ = ∆ipℓp✶ · · · pℓ−✶ ✰ ♥♦r♠❛❧ ❢♦r♠✳ ❙❛♠❡ r❡s✉❧ts✱ ❢♦r s✉♣✳
❊❧r✐❢❛✐✕▼♦rt♦♥ ❈♦♥✈❡①✐t②✳ ❙❙❙(b) ✐s ❝♦♥♥❡❝t❡❞ ❜② ❝♦♥❥✉❣❛t✐♦♥s ❜② ♣❡r♠✉t❛t✐♦♥ ❜r❛✐❞s✳ ❈♦♠♣❧❡①✐t②✿ | ❙❙❙(b)| · N!✳ ❋♦r a, b ≥ ✶✿ ∃ a ∧ b = ♠❛①✐♠❛❧ d ≤ a, b✳ ❋r❛♥❝♦✕●♦♥③❛❧❡③✲▼❡♥❡s❡s ✷✵✵✸✳ x, y ∈ P✱ g, gx, gy ∈ ❙❙❙(b) ⇒ gx∧y ∈ ❙❙❙(b)✳ ∴ ❊♥♦✉❣❤ t♦ ❝♦♥s✐❞❡r ♠✐♥✐♠❛❧ ♣❡r♠✉t❛t✐♦♥ ❜r❛✐❞s ❛❜♦✈❡ σ✶, . . . , σN−✶✳ ❈♦♠♣❧❡①✐t②✿ | ❙❙❙(b)| · N = N · | ❙❙❙(b)|✳ ❚②♣✐❝❛❧❧② ❤✉❣❡✦
■♥ t❤❡ ❢r❡❡ ❣r♦✉♣✱ Ig = ❝②❝❧❡ ♦❢ t❤❡ ❝②❝❧✐♥❣ ♦r❜✐t ♦❢ g✳ ❯❙❙(b) := ❛❧❧ ❝②❝❧❡s ♦❢ ❝②❝❧✐♥❣ ♦r❜✐ts ✐♥ ❙❙❙(g)✳
❈♦♠♣❧❡①✐t②✿ n · | ❯❙❙(b)|✳ ❚②♣✐❝❛❧❧②✱ | ❯❙❙(b)| ✐s ❧✐♥❡❛r ✐♥ |b|✳ ✭▼❛② ❜❡ ❡①♣♦♥❡♥t✐❛❧✳✮ ▲❡❡ ✷✵✵✵✳ ❘❙❙❙(b) ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❝②❝❧✐♥❣ ❛♥❞ ❞❡❝②❝❧✐♥❣ ♦r❜✐ts ✭♥♦ ♠✐♥✐♠❛❧ ♣❜✬s✮✳
♠✐♥✐♠❛❧ ♣❜✬s✮✳ ❙❈(❜) ⊆ ❙❙❙❘(b) ⊆ ❯❙❙(b) ⊆ ❙❙❙(b) ⊆ ❙❙(b). ✭❚②♣♦ ✐♥t❡♥t✐♦♥❛❧✳✮
❙❈(❜) ⊆ ❙❙❙❘(b) ⊆ ❯❙❙(b) ⊆ ❙❙❙(b) ⊆ ❙❙(b). ❆♥✕❑♦ ✷✵✶✷✿ ✶✳ ❈❙P ❢♦r ♣s❡✉❞♦✲❆♥♦s♦✈ ❜r❛✐❞s ❜♦✐❧s ❞♦✇♥ t♦ ❈❙P ❢♦r r✐❣✐❞ ♣s❡✉❞♦✲❆♥♦s♦✈ ❜r❛✐❞s✳ ✷✳ ❚❤❡r❡✱ ❙❈(❜) = ❘❙❙❙(b) = ❯❙❙(b)✳ ✸✳ ∃ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ✇✐t❤ | ❙❈(b)| ≥ ✷N/✷✳ ❚s✳ ❊①♣❡r✐♠❡♥t❛❧❧②✿ ❙✐♠♣❧❡✱ ❤✐❣❤✲❡♥tr♦♣② ❞✐str✐❜✉t✐♦♥ ♦♥ ❇N ✇✐t❤ | ❯❙❙(b)| ≥ ✷N−✷ ✐♥ ♣r♦❜❛❜✐❧✐t② ✶ − ✷−N/✷✿ P✐❝❦ b :=r♥❞ σ±✶
i✶ · · · σ±✶ iN
✉♥t✐❧ b ∈ ❯❙❙(b) ❛♥❞ ❤❛s ❝❛♥♦♥✐❝❛❧ ❧❡♥❣t❤ ≥ N
✹ ✳
❈♦♥❝❡♥tr❛t✐♦♥ ♦❢ ♠❡❛s✉r❡✳ ❇✷✵✱ ✶, ✵✵✵ tr✐❡s✿ | ❯❙❙(b)| ≥ ✷✶✼.✸✳ ❍✐❣❤ ❡♥tr♦♣②✳ ◆♦ ❜✐rt❤❞❛② ✐♥ ✷✶✹ s❛♠♣❧❡s✳
❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❯❙❙(b) ❢♦r b =r♥❞ σ±✶
i✶ · · · σ±✶ iN ∈ ❇N
❦✐❧❧s ♠② ✭✽✲❝♦r❡ ✽●❇ ❘❆▼✮ ❝♦♠♣✉t❡r ❛❧r❡❛❞② ❢♦r N = ✸✷✳ ❆♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ ▲❇❆✯✱ ❤♦✇❡✈❡r✱ s✉❝❝❡❡❞s t❤❡r❡✳ ❍♦♠♦♠♦r♣❤✐❝ ♣r❡✐♠❛❣❡ ✐♥✈❛r✐❛♥ts✳ ❖♥ ❜♦❛r❞✱ ■❨✧❍✿ ✶✳ ❱❡rs❤✐❦✬s ❣r♦✉♣ ❱❀ ✷✳ ▲✐♥❡❛r t✐♠❡ ♥♦r♠❛❧ ❢♦r♠ ✐♥ ❱❀ ✸✳ ▲✐♥❡❛r t✐♠❡ ❝♦♥❥✉❣❛❝② ♥♦r♠❛❧ ❢♦r♠ ✐♥ ❱❀ ✹✳ ❚❤❡ ❤②❜r✐❞ ✇✐t❤ ▲❇❆✯ ✐♥ ❇✳