rrs rtstrs - - PowerPoint PPT Presentation
rrs rtstrs rtrst rs r r rst Prrs
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✱ ❙❧♦✈❡♥✐❛✮
❏♦✐♥t ✇♦r❦ ✇✐t❤ Pr✐♠♦➸ P♦t♦↔♥✐❦ ❛♥❞ ●❛❜r✐❡❧ ❱❡rr❡t
✶✳✻✳✷✵✶✽
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ)✱ E(Γ) ❛♥❞ A(Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ✳ ❆✉t♦♠♦r♣❤✐s♠ ♦❢ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ✳ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ ✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r ✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✱ ♦r ✱ r❡s♣❡❝t✐✈❡❧②✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ)✱ E(Γ) ❛♥❞ A(Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ✳ ❆✉t♦♠♦r♣❤✐s♠ ♦❢ Γ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ V (Γ) ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t(Γ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ✳ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ ✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r ✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✱ ♦r ✱ r❡s♣❡❝t✐✈❡❧②✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ s✐♠♣❧❡✱ ✜♥✐t❡ ❛♥❞ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❲✐t❤ V (Γ)✱ E(Γ) ❛♥❞ A(Γ) ✇❡ ❞❡♥♦t❡ r❡s♣❡❝t✐✈❡❧② t❤❡ ✈❡rt❡①✲s❡t✱ t❤❡ ❡❞❣❡✲s❡t ❛♥❞ t❤❡ ❛r❝✲s❡t ♦❢ Γ✳ ❆✉t♦♠♦r♣❤✐s♠ ♦❢ Γ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ V (Γ) ♣r❡s❡r✈✐♥❣ t❤❡ ❛❞❥❛❝❡♥❝② r❡❧❛t✐♦♥✳ ❆✉t(Γ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ✳ ❆ ❣r❛♣❤ Γ ✐s s❛✐❞ t♦ ❜❡ G✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ G✲❡❞❣❡✲tr❛♥s✐t✐✈❡ ♦r G✲❛r❝✲tr❛♥s✐t✐✈❡ ✐❢ G ≤ ❆✉t(Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ V (Γ)✱ E(Γ) ♦r A(Γ)✱ r❡s♣❡❝t✐✈❡❧②✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ■❢ ✐s ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ■❢ ✐s ✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ ♥❛t✉r❛❧ ❛♣♣r♦❛❝❤ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ▲❛r❣❡r ♦❜❥❡❝ts ❤❛✈❡ t❤❡ ♣♦t❡♥t✐❛❧ t♦ ❛❞♠✐t ♠✉❝❤ ❧❛r❣❡r ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s✳ ■❢ Γ ✐s G✲✈❡rt❡①✲tr❛♥s✐t✐✈❡✱ t❤❡♥ |G| = |V (Γ)| · |Gv|✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ❝✉❜✐❝ ✐❢ ❡❛❝❤ ✈❡rt❡① ✐s ✐♥❝✐❞❡♥t ✇✐t❤ t❤r❡❡ ❡❞❣❡s✳ ❚❤❡♦r❡♠ ✭❚✉tt❡✱ ✶✾✹✻✮ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ✭✸✲r❡❣✉❧❛r✮ G✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❧❡t v ∈ V (Γ)✳ ❚❤❡♥ |Gv| = ✸ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s ♦❢ Γ❀ |Gv| = ✻ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✷✲❛r❝s ♦❢ Γ❀ |Gv| = ✶✷ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✸✲❛r❝s ♦❢ Γ❀ |Gv| = ✷✹ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✹✲❛r❝s ♦❢ Γ❀ |Gv| = ✹✽ ❛♥❞ G ❛❝ts r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ✺✲❛r❝s ♦❢ Γ✳ ❋♦r ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r s✱ ❛♥ s✲❛r❝ ✐♥ ❛ ❣r❛♣❤ Γ ✐s ❛♥ (s + ✶)✲t✉♣❧❡ ♦❢ ✈❡rt✐❝❡s v✵, v✶, . . . , vs✱ s✉❝❤ t❤❛t vi−✶ ✐s ❛❞❥❛❝❡♥t ✇✐t❤ vi ✭❢♦r ✶ ≤ i ≤ s✮ ❛♥❞ vi−✶ = vi+✶✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✐s t❤❡ ❣r❛♣❤ G[H] ✇✐t❤ ✈❡rt❡① s❡t V (G) × V (H)✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s (x✶, y✶) ❛♥❞ (x✷, y✷) ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r {x✶, x✷} ∈ E(G) ♦r x✶ = x✷ ❛♥❞ {y✶, y✷} ∈ E(H)✳ ❊①❛♠♣❧❡ C✻[K✷]
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊①❛♠♣❧❡ ▲❡t Γ = Cn[K✷] ✭❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ Cn ✇✐t❤ K✷✮✳ ❚❤❡♥ Γ ✐s ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❆✉t(Γ) = Z✷ ≀ D✷n ∼ = Zn
✷ ⋊ D✷n ✭❢♦r n ≥ ✺✮✳
❍❡♥❝❡ ❆✉t ✷ ✷ ❛♥❞ ❆✉t ✷ ✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❜♦✉♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ♦❢ ✹✲✈❛❧❡♥t✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ ❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ❛❝t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ♦♥ ✐ts ♥❡✐❣❤❜♦✉rs ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✽✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②
✽✳
■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ●❛r❞✐♥❡r ✭✶✾✼✸✮ t❤❛t ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t✱ ✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ✷✹✸✻✱ ✉♥❧❡ss t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ ✐s ✐s♦♠♦r♣❤✐❝ t♦
✽✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊①❛♠♣❧❡ ▲❡t Γ = Cn[K✷] ✭❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ Cn ✇✐t❤ K✷✮✳ ❚❤❡♥ Γ ✐s ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❆✉t(Γ) = Z✷ ≀ D✷n ∼ = Zn
✷ ⋊ D✷n ✭❢♦r n ≥ ✺✮✳
❍❡♥❝❡ |❆✉t(Γ)| = ✷n · ✷n ❛♥❞ |❆✉t(Γ)v| = ✷n✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❜♦✉♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ♦❢ ✹✲✈❛❧❡♥t✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ ❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ❛❝t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ♦♥ ✐ts ♥❡✐❣❤❜♦✉rs ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✽✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②
✽✳
■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ●❛r❞✐♥❡r ✭✶✾✼✸✮ t❤❛t ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t✱ ✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ✷✹✸✻✱ ✉♥❧❡ss t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ ✐s ✐s♦♠♦r♣❤✐❝ t♦
✽✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊①❛♠♣❧❡ ▲❡t Γ = Cn[K✷] ✭❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ Cn ✇✐t❤ K✷✮✳ ❚❤❡♥ Γ ✐s ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❆✉t(Γ) = Z✷ ≀ D✷n ∼ = Zn
✷ ⋊ D✷n ✭❢♦r n ≥ ✺✮✳
❍❡♥❝❡ |❆✉t(Γ)| = ✷n · ✷n ❛♥❞ |❆✉t(Γ)v| = ✷n✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❜♦✉♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ♦❢ ✹✲✈❛❧❡♥t✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ ❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ❛❝t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ♦♥ ✐ts ♥❡✐❣❤❜♦✉rs ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✽✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②
✽✳
■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ●❛r❞✐♥❡r ✭✶✾✼✸✮ t❤❛t ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t✱ ✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ✷✹✸✻✱ ✉♥❧❡ss t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ ✐s ✐s♦♠♦r♣❤✐❝ t♦
✽✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊①❛♠♣❧❡ ▲❡t Γ = Cn[K✷] ✭❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ Cn ✇✐t❤ K✷✮✳ ❚❤❡♥ Γ ✐s ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❆✉t(Γ) = Z✷ ≀ D✷n ∼ = Zn
✷ ⋊ D✷n ✭❢♦r n ≥ ✺✮✳
❍❡♥❝❡ |❆✉t(Γ)| = ✷n · ✷n ❛♥❞ |❆✉t(Γ)v| = ✷n✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❜♦✉♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ♦❢ ✹✲✈❛❧❡♥t✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ ❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ❛❝t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ♦♥ ✐ts ♥❡✐❣❤❜♦✉rs ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✽✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜② D✽✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ●❛r❞✐♥❡r ✭✶✾✼✸✮ t❤❛t ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t✱ ✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ✷✹✸✻✱ ✉♥❧❡ss t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ ✐s ✐s♦♠♦r♣❤✐❝ t♦
✽✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊①❛♠♣❧❡ ▲❡t Γ = Cn[K✷] ✭❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ Cn ✇✐t❤ K✷✮✳ ❚❤❡♥ Γ ✐s ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ ❛♥❞ ❆✉t(Γ) = Z✷ ≀ D✷n ∼ = Zn
✷ ⋊ D✷n ✭❢♦r n ≥ ✺✮✳
❍❡♥❝❡ |❆✉t(Γ)| = ✷n · ✷n ❛♥❞ |❆✉t(Γ)v| = ✷n✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❜♦✉♥❞ ♦♥ t❤❡ ♦r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ♦❢ ✹✲✈❛❧❡♥t✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ ❖❜s❡r✈❡ t❤❛t ✐♥ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡✱ t❤❡ ❛❝t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ♦♥ ✐ts ♥❡✐❣❤❜♦✉rs ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✐❤❡❞r❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r ✽✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜② D✽✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ✇♦r❦ ♦❢ ●❛r❞✐♥❡r ✭✶✾✼✸✮ t❤❛t ❢♦r ❛ ❝♦♥♥❡❝t❡❞ ✹✲✈❛❧❡♥t✱ G✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✱ |Gv| ≤ ✷✹✸✻✱ ✉♥❧❡ss t❤❡ ❛❝t✐♦♥ ♦❢ Gv ♦♥ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ v ✐s ✐s♦♠♦r♣❤✐❝ t♦ D✽✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ G✲✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ v ∈ V (Γ)✳ ▲❡t Γ(v) ❞❡♥♦t❡ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ v ✐♥ Γ✳ ❚❤❡ ❧♦❝❛❧ ❛❝t✐♦♥ G Γ(v)
v
♦❢ G ✐s t❤❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❝t✐♦♥ ♦❢ Gv ♦♥ Γv✳ ■❢ Γ ✐s ❛ G✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ L ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✇❤✐❝❤ ✐s ♣❡r♠✉t❛t✐♦♥ ✐s♦♠♦r♣❤✐❝ t♦ G Γ(v)
v
✱ t❤❡♥ ✇❡ s❛② t❤❛t t❤❡ ♣❛✐r (Γ, G) ✐s ❧♦❝❛❧❧②✲L✳ ❉❡✜♥✐t✐♦♥ ❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❝❛❧❧❡❞ ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧♦❝❛❧❧②✲ ♣❛✐r ❛♥❞ ❢♦r ❡✈❡r② ❛r❝ ✱ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ G✲✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ v ∈ V (Γ)✳ ▲❡t Γ(v) ❞❡♥♦t❡ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ v ✐♥ Γ✳ ❚❤❡ ❧♦❝❛❧ ❛❝t✐♦♥ G Γ(v)
v
♦❢ G ✐s t❤❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐♥❞✉❝❡❞ ❜② t❤❡ ❛❝t✐♦♥ ♦❢ Gv ♦♥ Γv✳ ■❢ Γ ✐s ❛ G✲❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ L ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✇❤✐❝❤ ✐s ♣❡r♠✉t❛t✐♦♥ ✐s♦♠♦r♣❤✐❝ t♦ G Γ(v)
v
✱ t❤❡♥ ✇❡ s❛② t❤❛t t❤❡ ♣❛✐r (Γ, G) ✐s ❧♦❝❛❧❧②✲L✳ ❉❡✜♥✐t✐♦♥ ❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ L ✐s ❝❛❧❧❡❞ ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c(L) s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧♦❝❛❧❧②✲L ♣❛✐r (Γ, G) ❛♥❞ ❢♦r ❡✈❡r② ❛r❝ v ∈ V (Γ)✱ t❤❡ ✐♥❡q✉❛❧✐t② |Gv| ≤ c(L) ❤♦❧❞s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s
✸ ❛♥❞ ✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳
■❢ ✐s ❧♦❝❛❧❧②
✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t
❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ ✸✳ ■❢ ✐s ❧♦❝❛❧❧②
✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t
✹✽✳
✹ ♦t❤❡r
t❤❛♥
✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s A✸ ❛♥❞ S✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ ✐s ❧♦❝❛❧❧②
✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t
❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ ✸✳ ■❢ ✐s ❧♦❝❛❧❧②
✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t
✹✽✳
✹ ♦t❤❡r
t❤❛♥
✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s A✸ ❛♥❞ S✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② A✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t G ❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ |Gv| = ✸✳ ■❢ ✐s ❧♦❝❛❧❧②
✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t
✹✽✳
✹ ♦t❤❡r
t❤❛♥
✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s A✸ ❛♥❞ S✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② A✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t G ❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ |Gv| = ✸✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② S✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t |Gv| ≤ ✹✽✳
✹ ♦t❤❡r
t❤❛♥
✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s A✸ ❛♥❞ S✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② A✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t G ❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ |Gv| = ✸✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② S✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t |Gv| ≤ ✹✽✳
t❤❛♥ D✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❊✈❡r② r❡❣✉❧❛r ❣r♦✉♣ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✱ ♥❛♠❡❧②✱ ✐❢ G Γ(v)
v
✐s r❡❣✉❧❛r✱ t❤❡♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛♥ ❛r❝ ✜①❡s ❛❧❧ ✈❡rt✐❝❡s ✐♥❝✐❞❡♥t ✇✐t❤ ✐t✱ ❛♥❞ ❝♦♥♥❡❝t❡❞♥❡ss ✐♠♣❧✐❡s t❤❛t |Gv| ❡q✉❛❧s t❤❡ ✈❛❧❡♥❝② ♦❢ t❤❡ ❣r❛♣❤✳ ❚✉tt❡✬s r❡s✉❧t ❛❜♦✉t ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✐♠♣❧✐❡s t❤❛t ❣r♦✉♣s A✸ ❛♥❞ S✸ ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② A✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t G ❛❝ts ✶✲r❡❣✉❧❛r❧② ♦♥ t❤❡ s❡t ♦❢ ❛r❝s✱ ❤❡♥❝❡ |Gv| = ✸✳ ■❢ (Γ, G) ✐s ❧♦❝❛❧❧② S✸✱ t❤❡♥ ❚✉tt❡✬s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t |Gv| ≤ ✹✽✳
t❤❛♥ D✽ ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❚r♦✜♠♦✈ ❛♥❞ ❲❡✐ss✮ ✷✲tr❛♥s✐t✐✈❡ ❣r♦✉♣s ❛♥❞ ❣r♦✉♣s ♦❢ ♣r✐♠❡ ❞❡❣r❡❡ ❛r❡ ❣r❛♣❤✲r❡str✐❝t✐✈❡✳ ❚❤❡♦r❡♠ ✭❙❛♠✐✱ ✷✵✵✻✮
❚❤❡♦r❡♠ ✭❱❡rr❡t✱ ✷✵✵✾✮
s♦♠❡ ♣r✐♠❡ p ✭♣✲s✉❜r❡❣✉❧❛r✮✳ ❚❤❡♦r❡♠ ✭❚r♦✜♠♦✈ ❛♥❞ ❲❡✐ss✱ ✷✵✶✵✮ Pr✐♠✐t✐✈❡ ❣r♦✉♣s ♦❢ ❧✐♥❡❛r t②♣❡ ✭♣r✐♠✐t✐✈❡ ❣r♦✉♣s ❛❞♠✐tt✐♥❣ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ P❙▲ ✐♥ ✐ts ♥❛t✉r❛❧ ❛❝t✐♦♥✮ ❛r❡ ❣r❛♣❤✲r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ L ≤ Sym(Ω) ✐s s❛✐❞ t♦ ❜❡ ♣r✐♠✐t✐✈❡✱ ✐❢ ✐t ✐s tr❛♥s✐t✐✈❡ ❛♥❞ ♣r❡s❡r✈❡s ♥♦ ♥♦♥✲tr✐✈✐❛❧ ♣❛rt✐t✐♦♥ ♦❢ Ω✳ ❈♦♥❥❡❝t✉r❡ ✭❲❡✐ss✱ ✶✾✼✽✮ Pr✐♠✐t✐✈❡ ❣r♦✉♣s ❛r❡ ❣r❛♣❤✲r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
Pr♦❜❧❡♠ S✸ ≀ S✷ ✐♥ t❤❡ ♣r♦❞✉❝t ❛❝t✐♦♥ ✭❞❡❣r❡❡ ✾✮❄ Pr♦❜❧❡♠
✺ ❛❝t✐♥❣ ♦♥ t❤❡ ✶✵ ✉♥♦r❞❡r❡❞ ♣❛✐rs ♦❢ ❛ ✺✲s❡t ✭❞❡❣r❡❡ ✶✵✮❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
Pr♦❜❧❡♠ S✸ ≀ S✷ ✐♥ t❤❡ ♣r♦❞✉❝t ❛❝t✐♦♥ ✭❞❡❣r❡❡ ✾✮❄ Pr♦❜❧❡♠ S✺ ❛❝t✐♥❣ ♦♥ t❤❡ ✶✵ ✉♥♦r❞❡r❡❞ ♣❛✐rs ♦❢ ❛ ✺✲s❡t ✭❞❡❣r❡❡ ✶✵✮❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❝❛❧❧❡❞ q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s tr❛♥s✐t✐✈❡✳ ■t ✐s ❝❛❧❧❡❞ ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ✐t ✐s ♥♦t q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ❤❛s ❛t ♠♦st t✇♦ ♦r❜✐ts✳ ❚❤❡♦r❡♠ ✭Pr❛❡❣❡r✱ ✷✵✵✵✮ ❆ q✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣ ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧♦❝❛❧❧②✲ ♣❛✐r ✇✐t❤ q✉❛s✐♣r✐♠✐t✐✈❡ ♦r ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s✳ ❈♦♥❥❡❝t✉r❡ ✭Pr❛❡❣❡r✮ ◗✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣s ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❝❛❧❧❡❞ q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s tr❛♥s✐t✐✈❡✳ ■t ✐s ❝❛❧❧❡❞ ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ✐t ✐s ♥♦t q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ❤❛s ❛t ♠♦st t✇♦ ♦r❜✐ts✳ ❚❤❡♦r❡♠ ✭Pr❛❡❣❡r✱ ✷✵✵✵✮ ❆ q✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣ L ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c′(L) s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧♦❝❛❧❧②✲L ♣❛✐r (Γ, G) ✇✐t❤ G q✉❛s✐♣r✐♠✐t✐✈❡ ♦r ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② |Gv| ≤ c′(L) ❤♦❧❞s✳ ❈♦♥❥❡❝t✉r❡ ✭Pr❛❡❣❡r✮ ◗✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣s ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❝❛❧❧❡❞ q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s tr❛♥s✐t✐✈❡✳ ■t ✐s ❝❛❧❧❡❞ ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ✐❢ ✐t ✐s ♥♦t q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ ❡❛❝❤ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ❤❛s ❛t ♠♦st t✇♦ ♦r❜✐ts✳ ❚❤❡♦r❡♠ ✭Pr❛❡❣❡r✱ ✷✵✵✵✮ ❆ q✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣ L ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c′(L) s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧♦❝❛❧❧②✲L ♣❛✐r (Γ, G) ✇✐t❤ G q✉❛s✐♣r✐♠✐t✐✈❡ ♦r ❜✐q✉❛s✐♣r✐♠✐t✐✈❡ ❛♥❞ t❤❡ ✐♥❡q✉❛❧✐t② |Gv| ≤ c′(L) ❤♦❧❞s✳ ❈♦♥❥❡❝t✉r❡ ✭Pr❛❡❣❡r✮ ◗✉❛s✐♣r✐♠✐t✐✈❡ ❣r♦✉♣s ❛r❡ ❣r❛♣❤ r❡str✐❝t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s s❛✐❞ t♦ ❜❡ s❡♠✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s ❡✐t❤❡r tr❛♥s✐t✐✈❡ ♦r s❡♠✐r❡❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❊✈❡r② ❣r❛♣❤✲r❡str✐❝t✐✈❡ ❣r♦✉♣ ✐s s❡♠✐♣r✐♠✐t✐✈❡✳ ❈♦♥❥❡❝t✉r❡ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❆ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s s❡♠✐♣r✐♠✐t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s s❛✐❞ t♦ ❜❡ s❡♠✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s ❡✐t❤❡r tr❛♥s✐t✐✈❡ ♦r s❡♠✐r❡❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❊✈❡r② ❣r❛♣❤✲r❡str✐❝t✐✈❡ ❣r♦✉♣ ✐s s❡♠✐♣r✐♠✐t✐✈❡✳ ❈♦♥❥❡❝t✉r❡ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❆ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s s❡♠✐♣r✐♠✐t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆ tr❛♥s✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s s❛✐❞ t♦ ❜❡ s❡♠✐♣r✐♠✐t✐✈❡ ✐❢ ❡❛❝❤ ♦❢ ✐ts ♥♦r♠❛❧ s✉❜❣r♦✉♣s ✐s ❡✐t❤❡r tr❛♥s✐t✐✈❡ ♦r s❡♠✐r❡❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❊✈❡r② ❣r❛♣❤✲r❡str✐❝t✐✈❡ ❣r♦✉♣ ✐s s❡♠✐♣r✐♠✐t✐✈❡✳ ❈♦♥❥❡❝t✉r❡ ✭P♦t♦↔♥✐❦✱ ❙♣✐❣❛✱ ❱❡rr❡t✮ ❆ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s ❣r❛♣❤✲r❡str✐❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s s❡♠✐♣r✐♠✐t✐✈❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■❢ L ∼ = D✽ ✐s tr❛♥s✐t✐✈❡ ❣r♦✉♣ ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✺✱ t❤❡♥ ✐t ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡ ■❢
✽✱ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r
✱ t❤❡r❡ ❡①✐sts ❛ ❣r❛♣❤ ♦❢ ♦r❞❡r ✷ ❛♥❞ ❛ ❣r♦✉♣ ✱ s✉❝❤ t❤❛t ✐s ❧♦❝❛❧❧②
✽ ❛♥❞
✷ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■❢ L ∼ = D✽ ✐s tr❛♥s✐t✐✈❡ ❣r♦✉♣ ♦❢ ❞❡❣r❡❡ ❛t ♠♦st ✺✱ t❤❡♥ ✐t ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡ ■❢ L = D✽✱ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ t❤❡r❡ ❡①✐sts ❛ ❣r❛♣❤ Γ ♦❢ ♦r❞❡r ✷n ❛♥❞ ❛ ❣r♦✉♣ G✱ s✉❝❤ t❤❛t (Γ, G) ✐s ❧♦❝❛❧❧② D✽ ❛♥❞ |Gv| = ✷n✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■❢ L ✐s ♣r✐♠✐t✐✈❡ ♦r r❡❣✉❧❛r✱ t❤❡♥ ✐t ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ ✐s ♥♦t ♣r✐♠✐t✐✈❡✱ t❤❡♥ ✐t ❤❛s ❜❧♦❝❦s ♦❢ s✐③❡ ✷ ♦r ✸✳ ❙✉♣♣♦s❡ t❤❛t ❛❞♠✐ts ❛ s②st❡♠ ♦❢ ✐♠♣r✐♠✐t✐✈✐t② ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ❛♥❞ ❧❡t ❜❡ ♦♥❡ ♦❢ t❤❡s❡ ❜❧♦❝❦s✳ ■❢ t❤❡ ♣♦✐♥t✇✐s❡ st❛❜✐❧✐③❡r ✐s tr✐✈✐❛❧✱ t❤❡♥ ✐s ♣❡r♠✉t❛t✐♦♥❛❧② ✐s♦♠♦r♣❤✐❝ t♦
✶✷✳
❖r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡r ✐♥ t❤✐s ❝❛s❡ ❝❛♥♥♦t ❜❡ ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t✱ ❜✉t ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❜② ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❣r❛♣❤✱ s♦ ✐t ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡✳ ■♥ ♦t❤❡r ❝❛s❡s ✇✐t❤ ✷ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ✐t ✐s ♦❢ ❡①♣♦♥❡♥t✐❛❧ t②♣❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■❢ L ✐s ♣r✐♠✐t✐✈❡ ♦r r❡❣✉❧❛r✱ t❤❡♥ ✐t ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ L ✐s ♥♦t ♣r✐♠✐t✐✈❡✱ t❤❡♥ ✐t ❤❛s ❜❧♦❝❦s ♦❢ s✐③❡ ✷ ♦r ✸✳ ❙✉♣♣♦s❡ t❤❛t ❛❞♠✐ts ❛ s②st❡♠ ♦❢ ✐♠♣r✐♠✐t✐✈✐t② ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ❛♥❞ ❧❡t ❜❡ ♦♥❡ ♦❢ t❤❡s❡ ❜❧♦❝❦s✳ ■❢ t❤❡ ♣♦✐♥t✇✐s❡ st❛❜✐❧✐③❡r ✐s tr✐✈✐❛❧✱ t❤❡♥ ✐s ♣❡r♠✉t❛t✐♦♥❛❧② ✐s♦♠♦r♣❤✐❝ t♦
✶✷✳
❖r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡r ✐♥ t❤✐s ❝❛s❡ ❝❛♥♥♦t ❜❡ ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t✱ ❜✉t ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❜② ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❣r❛♣❤✱ s♦ ✐t ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡✳ ■♥ ♦t❤❡r ❝❛s❡s ✇✐t❤ ✷ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ✐t ✐s ♦❢ ❡①♣♦♥❡♥t✐❛❧ t②♣❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■❢ L ✐s ♣r✐♠✐t✐✈❡ ♦r r❡❣✉❧❛r✱ t❤❡♥ ✐t ✐s ❣r❛♣❤ r❡str✐❝t✐✈❡✳ ■❢ L ✐s ♥♦t ♣r✐♠✐t✐✈❡✱ t❤❡♥ ✐t ❤❛s ❜❧♦❝❦s ♦❢ s✐③❡ ✷ ♦r ✸✳ ❙✉♣♣♦s❡ t❤❛t L ❛❞♠✐ts ❛ s②st❡♠ ♦❢ ✐♠♣r✐♠✐t✐✈✐t② ❝♦♥s✐st✐♥❣ ♦❢ t✇♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ❛♥❞ ❧❡t A ❜❡ ♦♥❡ ♦❢ t❤❡s❡ ❜❧♦❝❦s✳ ■❢ t❤❡ ♣♦✐♥t✇✐s❡ st❛❜✐❧✐③❡r L(A) ✐s tr✐✈✐❛❧✱ t❤❡♥ L ✐s ♣❡r♠✉t❛t✐♦♥❛❧② ✐s♦♠♦r♣❤✐❝ t♦ D✶✷✳ ❖r❞❡r ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡r ✐♥ t❤✐s ❝❛s❡ ❝❛♥♥♦t ❜❡ ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t✱ ❜✉t ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❜② ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❣r❛♣❤✱ s♦ ✐t ✐s ♦❢ ♣♦❧②♥♦♠✐❛❧ t②♣❡✳ ■♥ ♦t❤❡r ❝❛s❡s ✇✐t❤ ✷ ❜❧♦❝❦s ♦❢ s✐③❡ ✸ ✐t ✐s ♦❢ ❡①♣♦♥❡♥t✐❛❧ t②♣❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ✺ ✐♠♣r✐♠✐t✐✈❡ ❣r♦✉♣s ♦❢ ❞❡❣r❡❡ ✻ ✇✐t❤ ♥♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸✱ t❤❡② ❛r❡ ❛❧❧ ❝♦♥t❛✐♥❡❞ ✐♥ S✷ ≀ S✸ ✐♥ ✐ts ✐♠♣r✐♠✐t✐✈❡ ❛❝t✐♦♥ ♦♥ ✻ ♣♦✐♥ts✱ ❛♥❞ ❛r❡ ❞❡♥♦t❡❞ ❜② A✹(✻)✱ ✷A✹(✻)✱ S✹(✻d)✱ S✹(✻c) ❛♥❞ ✷S✹(✻)✳ ▲❡t✬s ❝♦♥s✐❞❡r
✷ ✷ ✸✳
❚❤❡ ❦❡r♥❡❧ ♦❢ t❤✐s ❛❝t✐♦♥ ✐s ✐s♦♠♦r♣❤✐❝ t♦
✸ ✷✱ ❛♥❞ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ✐t
✇✐t❤
✷
✳ ▲❡t ✵ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ❜❡ ❛♥ ✐♥❞❡① t✇♦ s✉❜❣r♦✉♣ ♦❢
✷
✳ ❑❡r♥❡❧s ♦❢
✹ ✻ ✱ ✹ ✻
❛♥❞
✹ ✻
❛r❡ ❛❧❧ ✐s♦♠♦r♣❤✐❝ t♦ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ✺ ✐♠♣r✐♠✐t✐✈❡ ❣r♦✉♣s ♦❢ ❞❡❣r❡❡ ✻ ✇✐t❤ ♥♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸✱ t❤❡② ❛r❡ ❛❧❧ ❝♦♥t❛✐♥❡❞ ✐♥ S✷ ≀ S✸ ✐♥ ✐ts ✐♠♣r✐♠✐t✐✈❡ ❛❝t✐♦♥ ♦♥ ✻ ♣♦✐♥ts✱ ❛♥❞ ❛r❡ ❞❡♥♦t❡❞ ❜② A✹(✻)✱ ✷A✹(✻)✱ S✹(✻d)✱ S✹(✻c) ❛♥❞ ✷S✹(✻)✳ ▲❡t✬s ❝♦♥s✐❞❡r Z✷ ≀ Sym({a, b, c}) ∼ = S✷ ≀ S✸✳
(a,0) (a,1) (b, 0) (b, 1) (c, 0) (c, 1)
❚❤❡ ❦❡r♥❡❧ ♦❢ t❤✐s ❛❝t✐♦♥ ✐s ✐s♦♠♦r♣❤✐❝ t♦ Z✸
✷✱ ❛♥❞ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ✐t
✇✐t❤ Z{a,b,c}
✷
✳ ▲❡t ✵ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ❜❡ ❛♥ ✐♥❞❡① t✇♦ s✉❜❣r♦✉♣ ♦❢
✷
✳ ❑❡r♥❡❧s ♦❢
✹ ✻ ✱ ✹ ✻
❛♥❞
✹ ✻
❛r❡ ❛❧❧ ✐s♦♠♦r♣❤✐❝ t♦ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ✺ ✐♠♣r✐♠✐t✐✈❡ ❣r♦✉♣s ♦❢ ❞❡❣r❡❡ ✻ ✇✐t❤ ♥♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸✱ t❤❡② ❛r❡ ❛❧❧ ❝♦♥t❛✐♥❡❞ ✐♥ S✷ ≀ S✸ ✐♥ ✐ts ✐♠♣r✐♠✐t✐✈❡ ❛❝t✐♦♥ ♦♥ ✻ ♣♦✐♥ts✱ ❛♥❞ ❛r❡ ❞❡♥♦t❡❞ ❜② A✹(✻)✱ ✷A✹(✻)✱ S✹(✻d)✱ S✹(✻c) ❛♥❞ ✷S✹(✻)✳ ▲❡t✬s ❝♦♥s✐❞❡r Z✷ ≀ Sym({a, b, c}) ∼ = S✷ ≀ S✸✳
(a,0) (a,1) (b, 0) (b, 1) (c, 0) (c, 1)
❚❤❡ ❦❡r♥❡❧ ♦❢ t❤✐s ❛❝t✐♦♥ ✐s ✐s♦♠♦r♣❤✐❝ t♦ Z✸
✷✱ ❛♥❞ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ✐t
✇✐t❤ Z{a,b,c}
✷
✳ ▲❡t K = {(✵, ✵, ✵), (✶, ✶, ✵), (✶, ✵, ✶), (✵, ✶, ✶)} ❜❡ ❛♥ ✐♥❞❡① t✇♦ s✉❜❣r♦✉♣ ♦❢ Z{a,b,c}
✷
✳ ❑❡r♥❡❧s ♦❢
✹ ✻ ✱ ✹ ✻
❛♥❞
✹ ✻
❛r❡ ❛❧❧ ✐s♦♠♦r♣❤✐❝ t♦ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡r❡ ❛r❡ ✺ ✐♠♣r✐♠✐t✐✈❡ ❣r♦✉♣s ♦❢ ❞❡❣r❡❡ ✻ ✇✐t❤ ♥♦ ❜❧♦❝❦s ♦❢ s✐③❡ ✸✱ t❤❡② ❛r❡ ❛❧❧ ❝♦♥t❛✐♥❡❞ ✐♥ S✷ ≀ S✸ ✐♥ ✐ts ✐♠♣r✐♠✐t✐✈❡ ❛❝t✐♦♥ ♦♥ ✻ ♣♦✐♥ts✱ ❛♥❞ ❛r❡ ❞❡♥♦t❡❞ ❜② A✹(✻)✱ ✷A✹(✻)✱ S✹(✻d)✱ S✹(✻c) ❛♥❞ ✷S✹(✻)✳ ▲❡t✬s ❝♦♥s✐❞❡r Z✷ ≀ Sym({a, b, c}) ∼ = S✷ ≀ S✸✳
(a,0) (a,1) (b, 0) (b, 1) (c, 0) (c, 1)
❚❤❡ ❦❡r♥❡❧ ♦❢ t❤✐s ❛❝t✐♦♥ ✐s ✐s♦♠♦r♣❤✐❝ t♦ Z✸
✷✱ ❛♥❞ ✇❡ ❝❛♥ ✐❞❡♥t✐❢② ✐t
✇✐t❤ Z{a,b,c}
✷
✳ ▲❡t K = {(✵, ✵, ✵), (✶, ✶, ✵), (✶, ✵, ✶), (✵, ✶, ✶)} ❜❡ ❛♥ ✐♥❞❡① t✇♦ s✉❜❣r♦✉♣ ♦❢ Z{a,b,c}
✷
✳ ❑❡r♥❡❧s ♦❢ A✹(✻)✱ S✹(✻d) ❛♥❞ S✹(✻c) ❛r❡ ❛❧❧ ✐s♦♠♦r♣❤✐❝ t♦ K✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
✐s♦♠♦r♣❤✐❝ t♦ S✷ ≀ S✸✱ s♦ t❤❡② ❛r❡ ❜♦t❤ ♦❢ ❡①♣♦♥❡♥t✐❛❧ t②♣❡✳ ❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ ❧❡t ∆ ❜❡ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t Γ[K✷]✳ ❙✐♥❝❡ Γ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✸✲✈❛❧❡♥t ❣r❛♣❤✱ ∆ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✻✲✈❛❧❡♥t ❣r❛♣❤✳ ◆♦t❡ t❤❛t ❆✉t ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ✭❜② ❛❝t✐♥❣ ♦♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ♦❢ ✈❡rt✐❝❡s ♦❢ ✮✳ ❙✐♠✐❧❛r❧②✱
✷
❛❧s♦ ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ✭❣✐✈❡♥ ❜② ✱ ❢♦r
✷
❛♥❞
✷✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ ❧❡t ∆ ❜❡ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t Γ[K✷]✳ ❙✐♥❝❡ Γ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✸✲✈❛❧❡♥t ❣r❛♣❤✱ ∆ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✻✲✈❛❧❡♥t ❣r❛♣❤✳ ◆♦t❡ t❤❛t ❆✉t(Γ) ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ∆ ✭❜② ❛❝t✐♥❣ ♦♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ♦❢ ✈❡rt✐❝❡s ♦❢ ∆✮✳ ❙✐♠✐❧❛r❧②✱
✷
❛❧s♦ ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ✭❣✐✈❡♥ ❜② ✱ ❢♦r
✷
❛♥❞
✷✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ❛♥❞ ❧❡t ∆ ❜❡ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t Γ[K✷]✳ ❙✐♥❝❡ Γ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✸✲✈❛❧❡♥t ❣r❛♣❤✱ ∆ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✻✲✈❛❧❡♥t ❣r❛♣❤✳ ◆♦t❡ t❤❛t ❆✉t(Γ) ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ∆ ✭❜② ❛❝t✐♥❣ ♦♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡ ♦❢ ✈❡rt✐❝❡s ♦❢ ∆✮✳ ❙✐♠✐❧❛r❧②✱ FV (Γ)
✷
❛❧s♦ ❤❛s ❛ ♥❛t✉r❛❧ ❛❝t✐♦♥ ❛s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ∆ ✭❣✐✈❡♥ ❜② (v, i)x = (v, i + x(v))✱ ❢♦r x ∈ FV (Γ)
✷
❛♥❞ (v, i) ∈ V (Γ) × F✷✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ❞❡s✐r❡❞ ❧♦❝❛❧ ❛❝t✐♦♥❄ ❘❡❝❛❧❧ t❤❛t t❤❡ ❦❡r♥❡❧ ✐♥ t❤❡ ❧♦❝❛❧ ❛❝t✐♦♥ ✐s
✷ ✷✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ❞❡s✐r❡❞ ❧♦❝❛❧ ❛❝t✐♦♥❄
v a b c (a, 0) (a, 1) (b, 0) (b, 1) (v, 0) (v, 1) (c,0) (c, 1) Γ(v) ∆((v, 0))
❘❡❝❛❧❧ t❤❛t t❤❡ ❦❡r♥❡❧ ✐♥ t❤❡ ❧♦❝❛❧ ❛❝t✐♦♥ ✐s
✷ ✷✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ❞❡s✐r❡❞ ❧♦❝❛❧ ❛❝t✐♦♥❄
v a b c (a, 0) (a, 1) (b, 0) (b, 1) (v, 0) (v, 1) (c,0) (c, 1) Γ(v) ∆((v, 0))
❘❡❝❛❧❧ t❤❛t t❤❡ ❦❡r♥❡❧ ✐♥ t❤❡ ❧♦❝❛❧ ❛❝t✐♦♥ ✐s Z✷
✷✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Γ ♦✈❡r F✷✱ t❤❛t ✐s E✶ = {x ∈ FV (Γ)
✷
: x(v) +
x(a) = ✵ (∀v ∈ V (Γ)}. ❚❤❡♥ E✶ ❆✉t(∆)✳ ❉❡✜♥❡
✶
✇❤❡r❡ ✐s ❛♥ ❛r❝✲tr❛♥s✐t✐✈❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t ✳ ❍♦✇ t♦ ✜♥❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✇✐t❤ ❧❛r❣❡ ✶✲❡✐❣❡♥s♣❛❝❡s❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Γ ♦✈❡r F✷✱ t❤❛t ✐s E✶ = {x ∈ FV (Γ)
✷
: x(v) +
x(a) = ✵ (∀v ∈ V (Γ)}. ❚❤❡♥ E✶ ❆✉t(∆)✳ ❉❡✜♥❡ G = E✶, A ✇❤❡r❡ A ✐s ❛♥ ❛r❝✲tr❛♥s✐t✐✈❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t(Γ)✳ ❍♦✇ t♦ ✜♥❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✇✐t❤ ❧❛r❣❡ ✶✲❡✐❣❡♥s♣❛❝❡s❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
▲❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Γ ♦✈❡r F✷✱ t❤❛t ✐s E✶ = {x ∈ FV (Γ)
✷
: x(v) +
x(a) = ✵ (∀v ∈ V (Γ)}. ❚❤❡♥ E✶ ❆✉t(∆)✳ ❉❡✜♥❡ G = E✶, A ✇❤❡r❡ A ✐s ❛♥ ❛r❝✲tr❛♥s✐t✐✈❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t(Γ)✳ ❍♦✇ t♦ ✜♥❞ ❝✉❜✐❝ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ✇✐t❤ ❧❛r❣❡ ✶✲❡✐❣❡♥s♣❛❝❡s❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
R = a, b, c, z |✶ = a✷ = b✷ = c✷ = z✷ = [a, z] = [b, z] = [c, z], [a, b] = [b, c] = [a, c] = z. ❚❤❡ ❈❛②❧❡② ❣r❛♣❤ M = ❈❛②(R, {a, b, c}) ✐s ❝❛❧❧❡❞ ▼ö❜✐✉s✕❑❛♥t♦r ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❛❜❝③ ❛❜ ❛ ❛❝ ❛❜❝ ❛❜③ ❛③ ❛❝③ ❜❝③ ❜ ✶ ❝ ❜❝ ❜③ ③ ❝③ e✶ e✷ e✸ e✹ e−✶
✶
e−✶
✷
e−✶
✸
e−✶
✹
❋✐❣✉r❡✿ ▼ö❜✐✉s✕❑❛♥t♦r ❣r❛♣❤
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■t ✐s ♦❜✈✐♦✉s ❢r♦♠ t❤❡ ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥ ♦❢ R t❤❛t ❛♥② ♣❡r♠✉t❛t✐♦♥ ♦❢ {a, b, c} ✐♥❞✉❝❡s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ R✳ ■t ❢♦❧❧♦✇s t❤❛t M ❛❞♠✐ts ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s B ✐s♦♠♦r♣❤✐❝ t♦ R ⋊ ❙②♠(✸)✳ ◆♦t❡ t❤❛t ✐s ✷✲❛r❝✲r❡❣✉❧❛r ❛♥❞ ❝♦♥t❛✐♥s ❛♥ ❛r❝✲r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ t❤❡ ❢♦r♠
✸✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
■t ✐s ♦❜✈✐♦✉s ❢r♦♠ t❤❡ ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥ ♦❢ R t❤❛t ❛♥② ♣❡r♠✉t❛t✐♦♥ ♦❢ {a, b, c} ✐♥❞✉❝❡s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ R✳ ■t ❢♦❧❧♦✇s t❤❛t M ❛❞♠✐ts ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s B ✐s♦♠♦r♣❤✐❝ t♦ R ⋊ ❙②♠(✸)✳ ◆♦t❡ t❤❛t B ✐s ✷✲❛r❝✲r❡❣✉❧❛r ❛♥❞ ❝♦♥t❛✐♥s ❛♥ ❛r❝✲r❡❣✉❧❛r s✉❜❣r♦✉♣ A ♦❢ t❤❡ ❢♦r♠ R ⋊ Z✸✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
n ❝♦✈❡rs ♦❢ ▼ö❜✐✉s✕❑❛♥t♦r ❣r❛♣❤
▲❡t n ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t N = Z✹
n = e✶, e✷, e✸, e✹✳ ▲❡t
Mn ❜❡ t❤❡ ❞❡r✐✈❡❞ ❝♦✈❡r✐♥❣ ❣r❛♣❤ ♦❢ M ✭✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈♦❧t❛❣❡ ❛ss✐❣♥♠❡♥t ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ✜❣✉r❡✮✳ ❚❤❡s❡ ❣r❛♣❤s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❛❧♥✐↔✱ ▼❛r✉➨✐↔✱ ▼✐❦❧❛✈✐↔ ❛♥❞ P♦t♦↔♥✐❦ ✐♥ ✷✵✵✼✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡✐r ✇♦r❦ t❤❛t ❧✐❢ts t♦ ✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ✐ts ❧✐❢t ❜② ✳ ❙✐♠✐❧❛r❧②✱ ❧❡t ❜❡ t❤❡ ❧✐❢t ♦❢ ✳ ◆♦t❡ t❤❛t ✐s ✷✲❛r❝✲r❡❣✉❧❛r ♦♥ ✱ ❛♥❞ ✐s ✶✲❛r❝✲r❡❣✉❧❛r✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
n ❝♦✈❡rs ♦❢ ▼ö❜✐✉s✕❑❛♥t♦r ❣r❛♣❤
▲❡t n ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t N = Z✹
n = e✶, e✷, e✸, e✹✳ ▲❡t
Mn ❜❡ t❤❡ ❞❡r✐✈❡❞ ❝♦✈❡r✐♥❣ ❣r❛♣❤ ♦❢ M ✭✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✈♦❧t❛❣❡ ❛ss✐❣♥♠❡♥t ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ✜❣✉r❡✮✳ ❚❤❡s❡ ❣r❛♣❤s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❛❧♥✐↔✱ ▼❛r✉➨✐↔✱ ▼✐❦❧❛✈✐↔ ❛♥❞ P♦t♦↔♥✐❦ ✐♥ ✷✵✵✼✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡✐r ✇♦r❦ t❤❛t B ❧✐❢ts t♦ Mn✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ✐ts ❧✐❢t ❜② ˜ B✳ ❙✐♠✐❧❛r❧②✱ ❧❡t ˜ A ❜❡ t❤❡ ❧✐❢t ♦❢ A✳ ◆♦t❡ t❤❛t ˜ B ✐s ✷✲❛r❝✲r❡❣✉❧❛r ♦♥ Mn✱ ❛♥❞ ˜ A ✐s ✶✲❛r❝✲r❡❣✉❧❛r✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮ ▲❡t n ≥ ✸ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Mn ♦✈❡r F✷✳ ❚❤❡♥ |E✶| ≥ ✷|V (Mn)|/✼✷✳ ❘❡♠❛r❦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s✉❣❣❡sts t❤❛t
✶
✷
✽ ✷ ✐❢
✐s ♦❞❞✱ ❛♥❞
✶
✷
✽ ✽ ✐❢
✐s ❡✈❡♥✳
✶ ✶ ✶
❆✉t
✷ ✶ ✶
❆✉t
✶ ✐s ❧♦❝❛❧❧② ✹ ✻ ❛♥❞ ✷ ✐s ❧♦❝❛❧❧② ✹ ✻
✳
✹ ✻
✐s s♦❧✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣
✶
✇❤❡r❡ ❛♥❞
✷
✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮ ▲❡t n ≥ ✸ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Mn ♦✈❡r F✷✳ ❚❤❡♥ |E✶| ≥ ✷|V (Mn)|/✼✷✳ ❘❡♠❛r❦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s✉❣❣❡sts t❤❛t |E✶| = ✷|V (Mn)|/✽+✷ ✐❢ n ✐s ♦❞❞✱ ❛♥❞ |E✶| = ✷|V (Mn)|/✽+✽ ✐❢ n ✐s ❡✈❡♥✳
✶ ✶ ✶
❆✉t
✷ ✶ ✶
❆✉t
✶ ✐s ❧♦❝❛❧❧② ✹ ✻ ❛♥❞ ✷ ✐s ❧♦❝❛❧❧② ✹ ✻
✳
✹ ✻
✐s s♦❧✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣
✶
✇❤❡r❡ ❛♥❞
✷
✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮ ▲❡t n ≥ ✸ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Mn ♦✈❡r F✷✳ ❚❤❡♥ |E✶| ≥ ✷|V (Mn)|/✼✷✳ ❘❡♠❛r❦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s✉❣❣❡sts t❤❛t |E✶| = ✷|V (Mn)|/✽+✷ ✐❢ n ✐s ♦❞❞✱ ❛♥❞ |E✶| = ✷|V (Mn)|/✽+✽ ✐❢ n ✐s ❡✈❡♥✳ G✶ := E✶, ˜ A = E✶ ⋊ ˜ A ≤ ❆✉t(∆) G✷ := E✶, ˜ B = E✶ ⋊ ˜ B ≤ ❆✉t(∆)
✶ ✐s ❧♦❝❛❧❧② ✹ ✻ ❛♥❞ ✷ ✐s ❧♦❝❛❧❧② ✹ ✻
✳
✹ ✻
✐s s♦❧✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣
✶
✇❤❡r❡ ❛♥❞
✷
✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮ ▲❡t n ≥ ✸ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Mn ♦✈❡r F✷✳ ❚❤❡♥ |E✶| ≥ ✷|V (Mn)|/✼✷✳ ❘❡♠❛r❦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s✉❣❣❡sts t❤❛t |E✶| = ✷|V (Mn)|/✽+✷ ✐❢ n ✐s ♦❞❞✱ ❛♥❞ |E✶| = ✷|V (Mn)|/✽+✽ ✐❢ n ✐s ❡✈❡♥✳ G✶ := E✶, ˜ A = E✶ ⋊ ˜ A ≤ ❆✉t(∆) G✷ := E✶, ˜ B = E✶ ⋊ ˜ B ≤ ❆✉t(∆) (∆, G✶) ✐s ❧♦❝❛❧❧② A✹(✻) ❛♥❞ (∆, G✷) ✐s ❧♦❝❛❧❧② S✹(✻d)✳
✹ ✻
✐s s♦❧✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣
✶
✇❤❡r❡ ❛♥❞
✷
✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❍✳✱ P♦t♦↔♥✐❦✱ ❱❡rr❡t✮ ▲❡t n ≥ ✸ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t E✶ ❜❡ t❤❡ ✶✲❡✐❣❡♥s♣❛❝❡ ❢♦r Mn ♦✈❡r F✷✳ ❚❤❡♥ |E✶| ≥ ✷|V (Mn)|/✼✷✳ ❘❡♠❛r❦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s✉❣❣❡sts t❤❛t |E✶| = ✷|V (Mn)|/✽+✷ ✐❢ n ✐s ♦❞❞✱ ❛♥❞ |E✶| = ✷|V (Mn)|/✽+✽ ✐❢ n ✐s ❡✈❡♥✳ G✶ := E✶, ˜ A = E✶ ⋊ ˜ A ≤ ❆✉t(∆) G✷ := E✶, ˜ B = E✶ ⋊ ˜ B ≤ ❆✉t(∆) (∆, G✶) ✐s ❧♦❝❛❧❧② A✹(✻) ❛♥❞ (∆, G✷) ✐s ❧♦❝❛❧❧② S✹(✻d)✳ S✹(✻c) ✐s s♦❧✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ G✶, τσ ✇❤❡r❡ τ ∈ ˜ B \ ˜ A ❛♥❞ σ ∈ FV (Γ)
✷
✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt✐❡s✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
Pr♦❜❧❡♠ ❉❡t❡r♠✐♥❡ t❤❡ ❣r♦✇t❤ ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ✽✲✈❛❧❡♥t ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s✳ Pr♦❜❧❡♠ ❉♦❡s t❤❡r❡ ❛ tr❛♥s✐t✐✈❡ ❣r♦✉♣ t❤❛t ✐s ♥❡✐t❤❡r ❣r❛♣❤✲r❡str✐❝t✐✈❡ ♥♦r ♦❢ ♣♦❧②♥♦♠✐❛❧ ♦r ❡①♣♦♥❡♥t✐❛❧ t②♣❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❖♥ ♦r❞❡rs ♦❢ ✈❡rt❡①✲st❛❜✐❧✐③❡rs ✐♥ ❛r❝✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s