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❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❑❧❛✈❞✐❥❛ ❑✉t♥❛r✱ P❛✇❡➟ P❡t❡❝❦✐ ❛♥❞ ❆♥❛st❛s✐②❛ ❚❛♥❛♥❛✳
■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ●r❛♣❤ ❚❤❡♦r②
✷✼✳✺✳✷✵✶✺
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❈❛②❧❡② ❣r❛♣❤s
❆✉t♦♠♦r♣❤✐s♠ ♦❢ ●❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ◆♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❆ ❣r❛♣❤ ✐s ❛♥ ♦r❞❡r❡❞ ♣❛✐r Γ = (V , E)✱ ✇❤❡r❡ V ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s✱ ❛♥❞ E ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❡❞❣❡s ♦❢ t❤❡ ❣r❛♣❤ Γ. ❆✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r❛♣❤ Γ ✐s ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ϕ : V (Γ) → V (Γ) s✉❝❤ t❤❛t {x, y} ∈ E(Γ) ⇔ {ϕ(x), ϕ(y)} ∈ E(Γ). ❲❡ ❞❡✜♥❡ t❤❡ s❡t Aut(Γ) t♦ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ t❤❡ ❣r❛♣❤ Γ. ■t ✐s ♥♦t ❞✐✣❝✉❧t t♦ s❡❡ t❤❛t Aut(Γ) ✐s ✐♥ ❢❛❝t t❤❡ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s✱ ❛♥❞ ✐t ✐s ❝❛❧❧❡❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ Γ.
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❲❤❡♥ ✇❡ s♣❡❛❦ ❛❜♦✉t ❣r❛♣❤s ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s ♦♥ t❤❡ ❣r❛♣❤✳ ❲❡ s❛② t❤❛t ❛ ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ✈❡rt❡① s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❡❞❣❡ s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❛r❝✲tr❛♥s✐t✐✈❡ ❆✉t ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❛r❝ s❡t ♦❢ t❤❡ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❲❤❡♥ ✇❡ s♣❡❛❦ ❛❜♦✉t ❣r❛♣❤s ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s ♦♥ t❤❡ ❣r❛♣❤✳ ❲❡ s❛② t❤❛t ❛ ❣r❛♣❤ Γ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t(Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ✈❡rt❡① s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❡❞❣❡✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t(Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❡❞❣❡ s❡t ♦❢ t❤❡ ❣r❛♣❤❀ ❛r❝✲tr❛♥s✐t✐✈❡ ⇔ ❆✉t(Γ) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ ❛r❝ s❡t ♦❢ t❤❡ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
✭✐✮ ✶ ∈ S✱ ✭✐✐✮ S−✶ = S❀ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❈❛②(G, S) ♦❢ G r❡❧❛t✐✈❡ t♦ S ❤❛s ✈❡rt❡① s❡t G ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ {g, gs} ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S✳ ❊①❛♠♣❧❡
✶✵✱
✶ ✺ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
✭✐✮ ✶ ∈ S✱ ✭✐✐✮ S−✶ = S❀ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ❈❛②(G, S) ♦❢ G r❡❧❛t✐✈❡ t♦ S ❤❛s ✈❡rt❡① s❡t G ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ {g, gs} ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S✳ ❊①❛♠♣❧❡ G = Z✶✵✱ S = {±✶, ✺}✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
■❢ X = Cay(G, S) t❤❡♥ t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ ✐ts❡❧❢ ❜② t❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❞✉❝❡s ❛ s✉❜❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ✇❤✐❝❤ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✈❡rt✐❝❡s✱ ❤❡♥❝❡ ❡✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✐s ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡ s♠❛❧❧❡st ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
■❢ X = Cay(G, S) t❤❡♥ t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ ✐ts❡❧❢ ❜② t❤❡ ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❞✉❝❡s ❛ s✉❜❣r♦✉♣ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ✇❤✐❝❤ ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ ✈❡rt✐❝❡s✱ ❤❡♥❝❡ ❡✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❍♦✇❡✈❡r✱ ♥♦t ❡✈❡r② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✐s ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡ s♠❛❧❧❡st ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ S ❛ ♥♦♥✲❡♠♣t② s✉❜s❡t ♦❢ G ❛♥❞ α ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ ✭✐✮ α✷ = ✶✱ ✭✐✐✮ α(g−✶)g ∈ S✱ ✭∀g ∈ G✮ ✭✐✐✐✮ α(S−✶) = S✳ ❚❤❡♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ X = GC(G, S, α) ♦♥ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r (S, α) ✐s ❛ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G✱ ❛♥❞ ❡❞❣❡s ♦❢ ❢♦r♠ {g, α(g)s}✱ ✇❤❡r❡ g ∈ G ❛♥❞ s ∈ S✳ ❊①❛♠♣❧❡ G = Z✶✵✱ α(x) = −x✱ S = {±✶, ✺}✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❊①❛♠♣❧❡ ▲❡t G = Z✸ × Z✸✱ S = {(✶, ✵), (✶, ✶), (✵, ✷), (✷, ✷)} ❛♥❞ α : (i, j) → (j, i)✳
01 21 11 00 10 20 22 02 12
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡ ❝♦♥❝❡♣t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦ ❛♥❞ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐ ✐♥ ✶✾✾✷✳ ❚❤❡② st✉❞✐❡❞ ♣r♦♣❡rt✐❡s ♦❢ s✉❝❤ ❣r❛♣❤s r❡❧❛t✐✈❡ t♦ ❞♦✉❜❧❡ ❝♦✈❡rs ♦❢ ❣r❛♣❤s ✭s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ❜✐♣❛rt✐t❡ ❞♦✉❜❧❡ ❝♦✈❡r ♦r ❝❛♥♦♥✐❝❛❧ ❞♦✉❜❧❡ ❝♦✈❡r✮✳ ❉♦✉❜❧❡ ❝♦✈❡r ♦❢ ❛ ❣r❛♣❤ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t
✷✳
❚❤✐s ♠❡❛♥s t❤❛t
✷ ❛♥❞ ❛❧❧ t❤❡ ❡❞❣❡s ♦❢
❛r❡ ✵ ✶ ❛♥❞ ✶ ✵ ✇❤❡r❡ ✐s ❛♥ ❡❞❣❡ ✐♥ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡ ❝♦♥❝❡♣t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦ ❛♥❞ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐ ✐♥ ✶✾✾✷✳ ❚❤❡② st✉❞✐❡❞ ♣r♦♣❡rt✐❡s ♦❢ s✉❝❤ ❣r❛♣❤s r❡❧❛t✐✈❡ t♦ ❞♦✉❜❧❡ ❝♦✈❡rs ♦❢ ❣r❛♣❤s ✭s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ❜✐♣❛rt✐t❡ ❞♦✉❜❧❡ ❝♦✈❡r ♦r ❝❛♥♦♥✐❝❛❧ ❞♦✉❜❧❡ ❝♦✈❡r✮✳ ❉♦✉❜❧❡ ❝♦✈❡r B(X) ♦❢ ❛ ❣r❛♣❤ X ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t X × K✷✳ ❚❤✐s ♠❡❛♥s t❤❛t V (B(X)) = V (X) × Z✷ ❛♥❞ ❛❧❧ t❤❡ ❡❞❣❡s ♦❢ B(X) ❛r❡ {(x, ✵), (y, ✶)} ❛♥❞ {(x, ✶), (y, ✵)} ✇❤❡r❡ {x, y} ✐s ❛♥ ❡❞❣❡ ✐♥ X✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
5
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
■t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ❆✉t(B(X)) ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷✳ ■❢ ❆✉t(B(X)) ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷ t❤❡♥ t❤❡ ❣r❛♣❤ X ✐s ❝❛❧❧❡❞ st❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✐t ✐s ❝❛❧❧❡❞ ✉♥st❛❜❧❡✳ ❚❤❡♦r❡♠ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t ❜❡ ❛ ♥♦♥✲❜✐♣❛rt✐t❡ ❣r❛♣❤✳ ❚❤❡♥ ✐ts ❞♦✉❜❧❡ ❝♦✈❡r ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♣♦s✐t✐♦♥ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ ■❢ ✐s st❛❜❧❡✱ t❤❡♥ ✐t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡r❡❢♦r❡✱ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ❣r❛♣❤ ✐s ✉♥st❛❜❧❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
■t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ❆✉t(B(X)) ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷✳ ■❢ ❆✉t(B(X)) ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷ t❤❡♥ t❤❡ ❣r❛♣❤ X ✐s ❝❛❧❧❡❞ st❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✐t ✐s ❝❛❧❧❡❞ ✉♥st❛❜❧❡✳ ❚❤❡♦r❡♠ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t X ❜❡ ❛ ♥♦♥✲❜✐♣❛rt✐t❡ ❣r❛♣❤✳ ❚❤❡♥ ✐ts ❞♦✉❜❧❡ ❝♦✈❡r ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ X ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♣♦s✐t✐♦♥ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ ■❢ ✐s st❛❜❧❡✱ t❤❡♥ ✐t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡r❡❢♦r❡✱ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ❣r❛♣❤ ✐s ✉♥st❛❜❧❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
■t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ❆✉t(B(X)) ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷✳ ■❢ ❆✉t(B(X)) ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❆✉t(X) × Z✷ t❤❡♥ t❤❡ ❣r❛♣❤ X ✐s ❝❛❧❧❡❞ st❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✐t ✐s ❝❛❧❧❡❞ ✉♥st❛❜❧❡✳ ❚❤❡♦r❡♠ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t X ❜❡ ❛ ♥♦♥✲❜✐♣❛rt✐t❡ ❣r❛♣❤✳ ❚❤❡♥ ✐ts ❞♦✉❜❧❡ ❝♦✈❡r ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ X ✐s ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♣♦s✐t✐♦♥ ✭▼❛r✉➨✐↔✱ ❙❝❛♣❡❧❧❛t♦✱ ❩❛❣❛❣❧✐❛ ❙❛❧✈✐✱ ✶✾✾✷✮ ▲❡t X ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤✳ ■❢ X ✐s st❛❜❧❡✱ t❤❡♥ ✐t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ ❚❤❡r❡❢♦r❡✱ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❈❛②❧❡② ❣r❛♣❤ ✐s ✉♥st❛❜❧❡✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡♠♠❛ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t X = GC(G, S, α) ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r (S, α)✱ ❛♥❞ ❧❡t Fix(α) = {g ∈ G | α(g) = g}✳ ❚❤❡♥ Fix(α)L ≤ ❆✉t(X) ❛♥❞ ♠♦r❡♦✈❡r ✐t ❛❝ts s❡♠✐r❡❣✉❧❛r❧② ♦♥ V (X)✳ ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲tr✐✈✐❛❧ ❡❧❡♠❡♥t ✱ ✇❤✐❝❤ ✐s ✜①❡❞ ❜② ✳ ▼♦r❡♦✈❡r✱ ❛❞♠✐ts ❛ s❡♠✐r❡❣✉❧❛r ❛✉t♦♠♦r♣❤✐s♠ ✇❤✐❝❤ ❧✐❡s ✐♥ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡♠♠❛ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t X = GC(G, S, α) ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r (S, α)✱ ❛♥❞ ❧❡t Fix(α) = {g ∈ G | α(g) = g}✳ ❚❤❡♥ Fix(α)L ≤ ❆✉t(X) ❛♥❞ ♠♦r❡♦✈❡r ✐t ❛❝ts s❡♠✐r❡❣✉❧❛r❧② ♦♥ V (X)✳ ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t X = GC(G, S, α) ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r (S, α)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲tr✐✈✐❛❧ ❡❧❡♠❡♥t g ∈ G✱ ✇❤✐❝❤ ✐s ✜①❡❞ ❜② α✳ ▼♦r❡♦✈❡r✱ X ❛❞♠✐ts ❛ s❡♠✐r❡❣✉❧❛r ❛✉t♦♠♦r♣❤✐s♠ ✇❤✐❝❤ ❧✐❡s ✐♥ GL ∩ Aut(X)✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t X = Cay(G, S) ❜❡ ❛ ❈❛②❧❡② ❣r❛♣❤✱ ❛♥❞ ❧❡t ❆✉t(G, S) ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G t❤❛t ✜① s❡t S✱ t❤❛t ✐s ❆✉t(G, S) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S}. ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❆✉t ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t ✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ✶ ✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ r❡s✉❧t✱ ❧❡t ✉s ❞❡✜♥❡ ❆✉t ❆✉t ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r ✳ ❚❤❡♥ ❆✉t ❆✉t ✇❤✐❝❤ ✜①❡s t❤❡ ✈❡rt❡① ✶ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t X = Cay(G, S) ❜❡ ❛ ❈❛②❧❡② ❣r❛♣❤✱ ❛♥❞ ❧❡t ❆✉t(G, S) ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G t❤❛t ✜① s❡t S✱ t❤❛t ✐s ❆✉t(G, S) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S}. ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❆✉t(G, S) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t(X)✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ✶G ∈ V (X)✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ r❡s✉❧t✱ ❧❡t ✉s ❞❡✜♥❡ ❆✉t ❆✉t ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r ✳ ❚❤❡♥ ❆✉t ❆✉t ✇❤✐❝❤ ✜①❡s t❤❡ ✈❡rt❡① ✶ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t X = Cay(G, S) ❜❡ ❛ ❈❛②❧❡② ❣r❛♣❤✱ ❛♥❞ ❧❡t ❆✉t(G, S) ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G t❤❛t ✜① s❡t S✱ t❤❛t ✐s ❆✉t(G, S) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S}. ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❆✉t(G, S) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t(X)✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ✶G ∈ V (X)✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ r❡s✉❧t✱ ❧❡t ✉s ❞❡✜♥❡ ❆✉t(G, S, α) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S, αϕ = ϕα}. ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r ✳ ❚❤❡♥ ❆✉t ❆✉t ✇❤✐❝❤ ✜①❡s t❤❡ ✈❡rt❡① ✶ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t X = Cay(G, S) ❜❡ ❛ ❈❛②❧❡② ❣r❛♣❤✱ ❛♥❞ ❧❡t ❆✉t(G, S) ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G t❤❛t ✜① s❡t S✱ t❤❛t ✐s ❆✉t(G, S) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S}. ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❆✉t(G, S) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❆✉t(X)✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ st❛❜✐❧✐③❡r ♦❢ ✶G ∈ V (X)✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ ❛❜♦✈❡ r❡s✉❧t✱ ❧❡t ✉s ❞❡✜♥❡ ❆✉t(G, S, α) = {ϕ ∈ ❆✉t(G) | ϕ(S) = S, αϕ = ϕα}. ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ▲❡t X = GC(G, S, α) ❜❡ ❛ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦r❞❡r❡❞ ♣❛✐r (S, α)✳ ❚❤❡♥ ❆✉t(G, S, α) ≤ ❆✉t(X) ✇❤✐❝❤ ✜①❡s t❤❡ ✈❡rt❡① ✶G ∈ V (X)✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ❋♦r ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r k ≥ ✶ ❧❡t n = ✷((✷k + ✶)✷ + ✶) ❛♥❞ ❧❡t X ❜❡ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ GC(Zn, S, α) ♦♥ t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ Zn ✇✐t❤ r❡s♣❡❝t t♦ S = {±✷, ±✹k✷, ✷k✷ + ✷k + ✶} ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ α ∈ ❆✉t(Zn) ❞❡✜♥❡❞ ❜② t❤❡ r✉❧❡ α(x) = ((✷k + ✶)✷ + ✷) · x✳ ❚❤❡♥ X ✐s ❛ ♥♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳ ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ❋♦r ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t ✷ ♠♦❞ ✺ ✱ ✷ ✶ ❛♥❞ ✷✵ ✱ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ ✷ ✹ ✺ ✶✵ ✺ ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❆✉t ❞❡✜♥❡❞ ❜② t❤❡ r✉❧❡ ✶✵ ✶ ✱ ✐s ❛ ♥♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ❋♦r ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r k ≥ ✶ ❧❡t n = ✷((✷k + ✶)✷ + ✶) ❛♥❞ ❧❡t X ❜❡ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ GC(Zn, S, α) ♦♥ t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ Zn ✇✐t❤ r❡s♣❡❝t t♦ S = {±✷, ±✹k✷, ✷k✷ + ✷k + ✶} ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ α ∈ ❆✉t(Zn) ❞❡✜♥❡❞ ❜② t❤❡ r✉❧❡ α(x) = ((✷k + ✶)✷ + ✷) · x✳ ❚❤❡♥ X ✐s ❛ ♥♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳ ❚❤❡♦r❡♠ ✭❍✱ ❑✉t♥❛r✱ ▼❛r✉➨✐↔✱ ✷✵✶✺✮ ❋♦r ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r k s✉❝❤ t❤❛t k ≡ ✷ (♠♦❞ ✺)✱ t = ✷k + ✶ ❛♥❞ n = ✷✵t✱ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ GC(Zn, S, α) ♦♥ t❤❡ ❝②❝❧✐❝ ❣r♦✉♣ Zn ✇✐t❤ r❡s♣❡❝t t♦ S = {±✷t, ±✹t, ✺, ✶✵t − ✺} ❛♥❞ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ α ∈ ❆✉t(Zn) ❞❡✜♥❡❞ ❜② t❤❡ r✉❧❡ α(x) = (✶✵t + ✶)x✱ ✐s ❛ ♥♦♥✲❈❛②❧❡② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t ωα : G → G ❜❡ t❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ❜② ωα(x) = α(x)x−✶ ❛♥❞ ❧❡t ωα(G) = {ωα(g) | g ∈ G}✳ ◆♦t✐❝❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ✭✐✐✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ ωα(G) ∩ S = ∅✳ Pr♦♣♦s✐t✐♦♥ ✭❛✮ ■❢ ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ t❤❡♥ ✐s ❛ s✉❜❣r♦✉♣ ♦❢ ❀ ✭❜✮ ❢♦r ❡✈❡r② ✱ ✐t ❤♦❧❞s
✶✳
❚❤❡♦r❡♠ ✭▼✐❧❧❡r✱ ✶✾✵✾✮ ■❢ ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r ❛♥❞ ❆✉t s✉❝❤ t❤❛t
✷
✶ t❤❡♥ ❋✐① ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
▲❡t ωα : G → G ❜❡ t❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ❜② ωα(x) = α(x)x−✶ ❛♥❞ ❧❡t ωα(G) = {ωα(g) | g ∈ G}✳ ◆♦t✐❝❡ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ✭✐✐✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ ωα(G) ∩ S = ∅✳ Pr♦♣♦s✐t✐♦♥ ✭❛✮ ■❢ G ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ t❤❡♥ ωα(G) ✐s ❛ s✉❜❣r♦✉♣ ♦❢ G❀ ✭❜✮ ❢♦r ❡✈❡r② x ∈ ωα(G)✱ ✐t ❤♦❧❞s α(x) = x−✶✳ ❚❤❡♦r❡♠ ✭▼✐❧❧❡r✱ ✶✾✵✾✮ ■❢ G ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r ❛♥❞ α ∈ ❆✉t(G) s✉❝❤ t❤❛t α✷ = ✶ t❤❡♥ G = ❋✐①(α) × ωα(G)✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡ ♣r❡✈✐♦✉s r❡s✉❧t ❡♥❛❜❧❡s ✉s t♦ ❞❡s❝r✐❜❡ ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤s ♦♥ ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r ✐♥ ❛ s✐♠♣❧❡ ✇❛②✳ ■❢ G ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r✱ ❛♥❞ α ∈ ❆✉t(G) s✉❝❤ t❤❛t α✷ = ✶✱ t❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ G = G✶ × G✷✱ ✇❤❡r❡ G✶ = ❋✐①(α)✱ G✷ = ωα(G)✳ ❚❤❡♥ ❢♦r (x, y) ∈ G✶ × G✷ ✇❡ ❤❛✈❡ α(x✶, x✷) = (x✶, x−✶
✷ )✳ ❚❤❡♥
GC(G, S, α) ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G✶ × G✷✳ ▲❡t S′ ⊆ G✶ × G✷ ❜❡ t❤❡ ✐♠❛❣❡ ♦❢ S✳ ❚❤❡♥ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t✿ ✭✐✮ S′ ∩ ({✶G✶} × G✷) = ∅❀ ✭✐✐✮ (s✶, s✷) ∈ S′ ⇔ (s−✶
✶ , s✷) ∈ S′✳
❚❤❡ ❡❞❣❡s ❛r❡ ♥♦✇ ❣✐✈❡♥ ✇✐t❤ (x✶, x✷) ∼ (x✶s✶, x−✶
✷ s✷)✱ ✇❤❡r❡
(s✶, s✷) ∈ S′✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ▲❡t G = Z✷n × H✱ ✇❤❡r❡ H ✐s ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r ❛♥❞ ❧❡t α ∈ ❆✉t(G) s✉❝❤ t❤❛t α✷ = ✶✳ ❚❤❡♥ H = H✶ × H✷✱ α(x, y✶, y✷) = (ax, y✶, y−✶
✷ )✱ ✇❤❡r❡ a ∈ {±✶, ✷n−✶ ± ✶}✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ▲❡t ❛♥❞ ❧❡t ❜❡ ❛ ✜♥✐t❡ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r✳ ❚❤❡♥ ❛♥② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤
✷
✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❉✐❤
✷
✶
✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ ✐♥✈❡rs✐♦♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✳ ❚❤❡♥ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ✐s ❈❛②❧❡② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✭✐✮ ✐s ❡❧❡♠❡♥t❛r② ❆❜❡❧✐❛♥ ✷✲❣r♦✉♣❀ ✭✐✐✮ ❙②❧♦✇ ✷✲s✉❜❣r♦✉♣ ♦❢ ✐s ❝②❝❧✐❝✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ▲❡t m ∈ N ❛♥❞ ❧❡t H ❜❡ ❛ ✜♥✐t❡ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r✳ ❚❤❡♥ ❛♥② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ X = GC(Z✷m × H, S, α) ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❉✐❤(Z✷m−✶ × H)✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ ✐♥✈❡rs✐♦♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✳ ❚❤❡♥ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ✐s ❈❛②❧❡② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✭✐✮ ✐s ❡❧❡♠❡♥t❛r② ❆❜❡❧✐❛♥ ✷✲❣r♦✉♣❀ ✭✐✐✮ ❙②❧♦✇ ✷✲s✉❜❣r♦✉♣ ♦❢ ✐s ❝②❝❧✐❝✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ▲❡t m ∈ N ❛♥❞ ❧❡t H ❜❡ ❛ ✜♥✐t❡ ❆❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦❞❞ ♦r❞❡r✳ ❚❤❡♥ ❛♥② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ X = GC(Z✷m × H, S, α) ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❉✐❤(Z✷m−✶ × H)✳ ❚❤❡♦r❡♠ ▲❡t G ❜❡ ❛♥ ❆❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ α ✐♥✈❡rs✐♦♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G✳ ❚❤❡♥ ❡✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦♥ G ✇✐t❤ r❡s♣❡❝t t♦ α ✐s ❈❛②❧❡② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✭✐✮ G ✐s ❡❧❡♠❡♥t❛r② ❆❜❡❧✐❛♥ ✷✲❣r♦✉♣❀ ✭✐✐✮ ❙②❧♦✇ ✷✲s✉❜❣r♦✉♣ ♦❢ G ✐s ❝②❝❧✐❝✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ❊✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ♦r❞❡r ✷p ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♦❢✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❣r♦✉♣s ♦❢ ♦r❞❡r ✷ ✱
✷
❛♥❞
✷ ✳
■❢
✷ ✱ t❤❡♥
♦r ✳ ▲❡t
✷ ✷ ✶ ✳
✇❤❡r❡ ✶
✷
✶ ✐♠♣❧✐❡s t❤❛t ✶ ♠♦❞ ❛♥❞ ✶ ✵ ♠♦❞
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ❊✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ♦r❞❡r ✷p ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♦❢✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❣r♦✉♣s ♦❢ ♦r❞❡r ✷p✱ Z✷p ❛♥❞ D✷p✳ ■❢
✷ ✱ t❤❡♥
♦r ✳ ▲❡t
✷ ✷ ✶ ✳
✇❤❡r❡ ✶
✷
✶ ✐♠♣❧✐❡s t❤❛t ✶ ♠♦❞ ❛♥❞ ✶ ✵ ♠♦❞
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ❊✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ♦r❞❡r ✷p ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♦❢✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❣r♦✉♣s ♦❢ ♦r❞❡r ✷p✱ Z✷p ❛♥❞ D✷p✳ ■❢ G = Z✷p✱ t❤❡♥ α = id ♦r α(x) = −x✳ ▲❡t
✷ ✷ ✶ ✳
✇❤❡r❡ ✶
✷
✶ ✐♠♣❧✐❡s t❤❛t ✶ ♠♦❞ ❛♥❞ ✶ ✵ ♠♦❞
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ❊✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ♦r❞❡r ✷p ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♦❢✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❣r♦✉♣s ♦❢ ♦r❞❡r ✷p✱ Z✷p ❛♥❞ D✷p✳ ■❢ G = Z✷p✱ t❤❡♥ α = id ♦r α(x) = −x✳ ▲❡t G = D✷p = τ, ρ | τ ✷ = ρp = id, τρτ = ρ−✶✳ α(ρ) = ρk ✇❤❡r❡ (k, p) = ✶; α(τ) = τρl.
✷
✶ ✐♠♣❧✐❡s t❤❛t ✶ ♠♦❞ ❛♥❞ ✶ ✵ ♠♦❞
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❚❤❡♦r❡♠ ❊✈❡r② ❣❡♥❡r❛❧✐③❡❞ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ♦r❞❡r ✷p ✐s ❛ ❈❛②❧❡② ❣r❛♣❤✳ Pr♦♦❢✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ❣r♦✉♣s ♦❢ ♦r❞❡r ✷p✱ Z✷p ❛♥❞ D✷p✳ ■❢ G = Z✷p✱ t❤❡♥ α = id ♦r α(x) = −x✳ ▲❡t G = D✷p = τ, ρ | τ ✷ = ρp = id, τρτ = ρ−✶✳ α(ρ) = ρk ✇❤❡r❡ (k, p) = ✶; α(τ) = τρl. α✷ = ✶ ✐♠♣❧✐❡s t❤❛t k ≡ ±✶ (♠♦❞ p) ❛♥❞ l(k + ✶) ≡ ✵ (♠♦❞ p).
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣