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❚❡st✐♥❣ ✐s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥❞r❡② ❱❛s✐❧✬❡✈✮ ■❧✐❛ P♦♥♦♠❛r❡♥❦♦ ❙t✳P❡t❡rs❜✉r❣ ❉❡♣❛rt♠❡♥t ♦❢ ❱✳❆✳❙t❡❦❧♦✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❲♦r❦s❤♦♣ ♦♥ ❛❧❣❡❜r❛✐❝ ❣r❛♣❤ t❤❡♦r②✱ P✐❧s❡♥✱ ❖❝t♦❜❡r ✸✕✼✱ ✷✵✶✻ ✶ ✴ ✶✺

❛♥❞ ❢♦r ❛♥❞ ✐s ✭✐❢ ♥♦t ❡♠♣t②✮ ✲❝♦s❡t ✐♥ ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ✭❈●■P✮ ❋♦r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ ✱ ✜♥❞ t❤❡ s❡t ✱ ✇❤❡r❡ ❛♥❞ ■♥♣✉t ❝♦♥s✐sts ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ♦❢ ❛♥❞ t❤❡ s❡ts ❖✉t♣✉t ✐s ❡✐t❤❡r ❡♠♣t② ♦r ❣✐✈❡♥ ❜② ❛ ♣❡r♠✉t❛t✐♦♥ ❢r♦♠ ❛♥❞ s♦♠❡ ❣❡♥❡r❛t✐♥❣ s❡t ♦❢ ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ G ✐s ❛ ✭✜♥✐t❡✮ ❣r♦✉♣✱ X ⊆ G ⇒ Γ = Cay( G , X ) ✿ V (Γ) = G ❛♥❞ E (Γ) = { ( g , xg ) | g ∈ G , x ∈ X } ✷ ✴ ✶✺

❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ✭❈●■P✮ ❋♦r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ ✱ ✜♥❞ t❤❡ s❡t ✱ ✇❤❡r❡ ❛♥❞ ■♥♣✉t ❝♦♥s✐sts ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ♦❢ ❛♥❞ t❤❡ s❡ts ❖✉t♣✉t ✐s ❡✐t❤❡r ❡♠♣t② ♦r ❣✐✈❡♥ ❜② ❛ ♣❡r♠✉t❛t✐♦♥ ❢r♦♠ ❛♥❞ s♦♠❡ ❣❡♥❡r❛t✐♥❣ s❡t ♦❢ ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ G ✐s ❛ ✭✜♥✐t❡✮ ❣r♦✉♣✱ X ⊆ G ⇒ Γ = Cay( G , X ) ✿ V (Γ) = G ❛♥❞ E (Γ) = { ( g , xg ) | g ∈ G , x ∈ X } Γ = Cay( G , X ) ❛♥❞ Γ ′ = Cay( G , X ′ ) Iso(Γ , Γ ′ ) = { f ∈ Sym( G ) | s f ∈ E (Γ ′ ) ❢♦r s ∈ E (Γ) } Aut(Γ) = Iso(Γ , Γ) ❛♥❞ G right ≤ Aut(Γ) ≤ Sym( G ) Iso(Γ , Γ ′ ) ✐s ✭✐❢ ♥♦t ❡♠♣t②✮ Aut(Γ) ✲❝♦s❡t ✐♥ Sym( G ) ✷ ✴ ✶✺

■♥♣✉t ❝♦♥s✐sts ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ♦❢ ❛♥❞ t❤❡ s❡ts ❖✉t♣✉t ✐s ❡✐t❤❡r ❡♠♣t② ♦r ❣✐✈❡♥ ❜② ❛ ♣❡r♠✉t❛t✐♦♥ ❢r♦♠ ❛♥❞ s♦♠❡ ❣❡♥❡r❛t✐♥❣ s❡t ♦❢ ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ G ✐s ❛ ✭✜♥✐t❡✮ ❣r♦✉♣✱ X ⊆ G ⇒ Γ = Cay( G , X ) ✿ V (Γ) = G ❛♥❞ E (Γ) = { ( g , xg ) | g ∈ G , x ∈ X } Γ = Cay( G , X ) ❛♥❞ Γ ′ = Cay( G , X ′ ) Iso(Γ , Γ ′ ) = { f ∈ Sym( G ) | s f ∈ E (Γ ′ ) ❢♦r s ∈ E (Γ) } Aut(Γ) = Iso(Γ , Γ) ❛♥❞ G right ≤ Aut(Γ) ≤ Sym( G ) Iso(Γ , Γ ′ ) ✐s ✭✐❢ ♥♦t ❡♠♣t②✮ Aut(Γ) ✲❝♦s❡t ✐♥ Sym( G ) ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ✭❈●■P✮ ❋♦r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ X , X ′ ⊆ G ✱ ✜♥❞ t❤❡ s❡t Iso(Γ , Γ ′ ) ✱ ✇❤❡r❡ Γ = Cay( G , X ) ❛♥❞ Γ ′ = Cay( G , X ′ ) ✷ ✴ ✶✺

❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ G ✐s ❛ ✭✜♥✐t❡✮ ❣r♦✉♣✱ X ⊆ G ⇒ Γ = Cay( G , X ) ✿ V (Γ) = G ❛♥❞ E (Γ) = { ( g , xg ) | g ∈ G , x ∈ X } Γ = Cay( G , X ) ❛♥❞ Γ ′ = Cay( G , X ′ ) Iso(Γ , Γ ′ ) = { f ∈ Sym( G ) | s f ∈ E (Γ ′ ) ❢♦r s ∈ E (Γ) } Aut(Γ) = Iso(Γ , Γ) ❛♥❞ G right ≤ Aut(Γ) ≤ Sym( G ) Iso(Γ , Γ ′ ) ✐s ✭✐❢ ♥♦t ❡♠♣t②✮ Aut(Γ) ✲❝♦s❡t ✐♥ Sym( G ) ❈❛②❧❡② ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ✭❈●■P✮ ❋♦r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ X , X ′ ⊆ G ✱ ✜♥❞ t❤❡ s❡t Iso(Γ , Γ ′ ) ✱ ✇❤❡r❡ Γ = Cay( G , X ) ❛♥❞ Γ ′ = Cay( G , X ′ ) ■♥♣✉t ❝♦♥s✐sts ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ♦❢ G ❛♥❞ t❤❡ s❡ts X , X ′ ❖✉t♣✉t Iso(Γ , Γ ′ ) ✐s ❡✐t❤❡r ❡♠♣t② ♦r ❣✐✈❡♥ ❜② ❛ ♣❡r♠✉t❛t✐♦♥ ❢r♦♠ Iso(Γ , Γ ′ ) ❛♥❞ s♦♠❡ ❣❡♥❡r❛t✐♥❣ s❡t ♦❢ Aut(Γ) ✷ ✴ ✶✺

❈●■P ●r♦✉♣ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ❈●■P ❢♦r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✱ ❛♥❞ ▼✉③②❝❤✉❦✱ ✷✵✵✹✮ ❈●■P ❢♦r t❤❡ ❈■✲❣r♦✉♣s ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ t✐♠❡ ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤✿ ❲❤❡t❤❡r ❛ ❣✐✈❡♥ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛ ❣✐✈❡♥ ❣r♦✉♣❄ ❙❛❜✐❞✉ss✐✬s ❝r✐t❡r✐♦♥✿ ❋♦r ❛ ❣r♦✉♣ ✱ t❤❡ ❣r❛♣❤ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ❝♦♥t❛✐♥s ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ ■♥ ❣❡♥❡r❛❧✱ t❤❡ r❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤s ✐s ♣r♦❜❛❜❧② ❡❛s✐❡r t❤❛♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❞❡t❡r♠✐♥✐♥❣ ✇❤❡t❤❡r ❛ ❣r❛♣❤ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ✐s ◆P✲❝♦♠♣❧❡t❡ ✭❆✳ ▲✉❜✐✇✱ ✶✾✽✶✮ ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✮ ❇❛❜❛✐✬s ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❈●■P ✐♥ q✉❛s✐♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✸ ✴ ✶✺

❈●■P ❢♦r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✱ ❛♥❞ ▼✉③②❝❤✉❦✱ ✷✵✵✹✮ ❈●■P ❢♦r t❤❡ ❈■✲❣r♦✉♣s ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ t✐♠❡ ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤✿ ❲❤❡t❤❡r ❛ ❣✐✈❡♥ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛ ❣✐✈❡♥ ❣r♦✉♣❄ ❙❛❜✐❞✉ss✐✬s ❝r✐t❡r✐♦♥✿ ❋♦r ❛ ❣r♦✉♣ ✱ t❤❡ ❣r❛♣❤ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ❝♦♥t❛✐♥s ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ ■♥ ❣❡♥❡r❛❧✱ t❤❡ r❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤s ✐s ♣r♦❜❛❜❧② ❡❛s✐❡r t❤❛♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❞❡t❡r♠✐♥✐♥❣ ✇❤❡t❤❡r ❛ ❣r❛♣❤ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠✱ ✇❤✐❝❤ ✐s ◆P✲❝♦♠♣❧❡t❡ ✭❆✳ ▲✉❜✐✇✱ ✶✾✽✶✮ ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✮ ❇❛❜❛✐✬s ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❈●■P ✐♥ q✉❛s✐♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❈●■P ⇒ ●r♦✉♣ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ✸ ✴ ✶✺