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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 ✸✕✼✱ ✷✵✶✻

✶ ✴ ✶✺

❈❛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} ❛♥❞ ❢♦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} Γ = Cay(G, X) ❛♥❞ Γ′ = Cay(G, X ′) Iso(Γ, Γ′) = {f ∈ Sym(G) | sf ∈ E(Γ′) ❢♦r s ∈ E(Γ)} Aut(Γ) = Iso(Γ, Γ) ❛♥❞ Gright ≤ 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♦✉♣ ❛♥❞ ✱ ✜♥❞ 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) | sf ∈ E(Γ′) ❢♦r s ∈ E(Γ)} Aut(Γ) = Iso(Γ, Γ) ❛♥❞ Gright ≤ 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❛❜❧❡ ♦❢ ❛♥❞ 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) | sf ∈ E(Γ′) ❢♦r s ∈ E(Γ)} Aut(Γ) = Iso(Γ, Γ) ❛♥❞ Gright ≤ 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)

✷ ✴ ✶✺

❇❛❜❛✐✬s ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❈●■P ✐♥ q✉❛s✐♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❈●■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♦✉♣ ■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♦✉♣ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠ ❈●■P ❢♦r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✱ ❛♥❞ ▼✉③②❝❤✉❦✱ ✷✵✵✹✮ ❈●■P ❢♦r t❤❡ ❈■✲❣r♦✉♣s ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ t✐♠❡ poly(| Aut(G)|) ❘❡❝♦❣♥✐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♦❜❧❡♠ ❈●■P ❢♦r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✱ ❛♥❞ ▼✉③②❝❤✉❦✱ ✷✵✵✹✮ ❈●■P ❢♦r t❤❡ ❈■✲❣r♦✉♣s ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ t✐♠❡ poly(| Aut(G)|) ❘❡❝♦❣♥✐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♦✉♣ G✱ t❤❡ ❣r❛♣❤ Γ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r G ⇔ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ) ❝♦♥t❛✐♥s ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ G ■♥ ❣❡♥❡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♦❜❧❡♠ ❈●■P ❢♦r t❤❡ ❝②❝❧✐❝ ❣r♦✉♣s ✐s s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭❊✈❞♦❦✐♠♦✈✲P✱ ✷✵✵✸✱ ❛♥❞ ▼✉③②❝❤✉❦✱ ✷✵✵✹✮ ❈●■P ❢♦r t❤❡ ❈■✲❣r♦✉♣s ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ t✐♠❡ poly(| Aut(G)|) ❘❡❝♦❣♥✐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♦✉♣ G✱ t❤❡ ❣r❛♣❤ Γ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r G ⇔ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ) ❝♦♥t❛✐♥s ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ G ■♥ ❣❡♥❡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✱ ✷✵✵✸✮

✸ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s

G ✐s ❛ ❣r♦✉♣✱ X ⊆ G✱ ❛♥❞ Γ = Cay(G, X) Γ ✐s s❛✐❞ t♦ ❜❡ ❝❡♥tr❛❧ ✐❢ X ✐s ❛ ♥♦r♠❛❧ s✉❜s❡t ✐♥ G✱ ✐✳❡✳✱ X g = X ❢♦r ❡✈❡r② g ∈ G✳

Pr♦♣♦s✐t✐♦♥

❆♥② ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✐s ❝❡♥tr❛❧ ■❢ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ ■❢ ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ ❜❡❝❛✉s❡ ◆♦t❡ t❤❛t ✭❛✮ ❛♥❞ ❝❡♥tr❛❧✐③❡ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ✭❜✮ ✱ s♦ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ ✳

✹ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s

G ✐s ❛ ❣r♦✉♣✱ X ⊆ G✱ ❛♥❞ Γ = Cay(G, X) Γ ✐s s❛✐❞ t♦ ❜❡ ❝❡♥tr❛❧ ✐❢ X ✐s ❛ ♥♦r♠❛❧ s✉❜s❡t ✐♥ G✱ ✐✳❡✳✱ X g = X ❢♦r ❡✈❡r② g ∈ G✳

Pr♦♣♦s✐t✐♦♥

❆♥② ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✐s ❝❡♥tr❛❧ ■❢ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ ■❢ ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ ❜❡❝❛✉s❡ ◆♦t❡ t❤❛t ✭❛✮ ❛♥❞ ❝❡♥tr❛❧✐③❡ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ✭❜✮ ✱ s♦ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ ✳

✹ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s

G ✐s ❛ ❣r♦✉♣✱ X ⊆ G✱ ❛♥❞ Γ = Cay(G, X) Γ ✐s s❛✐❞ t♦ ❜❡ ❝❡♥tr❛❧ ✐❢ X ✐s ❛ ♥♦r♠❛❧ s✉❜s❡t ✐♥ G✱ ✐✳❡✳✱ X g = X ❢♦r ❡✈❡r② g ∈ G✳

Pr♦♣♦s✐t✐♦♥

❆♥② ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✐s ❝❡♥tr❛❧ ■❢ Γ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ Gright ≤ Aut(Γ) ■❢ ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ ❜❡❝❛✉s❡ ◆♦t❡ t❤❛t ✭❛✮ ❛♥❞ ❝❡♥tr❛❧✐③❡ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ✭❜✮ ✱ s♦ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ ✳

✹ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s

G ✐s ❛ ❣r♦✉♣✱ X ⊆ G✱ ❛♥❞ Γ = Cay(G, X) Γ ✐s s❛✐❞ t♦ ❜❡ ❝❡♥tr❛❧ ✐❢ X ✐s ❛ ♥♦r♠❛❧ s✉❜s❡t ✐♥ G✱ ✐✳❡✳✱ X g = X ❢♦r ❡✈❡r② g ∈ G✳

Pr♦♣♦s✐t✐♦♥

❆♥② ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✐s ❝❡♥tr❛❧ ■❢ Γ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ Gright ≤ Aut(Γ) ■❢ Γ ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ GleftGright ≤ Aut(Γ) ❜❡❝❛✉s❡ h(g, xg) = (hg, xh−1hg) = (hg, x′(hg)) ◆♦t❡ t❤❛t ✭❛✮ ❛♥❞ ❝❡♥tr❛❧✐③❡ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ✭❜✮ ✱ s♦ ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ ✳

✹ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s

G ✐s ❛ ❣r♦✉♣✱ X ⊆ G✱ ❛♥❞ Γ = Cay(G, X) Γ ✐s s❛✐❞ t♦ ❜❡ ❝❡♥tr❛❧ ✐❢ X ✐s ❛ ♥♦r♠❛❧ s✉❜s❡t ✐♥ G✱ ✐✳❡✳✱ X g = X ❢♦r ❡✈❡r② g ∈ G✳

Pr♦♣♦s✐t✐♦♥

❆♥② ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❛❜❡❧✐❛♥ ❣r♦✉♣ ✐s ❝❡♥tr❛❧ ■❢ Γ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ Gright ≤ Aut(Γ) ■❢ Γ ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ t❤❡♥ GleftGright ≤ Aut(Γ) ❜❡❝❛✉s❡ h(g, xg) = (hg, xh−1hg) = (hg, x′(hg)) ◆♦t❡ t❤❛t ✭❛✮ Gleft ❛♥❞ Gright ❝❡♥tr❛❧✐③❡ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ ✭❜✮ Gleft ∩ Gright = {hright | h ∈ Z(G)}✱ s♦ Z(G) = 1 ⇒ GleftGright ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t✇♦ ❝♦♣✐❡s ♦❢ G✳

✹ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s ♦✈❡r ❆❧♠♦st ❙✐♠♣❧❡ ●r♦✉♣s

S ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✭S ≃ Inn(S)) G ✐s ❝❛❧❧❡❞ ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ ✐❢ S ≤ G ≤ Aut(S) S = Soc(G) ✐s t❤❡ s♦❝❧❡ ♦❢ G

❖✉r ●♦❛❧

❚❡st ✐s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛♥ ❛r❜✐tr❛r② ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛ s②♠♠❡tr✐❝ ❣r♦✉♣ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r♦✉♣ ■♥❞❡❡❞✱ ✐❢ ✱ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ✐s ❡q✉❛❧ t♦ ✱ ✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣❛rt✐t✐♦♥s ♦❢ ✳ ❙✐♥❝❡ ✐s ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ✱ t❤❡ ♥✉♠❜❡r ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥

✺ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s ♦✈❡r ❆❧♠♦st ❙✐♠♣❧❡ ●r♦✉♣s

S ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✭S ≃ Inn(S)) G ✐s ❝❛❧❧❡❞ ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ ✐❢ S ≤ G ≤ Aut(S) S = Soc(G) ✐s t❤❡ s♦❝❧❡ ♦❢ G

❖✉r ●♦❛❧

❚❡st ✐s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛♥ ❛r❜✐tr❛r② ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛ s②♠♠❡tr✐❝ ❣r♦✉♣ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r♦✉♣ ■♥❞❡❡❞✱ ✐❢ ✱ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ✐s ❡q✉❛❧ t♦ ✱ ✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣❛rt✐t✐♦♥s ♦❢ ✳ ❙✐♥❝❡ ✐s ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ✱ t❤❡ ♥✉♠❜❡r ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥

✺ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s ♦✈❡r ❆❧♠♦st ❙✐♠♣❧❡ ●r♦✉♣s

S ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✭S ≃ Inn(S)) G ✐s ❝❛❧❧❡❞ ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ ✐❢ S ≤ G ≤ Aut(S) S = Soc(G) ✐s t❤❡ s♦❝❧❡ ♦❢ G

❖✉r ●♦❛❧

❚❡st ✐s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛♥ ❛r❜✐tr❛r② ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛ s②♠♠❡tr✐❝ ❣r♦✉♣ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r♦✉♣ ■♥❞❡❡❞✱ ✐❢ ✱ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ✐s ❡q✉❛❧ t♦ ✱ ✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣❛rt✐t✐♦♥s ♦❢ ✳ ❙✐♥❝❡ ✐s ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ ✱ t❤❡ ♥✉♠❜❡r ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥

✺ ✴ ✶✺

❈❡♥tr❛❧ ❈❛②❧❡② ●r❛♣❤s ♦✈❡r ❆❧♠♦st ❙✐♠♣❧❡ ●r♦✉♣s

S ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✭S ≃ Inn(S)) G ✐s ❝❛❧❧❡❞ ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ ✐❢ S ≤ G ≤ Aut(S) S = Soc(G) ✐s t❤❡ s♦❝❧❡ ♦❢ G

❖✉r ●♦❛❧

❚❡st ✐s♦♠♦r♣❤✐s♠ ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛♥ ❛r❜✐tr❛r② ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

Pr♦♣♦s✐t✐♦♥

❚❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r ❛ s②♠♠❡tr✐❝ ❣r♦✉♣ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❣r♦✉♣ ■♥❞❡❡❞✱ ✐❢ G = Sym(n)✱ t❤❡♥ t❤❡ ♥✉♠❜❡r N(n) ♦❢ t❤❡ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s ♦✈❡r G ✐s ❡q✉❛❧ t♦ 2p(n)✱ ✇❤❡r❡ p(n) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣❛rt✐t✐♦♥s ♦❢ n✳ ❙✐♥❝❡ p(n) ✐s ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧ t♦ 2

√n✱ t❤❡

♥✉♠❜❡r N(n) ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ |G| = n!

✺ ✴ ✶✺

▼❛✐♥ ❘❡s✉❧ts✳ P❛rt ✶

❚❤❡♦r❡♠ ✶

❋♦r ❛♥② t✇♦ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s Γ ❛♥❞ Γ′ ♦✈❡r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ G ♦❢ ♦r❞❡r n✱ t❤❡ s❡t Iso(Γ, Γ′) ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t✐♠❡ poly(n)✳

❈♦r♦❧❧❛r②

❚❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤ ♦✈❡r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ G ♦❢ ♦r❞❡r n ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t✐♠❡ poly(n)✳

✻ ✴ ✶✺

❈❛②❧❡② ❘❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❘❡❣✉❧❛r ❙✉❜❣r♦✉♣s

Γ = Cay(G, X) ❛♥❞ Γ′ = Cay(G, X ′) IsoCay(Γ, Γ′) = Aut(G) ∩ Iso(Γ, Γ′) Γ ❛♥❞ Γ′ ❛r❡ ❝❛❧❧❡❞ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝ ✐❢ IsoCay(Γ, Γ′) = ∅ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❣r❛♣❤ ♦✈❡r ❛ ❣r♦✉♣ ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ✐s♦♠♦r♣❤✐❝ t♦ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝

• ✐✈❡♥ ❛ ❣r♦✉♣

✱ ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s s❛✐❞ t♦ ❜❡ ✲r❡❣✉❧❛r✱ ✐❢ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✳

Pr♦♣♦s✐t✐♦♥ ✭❇❛❜❛✐✱ ✶✾✼✼✮

❚❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♥♦♥❡q✉✐✈❛❧❡♥t ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❛ ❣r❛♣❤ ♦✈❡r ❛ ❣r♦✉♣ ❛♥❞ t❤❡ ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ ✲r❡❣✉❧❛r s✉❜❣r♦✉♣s ♦❢ ✳

✼ ✴ ✶✺

❈❛②❧❡② ❘❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❘❡❣✉❧❛r ❙✉❜❣r♦✉♣s

Γ = Cay(G, X) ❛♥❞ Γ′ = Cay(G, X ′) IsoCay(Γ, Γ′) = Aut(G) ∩ Iso(Γ, Γ′) Γ ❛♥❞ Γ′ ❛r❡ ❝❛❧❧❡❞ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝ ✐❢ IsoCay(Γ, Γ′) = ∅ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❣r❛♣❤ Γ ♦✈❡r ❛ ❣r♦✉♣ G ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ Cay(G, X) ✐s♦♠♦r♣❤✐❝ t♦ Γ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ Γ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝

• ✐✈❡♥ ❛ ❣r♦✉♣

✱ ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s s❛✐❞ t♦ ❜❡ ✲r❡❣✉❧❛r✱ ✐❢ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✳

Pr♦♣♦s✐t✐♦♥ ✭❇❛❜❛✐✱ ✶✾✼✼✮

❚❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♥♦♥❡q✉✐✈❛❧❡♥t ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❛ ❣r❛♣❤ ♦✈❡r ❛ ❣r♦✉♣ ❛♥❞ t❤❡ ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ ✲r❡❣✉❧❛r s✉❜❣r♦✉♣s ♦❢ ✳

✼ ✴ ✶✺

❈❛②❧❡② ❘❡♣r❡s❡♥t❛t✐♦♥s ❛♥❞ ❘❡❣✉❧❛r ❙✉❜❣r♦✉♣s

Γ = Cay(G, X) ❛♥❞ Γ′ = Cay(G, X ′) IsoCay(Γ, Γ′) = Aut(G) ∩ Iso(Γ, Γ′) Γ ❛♥❞ Γ′ ❛r❡ ❝❛❧❧❡❞ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝ ✐❢ IsoCay(Γ, Γ′) = ∅ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ❣r❛♣❤ Γ ♦✈❡r ❛ ❣r♦✉♣ G ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ Cay(G, X) ✐s♦♠♦r♣❤✐❝ t♦ Γ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ Γ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡② ❛r❡ ❈❛②❧❡② ✐s♦♠♦r♣❤✐❝

• ✐✈❡♥ ❛ ❣r♦✉♣ G✱ ❛ r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✐s

s❛✐❞ t♦ ❜❡ G✲r❡❣✉❧❛r✱ ✐❢ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ G✳

Pr♦♣♦s✐t✐♦♥ ✭❇❛❜❛✐✱ ✶✾✼✼✮

❚❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♥♦♥❡q✉✐✈❛❧❡♥t ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❛ ❣r❛♣❤ Γ ♦✈❡r ❛ ❣r♦✉♣ G ❛♥❞ t❤❡ ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s ♦❢ G✲r❡❣✉❧❛r s✉❜❣r♦✉♣s ♦❢ Aut(Γ)✳

✼ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ ❛ ❣r♦✉♣ ❛♥❞ K ≤ Sym(Ω)✳ ❆ s❡t B ♦❢ G✲r❡❣✉❧❛r ❣r♦✉♣s H ≤ K ✐s ❝❛❧❧❡❞ ❛ G✲❜❛s❡ ♦❢ K ✐❢ ❡✈❡r② G✲r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ K ✐s ❝♦♥❥✉❣❛t❡ ✐♥ K t♦ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ H✳ ❙❡t bG(K) = |B|✳ ❋♦r ♣✉t ■♥ t❤✐s ❝❛s❡ ❞✉❡ t♦ ❇❛❜❛✐✬s ❛r❣✉♠❡♥t ②✐❡❧❞s t❤❛t ✐s ❈■✲❣r❛♣❤ ❈●■P ✐s r❡❞✉❝✐❜❧❡ ✐♥ t✐♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ t♦ t❤❡ ♣r♦❜❧❡♠✿

• ✐✈❡♥ ❛ ❈❛②❧❡② ❣r❛♣❤

♦✈❡r ❛ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢

✽ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ ❛ ❣r♦✉♣ ❛♥❞ K ≤ Sym(Ω)✳ ❆ s❡t B ♦❢ G✲r❡❣✉❧❛r ❣r♦✉♣s H ≤ K ✐s ❝❛❧❧❡❞ ❛ G✲❜❛s❡ ♦❢ K ✐❢ ❡✈❡r② G✲r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ K ✐s ❝♦♥❥✉❣❛t❡ ✐♥ K t♦ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ H✳ ❙❡t bG(K) = |B|✳ ❋♦r Γ = Cay(G, X) ♣✉t bG(Γ) = bG(Aut(Γ)) ■♥ t❤✐s ❝❛s❡ bG(Γ) ≥ 1 ❞✉❡ t♦ Gright ≤ Aut(Γ) ❇❛❜❛✐✬s ❛r❣✉♠❡♥t ②✐❡❧❞s t❤❛t ✐s ❈■✲❣r❛♣❤ ❈●■P ✐s r❡❞✉❝✐❜❧❡ ✐♥ t✐♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ t♦ t❤❡ ♣r♦❜❧❡♠✿

• ✐✈❡♥ ❛ ❈❛②❧❡② ❣r❛♣❤

♦✈❡r ❛ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢

✽ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ ❛ ❣r♦✉♣ ❛♥❞ K ≤ Sym(Ω)✳ ❆ s❡t B ♦❢ G✲r❡❣✉❧❛r ❣r♦✉♣s H ≤ K ✐s ❝❛❧❧❡❞ ❛ G✲❜❛s❡ ♦❢ K ✐❢ ❡✈❡r② G✲r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ K ✐s ❝♦♥❥✉❣❛t❡ ✐♥ K t♦ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ H✳ ❙❡t bG(K) = |B|✳ ❋♦r Γ = Cay(G, X) ♣✉t bG(Γ) = bG(Aut(Γ)) ■♥ t❤✐s ❝❛s❡ bG(Γ) ≥ 1 ❞✉❡ t♦ Gright ≤ Aut(Γ) ❇❛❜❛✐✬s ❛r❣✉♠❡♥t ②✐❡❧❞s t❤❛t Γ ✐s ❈■✲❣r❛♣❤ ⇔ bG(Γ) = 1 ❈●■P ✐s r❡❞✉❝✐❜❧❡ ✐♥ t✐♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ t♦ t❤❡ ♣r♦❜❧❡♠✿

• ✐✈❡♥ ❛ ❈❛②❧❡② ❣r❛♣❤

♦✈❡r ❛ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢

✽ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣

❉❡✜♥✐t✐♦♥

▲❡t G ❜❡ ❛ ❣r♦✉♣ ❛♥❞ K ≤ Sym(Ω)✳ ❆ s❡t B ♦❢ G✲r❡❣✉❧❛r ❣r♦✉♣s H ≤ K ✐s ❝❛❧❧❡❞ ❛ G✲❜❛s❡ ♦❢ K ✐❢ ❡✈❡r② G✲r❡❣✉❧❛r s✉❜❣r♦✉♣ ♦❢ K ✐s ❝♦♥❥✉❣❛t❡ ✐♥ K t♦ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ H✳ ❙❡t bG(K) = |B|✳ ❋♦r Γ = Cay(G, X) ♣✉t bG(Γ) = bG(Aut(Γ)) ■♥ t❤✐s ❝❛s❡ bG(Γ) ≥ 1 ❞✉❡ t♦ Gright ≤ Aut(Γ) ❇❛❜❛✐✬s ❛r❣✉♠❡♥t ②✐❡❧❞s t❤❛t Γ ✐s ❈■✲❣r❛♣❤ ⇔ bG(Γ) = 1 ❈●■P ✐s r❡❞✉❝✐❜❧❡ ✐♥ t✐♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ bG(Γ) t♦ t❤❡ ♣r♦❜❧❡♠✿

• ✐✈❡♥ ❛ ❈❛②❧❡② ❣r❛♣❤ Γ ♦✈❡r ❛ ❣r♦✉♣ G✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ Aut(Γ)

✽ ✴ ✶✺

▼❛✐♥ ❘❡s✉❧ts✳ P❛rt ✷

▲❡t Gn st❛♥❞ ❢♦r t❤❡ s❡t ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s Γ ♦✈❡r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❣r♦✉♣ G ♦❢ ♦r❞❡r n ✇✐t❤ ❛ s✐♠♣❧❡ s♦❝❧❡ ❛♥❞ ❛ ❝②❝❧✐❝ q✉♦t✐❡♥t G/ Soc(G)✳

❚❤❡♦r❡♠ ✷

❋♦r ❡✈❡r② Γ ∈ Gn✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ Aut(Γ) ✐♥ t✐♠❡ poly(n)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❢✉❧❧ s②st❡♠ ♦❢ ♣❛✐r✇✐s❡ ♥♦♥❡q✉✐✈❛❧❡♥t ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ Γ ❝❛♥ ❜❡ ❢♦✉♥❞ ✇✐t❤✐♥ t❤❡ s❛♠❡ t✐♠❡✳ ❆ ❝❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ♦❢ ❡✈❡r② ❣r❛♣❤ ✐♥ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ✐♥ t✐♠❡ ✳

✾ ✴ ✶✺

▼❛✐♥ ❘❡s✉❧ts✳ P❛rt ✷

▲❡t Gn st❛♥❞ ❢♦r t❤❡ s❡t ♦❢ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤s Γ ♦✈❡r ❛♥ ❡①♣❧✐❝✐t❧② ❣✐✈❡♥ ❣r♦✉♣ G ♦❢ ♦r❞❡r n ✇✐t❤ ❛ s✐♠♣❧❡ s♦❝❧❡ ❛♥❞ ❛ ❝②❝❧✐❝ q✉♦t✐❡♥t G/ Soc(G)✳

❚❤❡♦r❡♠ ✷

❋♦r ❡✈❡r② Γ ∈ Gn✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ Aut(Γ) ✐♥ t✐♠❡ poly(n)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❢✉❧❧ s②st❡♠ ♦❢ ♣❛✐r✇✐s❡ ♥♦♥❡q✉✐✈❛❧❡♥t ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ Γ ❝❛♥ ❜❡ ❢♦✉♥❞ ✇✐t❤✐♥ t❤❡ s❛♠❡ t✐♠❡✳ ❆ ❝❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ♦❢ ❡✈❡r② ❣r❛♣❤ ✐♥ Gn ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ✐♥ t✐♠❡ poly(n)✳

✾ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣✳ ❘❡♠❛r❦s

Pr♦❜❧❡♠

❋♦r ❛ ❣r♦✉♣ G ❛♥❞ K ≤ Sym(G)✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ K

❊✈❞♦❦✐♠♦✈✱ ▼✉③②❝❤✉❦✱ P✱ ✷✵✶✻✿

❋♦r ❡✈❡r② ♣r✐♠❡ t❤❡r❡ ✐s s✉❝❤ t❤❛t ✱ ✇❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r ◆♦t❡ t❤❛t ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧② ✐♥ t❤❡ ♦r❞❡r ♦❢ ❛s ❣r♦✇s✱ ❜✉t t❤❡ ❣r♦✉♣ ❝❛♥♥♦t ❜❡ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛♥② ❣r❛♣❤✳

Pr♦❜❧❡♠ ✭❝♦rr❡❝t❡❞ ✈❡rs✐♦♥✮

❋♦r ❛ ❣r♦✉♣ ❛♥❞ ❛ ✲❝❧♦s❡❞ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢ ✐s ✲❝❧♦s❡❞ ✐❢

✶✵ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣✳ ❘❡♠❛r❦s

Pr♦❜❧❡♠

❋♦r ❛ ❣r♦✉♣ G ❛♥❞ K ≤ Sym(G)✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ K

❊✈❞♦❦✐♠♦✈✱ ▼✉③②❝❤✉❦✱ P✱ ✷✵✶✻✿

❋♦r ❡✈❡r② ♣r✐♠❡ p t❤❡r❡ ✐s K ≤ Sym(p3) s✉❝❤ t❤❛t bG(K) ≥ pp−2✱ ✇❤❡r❡ G ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r p3 ◆♦t❡ t❤❛t ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧② ✐♥ t❤❡ ♦r❞❡r ♦❢ ❛s ❣r♦✇s✱ ❜✉t t❤❡ ❣r♦✉♣ ❝❛♥♥♦t ❜❡ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛♥② ❣r❛♣❤✳

Pr♦❜❧❡♠ ✭❝♦rr❡❝t❡❞ ✈❡rs✐♦♥✮

❋♦r ❛ ❣r♦✉♣ ❛♥❞ ❛ ✲❝❧♦s❡❞ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢ ✐s ✲❝❧♦s❡❞ ✐❢

✶✵ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣✳ ❘❡♠❛r❦s

Pr♦❜❧❡♠

❋♦r ❛ ❣r♦✉♣ G ❛♥❞ K ≤ Sym(G)✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ K

❊✈❞♦❦✐♠♦✈✱ ▼✉③②❝❤✉❦✱ P✱ ✷✵✶✻✿

❋♦r ❡✈❡r② ♣r✐♠❡ p t❤❡r❡ ✐s K ≤ Sym(p3) s✉❝❤ t❤❛t bG(K) ≥ pp−2✱ ✇❤❡r❡ G ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r p3 ◆♦t❡ t❤❛t bG(K) ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧② ✐♥ t❤❡ ♦r❞❡r ♦❢ G ❛s p ❣r♦✇s✱ ❜✉t t❤❡ ❣r♦✉♣ ❝❛♥♥♦t ❜❡ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛♥② ❣r❛♣❤✳

Pr♦❜❧❡♠ ✭❝♦rr❡❝t❡❞ ✈❡rs✐♦♥✮

❋♦r ❛ ❣r♦✉♣ ❛♥❞ ❛ ✲❝❧♦s❡❞ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✱ ✜♥❞ ❛ ✲❜❛s❡ ♦❢ ✐s ✲❝❧♦s❡❞ ✐❢

✶✵ ✴ ✶✺

G✲❜❛s❡ ♦❢ ❛ P❡r♠✉t❛t✐♦♥ ●r♦✉♣✳ ❘❡♠❛r❦s

Pr♦❜❧❡♠

❋♦r ❛ ❣r♦✉♣ G ❛♥❞ K ≤ Sym(G)✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ K

❊✈❞♦❦✐♠♦✈✱ ▼✉③②❝❤✉❦✱ P✱ ✷✵✶✻✿

❋♦r ❡✈❡r② ♣r✐♠❡ p t❤❡r❡ ✐s K ≤ Sym(p3) s✉❝❤ t❤❛t bG(K) ≥ pp−2✱ ✇❤❡r❡ G ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r p3 ◆♦t❡ t❤❛t bG(K) ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧② ✐♥ t❤❡ ♦r❞❡r ♦❢ G ❛s p ❣r♦✇s✱ ❜✉t t❤❡ ❣r♦✉♣ K ❝❛♥♥♦t ❜❡ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ ❛♥② ❣r❛♣❤✳

Pr♦❜❧❡♠ ✭❝♦rr❡❝t❡❞ ✈❡rs✐♦♥✮

❋♦r ❛ ❣r♦✉♣ G ❛♥❞ ❛ 2✲❝❧♦s❡❞ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ K ≤ Sym(G)✱ ✜♥❞ ❛ G✲❜❛s❡ ♦❢ K H ≤ Sym(Ω) ✐s 2✲❝❧♦s❡❞ ✐❢ H = H(2) = Aut(Orb(H, Ω × Ω))

✶✵ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ❆♥❛❧②s✐s

G ✐s ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ |G| = n✱ S = Soc(G)✱ X ⊆ G Γ = Cay(G, X) ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤✱ K = Aut(Γ) ❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s✿

✶

✐s ♣r✐♠✐t✐✈❡✱ t❤❡♥ ♦r

✷

✐s ✐♠♣r✐♠✐t✐✈❡✱ t❤❡♥ ♦r ✐s ❛ ♥♦♥tr✐✈✐❛❧ ❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❉✳❑✳ ❋❛❞❞❡❡✈ ✐♥ ✶✾✺✵✬s ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ✐♥✈❡rs❡

• ❛❧♦✐s ♣r♦❜❧❡♠

✶✶ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ❆♥❛❧②s✐s

G ✐s ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ |G| = n✱ S = Soc(G)✱ X ⊆ G Γ = Cay(G, X) ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤✱ K = Aut(Γ) ❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s✿

✶ K ✐s ♣r✐♠✐t✐✈❡✱ t❤❡♥ K = Sym(G) ♦r |K| = poly(n) ✷ K ✐s ✐♠♣r✐♠✐t✐✈❡✱ t❤❡♥ |K| = poly(n) ♦r K ✐s ❛ ♥♦♥tr✐✈✐❛❧

❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❉✳❑✳ ❋❛❞❞❡❡✈ ✐♥ ✶✾✺✵✬s ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ✐♥✈❡rs❡

• ❛❧♦✐s ♣r♦❜❧❡♠

✶✶ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ❆♥❛❧②s✐s

G ✐s ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✱ |G| = n✱ S = Soc(G)✱ X ⊆ G Γ = Cay(G, X) ✐s ❛ ❝❡♥tr❛❧ ❈❛②❧❡② ❣r❛♣❤✱ K = Aut(Γ) ❲❡ ❤❛✈❡ t✇♦ ❝❛s❡s✿

✶ K ✐s ♣r✐♠✐t✐✈❡✱ t❤❡♥ K = Sym(G) ♦r |K| = poly(n) ✷ K ✐s ✐♠♣r✐♠✐t✐✈❡✱ t❤❡♥ |K| = poly(n) ♦r K ✐s ❛ ♥♦♥tr✐✈✐❛❧

❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❉✳❑✳ ❋❛❞❞❡❡✈ ✐♥ ✶✾✺✵✬s ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ✐♥✈❡rs❡

• ❛❧♦✐s ♣r♦❜❧❡♠

✶✶ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ Pr✐♠✐t✐✈❡ ❝❛s❡

▲❡t K = Aut(Γ) ❜❡ ♣r✐♠✐t✐✈❡✳ ❚❤❡♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❛❧♠♦st s✐♠♣❧❡ s✉❜❣r♦✉♣s ♦❢ ♣r✐♠✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣s ✭▲✐❡❜❡❝❦✱ Pr❛❡❣❡r✱ ❙❛①❧✱ ✷✵✶✵✮ ⇒ G = S ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✱ ✇❤❡r❡ ✐s ❡①t❡♥❞❡❞ ❜② t❤❡ ✐♥✈♦❧✉t✐♦♥ ✱ ✳ ❛♥❞ ✐s ❝❡♥tr❛❧ ■❢ ✱ t❤❡♥ t❤❡ ❖✬◆❛♥✲❙❝♦tt ❚❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t ■t ❢♦❧❧♦✇s t❤❛t ✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛ ✲❜❛s❡ ♦❢ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

✶✷ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ Pr✐♠✐t✐✈❡ ❝❛s❡

▲❡t K = Aut(Γ) ❜❡ ♣r✐♠✐t✐✈❡✳ ❚❤❡♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❛❧♠♦st s✐♠♣❧❡ s✉❜❣r♦✉♣s ♦❢ ♣r✐♠✐t✐✈❡ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣s ✭▲✐❡❜❡❝❦✱ Pr❛❡❣❡r✱ ❙❛①❧✱ ✷✵✶✵✮ ⇒ G = S G ✐s ♥♦♥❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ D(2, G) = Hol(G).2 ≤ Sym(G)✱ ✇❤❡r❡ Hol(G) = G Aut(G) ✐s ❡①t❡♥❞❡❞ ❜② t❤❡ ✐♥✈♦❧✉t✐♦♥ g → g−1✱ g ∈ G✳ Z(G) = 1 ❛♥❞ Γ ✐s ❝❡♥tr❛❧ ⇒ Gleft × Gright = G Inn(G) ≤ K ■❢ K = Sym(G)✱ t❤❡♥ t❤❡ ❖✬◆❛♥✲❙❝♦tt ❚❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t K ≤ D(2, G) ■t ❢♦❧❧♦✇s t❤❛t |K| = poly(n)✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛ G✲❜❛s❡ ♦❢ K ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡

✶✷ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ■♠♣r✐♠✐t✐✈❡ ❝❛s❡

▲❡t K = Aut(Γ) ❜❡ ✐♠♣r✐♠✐t✐✈❡✳ ❙❡t L t♦ ❜❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛❧❧ ♥♦♥✲s✐♥❣❧❡t♦♥ K✲❜❧♦❝❦s ❝♦♥t❛✐♥✐♥❣ t❤❡ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ♦❢ G ❚❤❡♥ S ≤ L G ✭❜❡❝❛✉s❡ Γ ✐s ❝❡♥tr❛❧✮ ▼♦r❡♦✈❡r✱ ✐❢ K0 ✐s t❤❡ s❡t✇✐s❡ st❛❜✐❧✐③❡r ♦❢ t❤❡ ✐♠♣r✐♠✐t✐✈✐t② s②st❡♠ L ❝♦♥t❛✐♥✐♥❣ L✱ t❤❡♥ K0 =

• Y ∈G/U

(K0)Y ❢♦r ❛ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ K✲❜❧♦❝❦ U ❝♦♥t❛✐♥✐♥❣ L ■❢ U = G✱ t❤❡♥ K ≤ NSym(G)(Sleft × Sright)✱ s♦ |K| = poly(n) ■❢ U < G✱ t❤❡♥ K ✐s ♣❡r♠✉t❛t✐♦♥ ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ✇r❡❛t❤ ♣r♦❞✉❝t ♦❢ t❤❡ ❣r♦✉♣s K U ❛♥❞ K G/L

✶✸ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ❆❧❣♦r✐t❤♠

❆s ✐♥ ♠❛♥② ♠♦❞❡r♥ ❛❧❣♦r✐t❤♠ ❢♦r t❡st✐♥❣ ✐s♦♠♦r♣❤✐s♠ t❤❡ ♠❛✐♥ t♦♦❧ ✐s t❤❡ ❲❡✐s❢❡✐❧❡r✕▲❡♠❛♥ ❛❧❣♦r✐t❤♠✳ ❇✐r❞✬s✲❡②❡ ✈✐❡✇ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠

✶ ❋✐♥❞ t❤❡ s❡❝t✐♦♥s

❛♥❞ ❜② ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ✭ ❛♥❞ ✮

✷ ❋✐♥❞

✱ ✇❤❡r❡ ❛♥❞ ❛r❡ t❤❡ ❵r❡str✐❝t✐♦♥s✬ ♦❢ ❛♥❞ t♦ ❛♥❞ ✭ ❛♥❞ ✮

✸ ❋✐♥❞

✱ ✇❤❡r❡ ❛♥❞ ❛r❡ t❤❡ ❵q✉♦t✐❡♥ts✬ ♦❢ ❛♥❞ ♠♦❞✉❧♦ ❛♥❞ ✭t❤❡ ❇❛❜❛✐ ❛❧❣♦r✐t❤♠ ❢♦r ✐s♦♠♦r♣❤✐s♠ t❡st✐♥❣✮

✹ ❖✉t♣✉t

♦❜t❛✐♥❡❞ ❜② ❵❣❧✉✐♥❣✬ ❛♥❞ ✭t❤❡ ❇❛❜❛✐ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦s❡t ✐♥t❡rs❡❝t✐♦♥✮

✶✹ ✴ ✶✺

❙❦❡t❝❤ ♦❢ t❤❡ Pr♦♦❢✳ ❆❧❣♦r✐t❤♠

❆s ✐♥ ♠❛♥② ♠♦❞❡r♥ ❛❧❣♦r✐t❤♠ ❢♦r t❡st✐♥❣ ✐s♦♠♦r♣❤✐s♠ t❤❡ ♠❛✐♥ t♦♦❧ ✐s t❤❡ ❲❡✐s❢❡✐❧❡r✕▲❡♠❛♥ ❛❧❣♦r✐t❤♠✳ ❇✐r❞✬s✲❡②❡ ✈✐❡✇ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠

✶ ❋✐♥❞ t❤❡ s❡❝t✐♦♥s U/L ❛♥❞ U′/L′ ❜② ❡①❤❛✉st✐✈❡ s❡❛r❝❤

✭S ≤ L ≤ U ≤ G ❛♥❞ |G/S| ≤ log n✮

✷ ❋✐♥❞ Iso(ΓU, Γ′

U′)✱ ✇❤❡r❡ ΓU ❛♥❞ Γ′ U′ ❛r❡ t❤❡ ❵r❡str✐❝t✐♦♥s✬ ♦❢ Γ

❛♥❞ Γ′ t♦ U ❛♥❞ U′ ✭|K U| = poly(n) ❛♥❞ |(K ′)U′| = poly(n)✮

✸ ❋✐♥❞ Iso(ΓL, Γ′

L′)✱ ✇❤❡r❡ ΓL ❛♥❞ Γ′ L′ ❛r❡ t❤❡ ❵q✉♦t✐❡♥ts✬ ♦❢ Γ

❛♥❞ Γ′ ♠♦❞✉❧♦ L ❛♥❞ L′ ✭t❤❡ ❇❛❜❛✐ ❛❧❣♦r✐t❤♠ ❢♦r ✐s♦♠♦r♣❤✐s♠ t❡st✐♥❣✮

✹ ❖✉t♣✉t Iso(Γ, Γ′) ♦❜t❛✐♥❡❞ ❜② ❵❣❧✉✐♥❣✬ Iso(ΓU, Γ′

U′) ❛♥❞

Iso(ΓL, Γ′

L′) ✭t❤❡ ❇❛❜❛✐ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦s❡t ✐♥t❡rs❡❝t✐♦♥✮

✶✹ ✴ ✶✺

❈❛②❧❡② ❣r❛♣❤s ❛♥❞ ❙❝❤✉r r✐♥❣s

❆ ❈❛②❧❡② ❣r❛♣❤ Γ = Cay(G, X) ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❡❧❡♠❡♥t

x∈X x ♦❢ ❛ ❙❝❤✉r r✐♥❣ ♦✈❡r G

■♥ t❤✐s ❧❛♥❣✉❛❣❡ t❤❡ ❛♥❛❧②s✐s ✇❡ ♠❛❞❡ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ♦✉r r❡s✉❧t ❣✐✈❡s t❤❡ str✉❝t✉r❡ t❤❡♦r❡♠ ❢♦r t❤❡ ❝❡♥tr❛❧ ❙❝❤✉r r✐♥❣s ♦✈❡r ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣s

✶✺ ✴ ✶✺