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  1. ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❯♥✐✈❡rs✐t② ♦❢ ❆❜❡r❞❡❡♥ ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ❇✐r♠✐♥❣❤❛♠✱ ✼t❤ ❆✉❣✉st ✷✵✶✼ ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✶ ✴ ✶✺

  2. ❉❡✜♥✐t✐♦♥ ▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ ❙②❧♦✇ ✲s✉❜❣r♦✉♣ ♦❢ ✳ ❚❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ♦❢ ♦♥ ✐s t❤❡ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠ s❡ts ❛r❡✿ ❢♦r ❡✈❡r② ✳ ❋✉s✐♦♥ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❚✇♦ s✉❜❣r♦✉♣s P, Q ≤ G ❛r❡ ❢✉s❡❞ ✐♥ G ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t P g = Q ✳ ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✷ ✴ ✶✺

  3. ❋✉s✐♦♥ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❚✇♦ s✉❜❣r♦✉♣s P, Q ≤ G ❛r❡ ❢✉s❡❞ ✐♥ G ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t P g = Q ✳ ❉❡✜♥✐t✐♦♥ ▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ ❙②❧♦✇ p ✲s✉❜❣r♦✉♣ ♦❢ G ✳ ❚❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ♦❢ G ♦♥ S ✐s t❤❡ ❝❛t❡❣♦r② F S ( G ) ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠ s❡ts ❛r❡✿ Mor F S ( G ) ( P, Q ) = Hom G ( P, Q ) = { c g | P : P → Q | g ∈ G, P g ≤ Q } , ❢♦r ❡✈❡r② P, Q ≤ S ✳ ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✷ ✴ ✶✺

  4. P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ✳ ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❊ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ✭ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ S = D 8 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � D 8 a 3 a a 3 x x ax a 2 x a 2 1 ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

  5. ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❊ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ✭ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧ p ( G ) ✳ S = D 8 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � D 8 G = D 8 a 3 a a 3 x x ax a 2 x a 2 1 ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

  6. ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✳ ❊ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧ p ( G ) ✳ S = D 8 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � D 8 G = D 8 a 3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s a ❜② ❡❧❡♠❡♥ts g ∈ G t❤❛t ❢✉s❡ a 3 x x s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ S ✳ ax a 2 x a 2 ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② Inn(D 8 ) : 1 F D 8 (D 8 ) = � Inn(D 8 ) � ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

  7. ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❊ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ✭ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧ p ( G ) ✳ S = D 8 S 4 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � b D 8 G = Sym(4) = S 4 a = (1234) , x = (13) . a 3 a a 3 x x ax a 2 x a 2 1 ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

  8. ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧ p ( G ) ✳ S = D 8 S 4 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � b D 8 G = Sym(4) = S 4 a = (1234) , x = (13) . a 3 ❊ a b = (123) ( a 2 ) b = ((13)(24)) (123) = (12)(34) = ax a 3 x x ax a 2 x ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② a 2 Inn(D 8 ) ❛♥❞ Aut( E ) ∼ = SL 2 (2) ∼ = S 3 1 F D 8 ( S 4 ) = � Inn(D 8 ) , Aut( E ) � ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

  9. ❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❊ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ✭ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✱ ❛♥❞ P✐❝❦ ❛ p ✲❣r♦✉♣ S ✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧ p ( G ) ✳ A 6 c S = D 8 S 4 D 8 := � a, x | a 4 = x 2 = 1 , a x = a 3 � b D 8 G = A 6 a = (1234)(56) , x = (13)(56) ✳ a 3 a c = (25)(46) ( a 2 ) c = ((13)(24)) (25)(46) a 3 x x = (13)(56) = x ax a 2 x a 2 1 ❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼ ✸ ✴ ✶✺

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