t rs t s r str - PowerPoint PPT Presentation

❖♥ ✜♥✐t❡ ❣r♦✉♣s ✇✐t❤ s♠❛❧❧ ♣r✐♠❡ s♣❡❝tr✉♠ ❑♦♥❞r❛t✐❡✈ ❆✳❙✳ ❑❤r❛♠ts♦✈ ■✳❱✳ ◆✳◆✳❑r❛s♦✈s❦✐✐ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❡❝❤❛♥✐❝s ♦❢ ❯❇ ❘❆❙ ❊❦❛t❡r✐♥❜✉r❣✱ ❘✉ss✐❛ ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✸ ❆✉❣✉st✱ ✷✵✶✸

❉❡✜♥✐t✐♦♥s ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❉❡♥♦t❡ ❜② π ( G ) t❤❡ s❡t ♦❢ ♣r✐♠❡ ❞✐✈✐s♦rs ♦❢ t❤❡ ♦r❞❡r ♦❢ G ✳ ■❢ | π ( G ) | = n t❤❡♥ G ✐s ❝❛❧❧❡❞ n ✲♣r✐♠❛r②✳ Γ( G ) ✐s ❛ ♣r✐♠❡ ❣r❛♣❤ ♦❢ t❤❡ ❣r♦✉♣ G ❀ ✐♥ t❤✐s ❣r❛♣❤✱ t❤❡ ✈❡rt❡① s❡t ✐s π ( G ) ❛♥❞ t✇♦ ❞✐✛❡r❡♥t ✈❡rt✐❝❡s p ❛♥❞ q ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐♥ G t❤❡r❡ ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r pq ✳ s = s ( G ) ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❣r❛♣❤ Γ( G ) ✳ { π i ✭●✮ | ✶ ≤ ✐ ≤ s } ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❣r❛♣❤ Γ( G ) ✳ ■❢ 2 ∈ π ( G ) t❤❡♥ 2 ∈ π 1 ( G ) ✳

❘❡❝♦❣♥✐t✐♦♥ ❣r♦✉♣s ❜② ♣r✐♠❡ ❣r❛♣❤ ❆ ❣r♦✉♣ G ✐s ❝❛❧❧❡❞ r❡❝♦❣♥✐③❛❜❧❡ ❜② ♣r✐♠❡ ❣r❛♣❤ ✐❢✱ ❢♦r ❛♥② ✜♥✐t❡ ❣r♦✉♣ H ✱ t❤❡ ❣r❛♣❤ ❡q✉❛❧✐t② Γ( H ) = Γ( G ) ✐♠♣❧✐❡s ❛ ❣r♦✉♣ ✐s♦♠♦r♣❤✐s♠ H ∼ = G ✳ ❚❤❡ ✜rst ❡①❛♠♣❧❡ ♦❢ ❣r♦✉♣s r❡❝♦❣♥✐③❛❜❧❡ ❜② ♣r✐♠❡ ❣r❛♣❤ ✇❡r❡ ♣r♦✈✐❞❡❞ ❜② ▼✳ ❍❛❣✐❡ ✐♥ 2003 ✱ ♥❛♠❡❧② s♦♠❡ s♣♦r❛❞✐❝ s✐♠♣❧❡ ❣r♦✉♣s✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ r❡❝♦❣♥✐t✐♦♥ ♦❢ ❛ ❣r♦✉♣ ❜② ♣r✐♠❡ ❣r❛♣❤ ✐s ❛ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♦❢ ♠♦r❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠✿ st✉❞② ✜♥✐t❡ ❣r♦✉♣s ❜② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡✐r ♣r✐♠❡ ❣r❛♣❤s✳

●r♦✉♣s ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ❲❡ ♠❛✐♥❧② ❝♦♥s✐❞❡r ❣r♦✉♣s ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ❚❤❡ ❝❧❛ss ♦❢ ✜♥✐t❡ ❣r♦✉♣s ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ✶✮ ❣❡♥❡r❛❧✐③❡s ✇✐❞❡❧② t❤❡ ❝❧❛ss ♦❢ ✜♥✐t❡ ❋r♦❜❡♥✐✉s ❣r♦✉♣s❀ ✷✮ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❝❧❛ss ♦❢ ✜♥✐t❡ ❣r♦✉♣s ❤❛✈✐♥❣ ❛♥ ✐s♦❧❛t❡❞ s✉❜❣r♦✉♣❀ ✸✮ ✐♥❝❧✉❞❡s ✐♠♣♦rt❛♥t ✑s♠❛❧❧✑ ❣r♦✉♣s✳

●r♦✉♣s ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ❙✉♣♣♦s❡ t❤❛t Γ( G ) ✐s ❞✐s❝♦♥♥❡❝t❡❞✳ ▲❡t G = G / F ( G ) ❜❡ ❛♥ ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣✳ ❚❤❡r❡ ❡①✐sts g ∈ G \ F ( G ) s✉❝❤ t❤❛t | g | ∈ π ( G ) ❛♥❞ g ❛❝ts ❢r❡❡❧② ♦♥ p ✲❝❤✐❡❢ ❢❛❝t♦rs ♦❢ t❤❡ ❣r♦✉♣ G ✱ p ∈ π ( F ( G )) ✳

Pr♦❜❧❡♠ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ Q ❜❡ ❛ ♥♦r♠❛❧ ♥♦♥tr✐✈✐❛❧ s✉❜❣r♦✉♣ ❢r♦♠ G ✱ G = G / Q ❜❡ ❛ ❦♥♦✇♥ ❣r♦✉♣ ❛♥❞ ❛♥ ❡❧❡♠❡♥t ♦❢ ♣r✐♠❡ ♦r❞❡r ❢r♦♠ G \ Q ❛❝ts ♦♥ Q ✜①❡❞✲♣♦✐♥ts✲❢r❡❡❧②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥s ❛r✐s❡✳ 1) ❲❤❛t ❛r❡ t❤❡ ❝❤✐❡❢ ❢❛❝t♦rs ♦❢ t❤❡ ❣r♦✉♣ G ✐♥ Q ❄ 2) ❲❤❛t ✐s t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❣r♦✉♣ Q ❄ 3) ■❢ Q ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣✱ ✐s t❤❡ ❛❝t✐♦♥ ♦❢ G ♦♥ Q ❝♦♠♣❧❡t❡❧② ✐rr❡❞✉❝✐❜❧❡❄ 4) ■s t❤❡ ❡①t❡♥s✐♦♥ ♦❢ G ♦✈❡r Q s♣❧✐tt❛❜❧❡❄

✶✳ ●✳ ❍✐❣♠❛♥ ✭✶✾✻✽✮✿ G ∼ = L 2 (2 n ) , n ≥ 2 ✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 3 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q ✳ ✷✳ ❘✳P✳ ▼❛rt✐♥❡❛✉ ✭✶✾✼✷✮✿ G ∼ = Sz (2 n ) ✱ n ✐s ♦❞❞✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q ✳ ✸✳ ❲✳❇✳ ❙t❡✇❛r❞ ✭✶✾✼✸✮✿ G ∼ = L 2 ( q ) ✱ q ✐s ♦❞❞✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 3 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q ✳ ✹✳ ❆✳ Pr✐♥❝❡ ✭✶✾✼✼✮✱ ●✳ ❩✉r❡❦ ✭✶✾✽✷✮✱ ❉✳❋✳ ❍♦❧t ❛♥❞ ❲✳ P❧❡s❦❡♥ ✭✶✾✽✻✮✿ G ∼ = A 5 ✱ Q = O 2 ( G ) ✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q ✳ ✺✳ ❆✳ Pr✐♥❝❡ ✭✶✾✽✷✮✿ G ∼ = A 6 ✱ Q = O 2 ( G ) ✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q ✳

●r♦✉♣s ✇✐t❤ s♠❛❧❧ ♣r✐♠❡ s♣❡❝tr✉♠ ❲❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ✜rst ✐t❡♠ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❢♦r ❣r♦✉♣s ✇✐t❤ s♠❛❧❧ ♣r✐♠❡ s♣❡❝tr✉♠✳ ■♥ ✷✵✶✵ ✖ ✷✵✶✸ ❑♦♥❞r❛t✐❡✈ ❛♥❞ ❑❤r❛♠ts♦✈ ❞❡s❝r✐❜❡❞ t❤❡ ❝❤✐❡❢ ❢❛❝t♦rs ♦❢ 3 ✲♣r✐♠❛r② ❛♥❞ ♠♦st ♣❛rt ♦❢ 4 ✲♣r✐♠❛r② ❣r♦✉♣s ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤✳

4 ✲♣r✐♠❛r② ❣r♦✉♣s ❚❤❡♦r❡♠ ✭❑♦♥❞r❛t✐❡✈✱ ❑❤r❛♠ts♦✈✱ ✷✵✶✶✮ ▲❡t G ❜❡ ❛ ✜♥✐t❡ 4 ✲♣r✐♠❛r② ❣r♦✉♣ ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤✱ ❛♥❞ ❧❡t G = G / F ( G ) ✳ ❚❤❡♥✱ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❤♦❧❞s✿ (1) G ✐s ❛ ❋r♦❜❡♥✐✉s ❣r♦✉♣❀ (2) G ✐s ❛ 2 ✲❋r♦❜❡♥✐✉s ❣r♦✉♣❀ (3) G ✐s ❛♥ ❛❧♠♦st s✐♠♣❧❡ tr✐♣r✐♠❛r② ❣r♦✉♣❀ = L 2 (2 m ) ✱ ✇❤❡r❡ m ≥ 5 ✱ 2 m − 1 ✱ ❛♥❞ (2 m + 1) / 3 ❛r❡ ♣r✐♠❡s❀ (4) G ∼ = L 2 (3 m ) ♦r PGL 2 (3 m ) ✱ ✇❤❡r❡ m ❛♥❞ (3 m − 1) / 2 ❛r❡ ♦❞❞ (5) G ∼ ♣r✐♠❡s ❛♥❞ (3 m + 1) / 4 ✐s ❡✐t❤❡r ❛ ♣r✐♠❡ ♦r 11 2 ✭❢♦r m = 5 ✮❀ (6) G ∼ = L 2 ( r ) ♦r PGL 2 ( r ) ✱ ✇❤❡r❡ r ✐s ❛ ♣r✐♠❡✱ 17 � = r ≥ 11 ✱ r 2 − 1 = 2 a 3 b s c ✱ s > 3 ✐s ❛ ♣r✐♠❡✱ a , b ∈ N ✱ ❛♥❞ c ✐s ❡✐t❤❡r 1 ♦r 2 ❢♦r r ∈ { 97 , 577 } ;

4 ✲♣r✐♠❛r② ❣r♦✉♣s (7) G ∼ = A 7 ✱ S 7 ✱ A 8 ✱ S 8 ✱ A 9 ✱ L 2 (16) ✱ L 2 (16): 2 ✱ Aut ( L 2 (16)) ✱ L 2 (25) ✱ L 2 (25): 2 ✱ L 2 (27): 3 ✱ L 2 (49) ✱ L 2 (49): 2 1 ✱ L 2 (49): 2 3 ✱ L 2 (81) ✱ L 2 (81): 2 ✱ L 2 (81): 4 ✱ L 3 (4) ✱ L 3 (4): 2 1 ✱ L 3 (4): 2 3 ✱ L 3 (5) ✱ Aut ( L 3 (5)) ✱ L 3 (7) ✱ L 3 (7): 2 ✱ L 3 (8) ✱ L 3 (8): 2 ✱ L 3 (8): 3 ✱ Aut ( L 3 (8)) ✱ L 3 (17) ✱ Aut ( L 3 (17)) ✱ L 4 (3) ✱ L 4 (3): 2 2 ✱ L 4 (3): 2 3 ✱ U 3 (4) ✱ U 3 (4): 2 ✱ Aut ( U 3 (4)) ✱ U 3 (5) ✱ U 3 (5): 2 ✱ U 3 (7) ✱ Aut ( U 3 (7)) ✱ U 3 (8) ✱ U 3 (8): 2 ✱ U 3 (8): 3 1 ✱ U 3 (8): 3 3 ✱ U 3 (8): 6 ✱ U 3 (9) ✱ U 3 (9): 2 ✱ Aut ( U 3 (9)) ✱ U 4 (3) ✱ U 4 (3): 2 2 ✱ U 4 (3): 2 3 ✱ U 5 (2) ✱ Aut ( U 5 (2)) ✱ S 4 (4) ✱ S 4 (4): 2 ✱ Aut ( S 4 (4)) ✱ S 4 (5) ✱ S 4 (7) ✱ S 4 (9) ✱ S 4 (9): 2 1 ✱ S 4 (9): 2 3 ✱ S 6 (2) ✱ G 2 (3) ✱ Aut ( G 2 (3)) ✱ O + 8 (2) ✱ 3 D 4 (2) ✱ Aut ( 3 D 4 (2)) ✱ Sz (8) ✱ Sz (32) ✱ Aut ( Sz (32)) ✱ 2 F 4 (2) ′ ✱ 2 F 4 (2) ✱ M 11 ✱ M 12 ✱ Aut ( M 12 ) ✱ ♦r J 2 ✳