SLIDE 1
❖♥ ✜♥✐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 ✷✵✶✸
t rs t s r str - - PowerPoint PPT Presentation
t rs t s r str - - PowerPoint PPT Presentation
t rs t s r str rt rts rss sttt
SLIDE 2
SLIDE 3
❘❡❝♦❣♥✐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✳
SLIDE 4
- 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✳
SLIDE 5
- 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))✳
SLIDE 6
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❛❜❧❡❄
SLIDE 7
✶✳ ●✳ ❍✐❣♠❛♥ ✭✶✾✻✽✮✿ G ∼ = L2(2n), n ≥ 2✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 3 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q✳ ✷✳ ❘✳P✳ ▼❛rt✐♥❡❛✉ ✭✶✾✼✷✮✿ G ∼ = Sz(2n)✱ n ✐s ♦❞❞✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q✳ ✸✳ ❲✳❇✳ ❙t❡✇❛r❞ ✭✶✾✼✸✮✿ G ∼ = L2(q)✱ q ✐s ♦❞❞✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 3 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q✳ ✹✳ ❆✳ Pr✐♥❝❡ ✭✶✾✼✼✮✱ ●✳ ❩✉r❡❦ ✭✶✾✽✷✮✱ ❉✳❋✳ ❍♦❧t ❛♥❞ ❲✳ P❧❡s❦❡♥ ✭✶✾✽✻✮✿ G ∼ = A5✱ Q = O2(G)✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q✳ ✺✳ ❆✳ Pr✐♥❝❡ ✭✶✾✽✷✮✿ G ∼ = A6✱ Q = O2(G)✱ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❛❝ts ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧② ♦♥ Q✳
SLIDE 8
- 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❛♣❤✳
SLIDE 9
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♦✉♣❀ (4) G ∼ = L2(2m)✱ ✇❤❡r❡ m ≥ 5✱ 2m − 1✱ ❛♥❞ (2m + 1)/3 ❛r❡ ♣r✐♠❡s❀ (5) G ∼ = L2(3m) ♦r PGL2(3m)✱ ✇❤❡r❡ m ❛♥❞ (3m − 1)/2 ❛r❡ ♦❞❞ ♣r✐♠❡s ❛♥❞ (3m + 1)/4 ✐s ❡✐t❤❡r ❛ ♣r✐♠❡ ♦r 112 ✭❢♦r m = 5✮❀ (6) G ∼ = L2(r) ♦r PGL2(r)✱ ✇❤❡r❡ r ✐s ❛ ♣r✐♠❡✱ 17 = r ≥ 11✱ r2 − 1 = 2a3bsc✱ s > 3 ✐s ❛ ♣r✐♠❡✱ a, b ∈ N✱ ❛♥❞ c ✐s ❡✐t❤❡r 1 ♦r 2 ❢♦r r ∈ {97, 577};
SLIDE 10
4✲♣r✐♠❛r② ❣r♦✉♣s
(7) G ∼ = A7✱ S7✱ A8✱ S8✱ A9✱ L2(16)✱ L2(16): 2✱ Aut(L2(16))✱ L2(25)✱ L2(25): 2✱ L2(27): 3✱ L2(49)✱ L2(49): 21✱ L2(49): 23✱ L2(81)✱ L2(81): 2✱ L2(81): 4✱ L3(4)✱ L3(4): 21✱ L3(4): 23✱ L3(5)✱ Aut(L3(5))✱ L3(7)✱ L3(7): 2✱ L3(8)✱ L3(8): 2✱ L3(8): 3✱ Aut(L3(8))✱ L3(17)✱ Aut(L3(17))✱ L4(3)✱ L4(3): 22✱ L4(3): 23✱ U3(4)✱ U3(4): 2✱ Aut(U3(4))✱ U3(5)✱ U3(5): 2✱ U3(7)✱ Aut(U3(7))✱ U3(8)✱ U3(8): 2✱ U3(8): 31✱ U3(8): 33✱ U3(8): 6✱ U3(9)✱ U3(9): 2✱ Aut(U3(9))✱ U4(3)✱ U4(3): 22✱ U4(3): 23✱ U5(2)✱ Aut(U5(2))✱ S4(4)✱ S4(4): 2✱ Aut(S4(4))✱ S4(5)✱ S4(7)✱ S4(9)✱ S4(9): 21✱ S4(9): 23✱ S6(2)✱ G2(3)✱ Aut(G2(3))✱ O+
8 (2)✱ 3D4(2)✱
Aut(3D4(2))✱ Sz(8)✱ Sz(32)✱ Aut(Sz(32))✱ 2F4(2)′✱ 2F4(2)✱ M11✱ M12✱ Aut(M12)✱ ♦r J2✳
SLIDE 11
❘❡❝♦❣♥✐③❛❜❧❡ ❣r♦✉♣s ✇✐t❤ s♠❛❧❧ ♣r✐♠❡ s♣❡❝tr✉♠
❆s ❛ ❝♦r♦❧❧❛r②✱ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts ✇❡r❡ ♦❜t❛✐♥❡❞✳ ❈♦r♦❧❧❛r② ✶ ❆ ✜♥✐t❡ 3✲♣r✐♠❛r② ❛❧♠♦st s✐♠♣❧❡ ❣r♦✉♣ ✇✐t❤ ❞✐s❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ✐s r❡❝♦❣♥✐③❛❜❧❡ ❜② ♣r✐♠❡ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ L2(17)✳ ❈♦r♦❧❧❛r② ✷ ❆ ✜♥✐t❡ 4✲♣r✐♠❛r② s✐♠♣❧❡ ❣r♦✉♣ ✐s r❡❝♦❣♥✐③❛❜❧❡ ❜② ♣r✐♠❡ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✐s♦♠♦r♣❤✐❝ t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r♦✉♣s✿ A8✱ L3(4)✱ ❛♥❞ L2(q)✱ ✇❤❡r❡ |π(q2 − 1)| = 3✱ q > 17✱ ❛♥❞ ❡✐t❤❡r q = 3m ❛♥❞ m ✐s ❛♥ ♦❞❞ ♣r✐♠❡ ♦r q ✐s ❛ ♣r✐♠❡ ❛♥❞ q ≡ 1 (mod 12) ♦r q ∈ {97, 577}✳
SLIDE 12
❚❤❡♦r❡♠ ❢♦r A7
❲❡ ✐♥✈❡st✐❣❛t❡❞ t❤❡ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❝❛s❡ ✇❤❡♥ Q = O2(G)✱ G ∼ = A7 ❛♥❞ ❛♥ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❢r♦♠ G ❛❝ts ♦♥ Q ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧②✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠ ✇❛s ♣r♦✈❡❞✳ ❚❤❡♦r❡♠ ✭❑♦♥❞r❛t✐❡✈✱ ❑❤r❛♠ts♦✈✱ ✷✵✶✷✮ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ ❛ ♥♦♥tr✐✈✐❛❧ ♥♦r♠❛❧ 2✲s✉❜❣r♦✉♣ Q ❛♥❞ G/Q ∼ = A7✳ ❙✉♣♣♦s❡ t❤❛t ❛♥ ❡❧❡♠❡♥t ♦❢ ♦r❞❡r 5 ❢r♦♠ G ❛❝ts ♦♥ Q ✜①❡❞ ♣♦✐♥ts ❢r❡❡❧②✳ ❚❤❡♥ t❤❡ ❡①t❡♥s✐♦♥ G ♦✈❡r Q ✐s s♣❧✐t✱ Q ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ Q ✐s t❤❡ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ♠✐♥✐♠❛❧ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ❡❛❝❤ ♦❢ ✇❤✐❝❤ ❛s GF(2)G/Q✲♠♦❞✉❧❡ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ♦♥❡ ♦❢ t❤❡ t✇♦ 4✲❞✐♠❡♥s✐♦♥❛❧ ✐rr❡❞✉❝✐❜❧❡ GF(2)A7✲♠♦❞✉❧❡s t❤❛t ❛r❡ ❝♦♥❥✉❣❛t❡❞ ❜② ♦✉t❡r ❛✉t♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r♦✉♣ A7✳
SLIDE 13
❋✐♥✐t❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ❣r❛♣❤ ❛s t❤❡ ❣r♦✉♣ Aut(J2)
❑❤♦sr❛✈✐ ✐♥ ✷✵✵✾ ♦❜t❛✐♥❡❞ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛ ❣r♦✉♣ ❤❛✈✐♥❣ t❤❡ s❛♠❡ ♣r✐♠❡ ❣r❛♣❤ ❛s t❤❡ ❣r♦✉♣ Aut(S) ❢♦r ❛♥② s♣♦r❛❞✐❝ s✐♠♣❧❡ ❣r♦✉♣ S ❡①❝❡♣t ❢♦r t❤❡ ❣r♦✉♣ J2✳ ❍❡ ♣♦s❡❞ t❤❡ ♣r♦❜❧❡♠✿ ❞❡s❝r✐❜❡ ❛❧❧ ❣r♦✉♣s G s✉❝❤ t❤❛t Γ(G) = Γ(Aut(J2))✳ ■t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ t❤❛t Γ(Aut(J2)) ✐s ❝♦♥♥❡❝t❡❞✳ ❑♦♥❞r❛t✐❡✈ s♦❧✈❡❞ t❤❡ ❑❤♦sr❛✈✐✬s ♣r♦❜❧❡♠ ✐♥ ✷✵✶✷✳ Γ(Aut(J2)) :
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3 2 5 7
SLIDE 14
❋✐♥✐t❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ❣r❛♣❤ ❛s t❤❡ ❣r♦✉♣ A10
❚❤❡ ❣r♦✉♣ A10 ✐s ❡①❝❡♣t✐♦♥❛❧ ✐♥ s♦♠❡ s❡♥s❡s✳ ■t ✐s t❤❡ ♦♥❧② ❣r♦✉♣ ✇✐t❤ ❝♦♥♥❡❝t❡❞ ♣r✐♠❡ ❣r❛♣❤ ❛♠♦♥❣ ❛❧❧ ✜♥✐t❡ s✐♠♣❧❡ ❣r♦✉♣s ❢r♦♠ ✧❆t❧❛s ♦❢ ✜♥✐t❡ ❣r♦✉♣s✧❛♥❞ ❛❧s♦ ❛♠♦♥❣ ❛❧❧ 4✲♣r✐♠❛r② s✐♠♣❧❡ ❣r♦✉♣s✳ ❘❡❝❡♥t❧② ❑♦♥❞r❛t✐❡✈ ❞❡s❝r✐❜❡❞ ❛❧❧ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ t❤❡ s❛♠❡ ♣r✐♠❡ ❣r❛♣❤ ❛s t❤❡ ❣r♦✉♣ A10✳ Γ(A10) :
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