SLIDE 1 ❖◆ P❘❖❏❊❈❚■❱❊ ▼❖❉❯▲❊❙ ❋❖❘ ❋■◆■❚❊ ●❘❖❯P❙ ❲❍❖❙❊ ❉■▼❊◆❙■❖◆ ❊◗❯❆▲❙ ❚❍❊ ❖❘❉❊❘ ❖❋ ❆ ❙❨▲❖❲ p✲❙❯❇●❘❖❯P
❆✳❊✳ ❩❛❧❡ss❦✐ ❚♦♣✐❝s ♦♥ ●r♦✉♣s ❛♥❞ ❚❤❡✐r ❘❡♣r❡s❡♥t❛t✐♦♥s
- ❛r❣♥❛♥♦✱ ■t❛❧②✱ ❖❝t♦❜❡r ✷✵✶✼
❆ ❝♦♥❢❡r❡♥❝❡ ✐♥ ❤♦♥♦✉r ♦❢
Pr♦❢❡ss♦r ▲✐♥♦ ❉✐ ▼❛rt✐♥♦
♦♥ ♦❝❝❛s✐♦♥ ♦❢ ❤✐s ✼✵t❤ ❜✐rt❤❞❛②
✶
SLIDE 2
■♥tr♦❞✉❝t✐♦♥ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ p ❛ ♣r✐♠❡✳ ▲❡t ❋ ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞ ♦❢ ❝❤❛r p > 0✳ Pr♦❥❡❝t✐✈❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ FG✲♠♦❞✉❧❡s ✭P■▼✮ ❛r❡ ❡①❛❝t❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❞✐r❡❝t s✉♠♠❛♥❞s ♦❢ t❤❡ r❡❣✉❧❛r FG✲♠♦❞✉❧❡✳ ❚❤❡s❡ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❇r❛✉❡r ❛♥❞ ◆❡s❜✐tt ✐♥ ✶✾✹✵ ❛♥❞ r❡♠❛✐♥ ✐♠♣♦rt❛♥t ♦❜❥❡❝ts ♦❢ st✉❞② ✐♥ r❡♣r❡s❡♥✲ t❛t✐♦♥ t❤❡♦r② ♦❢ ✜♥✐t❡ ❣r♦✉♣s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ✈❡r② ♣♦♦r ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡✐r ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ♦♥❧② ✇❡❧❧ ❦♥♦✇♥ ❢❛❝t ✐s t❤❛t t❤❡ ❞✐♠❡♥s✐♦♥ ✐s ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ♦r❞❡r ♦❢ ❛ ❙②❧♦✇ p✲s✉❜❣r♦✉♣ ♦❢ G✳
SLIDE 3
❚❤✐s ♥❛t✉r❛❧❧② ❧❡❛❞s t♦ Pr♦❜❧❡♠ ✶✳ ❋♦r ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ ❛ ♣r✐♠❡ p✱ ❞❡t❡r♠✐♥❡ ♣r♦❥❡❝t✐✈❡ FG✲♠♦❞✉❧❡ ♦❢ ❞✐♠❡♥s✐♦♥ |G|p✳ ❍❡r❡ |G|p ❞❡♥♦t❡s t❤❡ p✲♣❛rt ♦❢ t❤❡ ♦r❞❡r ♦❢ G✱ ❛♥❞ t❤❡r❡❢♦r❡ ♦❢ t❤❡ ♦r❞❡r ♦❢ ❡✈❡r② ❙②❧♦✇ p✲s✉❜❣r♦✉♣ ♦❢ G✳ ❊①♣❧✐❝✐t❧②✱ Pr♦❜❧❡♠ ✶ ✇❛s ✜rst st❛t❡❞ ❜② ▼❛❧❧❡ ❛♥❞ ❲✐❡❣❡❧ ✭✷✵✵✽✮✳ ■t ❝♦✉❧❞ ❜❡ ✇r♦♥❣ t♦ ❡①♣❡❝t t❤❛t t❤✐s ❜♦✉♥❞ ❛tt❛✐♥s ❢♦r ❡✈❡r② ❣r♦✉♣ G✳ ❚❤❡r❡ ❛r❡ t✇♦ ✇❡❧❧ ❦♥♦✇♥ ❝❛s❡s ✇❤❡r❡ t❤✐s ✐s tr✉❡✿ ✶✮ G ❤❛s ❛ s✉❜❣r♦✉♣ ♦❢ ✐♥❞❡① |G|p✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ G ✐s p✲s♦❧✈❛❜❧❡ ❛♥❞ ✷✮ G ✐s ❛ ❈❤❡✈❛❧❧❡② ❣r♦✉♣ ✭♦r ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❛ ✜♥✐t❡ r❡❞✉❝t✐✈❡ ❣r♦✉♣✮ ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✲ ✐st✐❝ p✳
SLIDE 4
◆♦t❡ t❤❛t ❡✈❡r② P■▼ Φ ❧✐❢ts t♦ ❝❤❛r❛❝t❡r✐st✐❝ ✵✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡r❡ ✐s ❛ ♣r♦❥❡❝t✐✈❡ ✐♥❞❡✲ ❝♦♠♣♦s❛❜❧❡ KpG✲♠♦❞✉❧❡ Φ ✇❤♦s❡ r❡❞✉❝t✐♦♥ ♠♦❞✉❧♦ p ②✐❡❧❞s Φ✳ ❍❡r❡ Kp ✐s t❤❡ r✐♥❣ ♦❢ ✐♥t❡✲ ❣❡rs ♦❢ s♦♠❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥ K ♦❢ t❤❡ ✜❡❧❞ ♦❢ p✲❛❞✐❝ ♥✉♠❜❡rs✳ ❈❧❡❛r❧②✱ Φ ✈✐❡✇❡❞ ❛s ❛ KG✲♠♦❞✉❧❡ ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ ✐rr❡❞✉❝✐❜❧❡ KG✲♠♦❞✉❧❡s✱ ✇❤♦s❡ ♠✉❧t✐✲ ♣❧✐❝✐t✐❡s ❛r❡ ❝❛❧❧❡❞ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♥✉♠❜❡rs✳ ❚❤❡ tr✐✈✐❛❧ KG✲♠♦❞✉❧❡ 1G ♦❝❝✉rs ✐♥ ❛ ✉♥✐q✉❡ P■▼ Φ1✱ ✇❤✐❝❤ ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♥❝✐♣❛❧ P■▼✳ ▼❛❧❧❡ ❛♥❞ ❲✐❡❣❡❧ ✭✷✵✵✽✮ ❤❛✈❡ ❞❡t❡r♠✐♥❡❞ s✐♠♣❧❡ ❣r♦✉♣s G ✇❤♦s❡ ♣r✐♥❝✐♣❛❧ P■▼ ✐s ♦❢ ❞✐♠❡♥s✐♦♥ |G|p✳ ■t ❜❡❝❛♠❡ ❝❧❡❛r ❢r♦♠ t❤❡✐r ✇♦r❦ t❤❛t Pr♦❜❧❡♠ ✶ ✐s ♥♦♥✲tr✐✈✐❛❧ ❛♥❞ r❡✲ q✉✐r❡s s✉❜st❛♥t✐❛❧ ♠❛❝❤✐♥❡r② t♦ ❜❡ ❞❡✈❡❧♦♣❡❞✳
SLIDE 5
◆❡①t st❡♣ ✐s ❞♦♥❡ ✐♥ ♠② ♣❛♣❡r ❏✳ ❆❧❣✳ ✷✵✶✸ ❢♦r G t♦ ❜❡ ❛♥ ❛r❜✐tr❛r② s✐♠♣❧❡ ❣r♦✉♣ ♦❢ ▲✐❡ t②♣❡ ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ p✳ ◆♦ ❡①tr❛ ❡①❛♠♣❧❡ ♦❝❝✉rr❡❞✱ ❛♣❛rt ❢r♦♠ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ❙t❡✐♥❜❡r❣ ♠♦❞✉❧❡s✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦♦❢ ✐♥ t❤✐s ❝❛s❡ ✐s ❧♦♥❣❡r t❤❛♥ t❤❡ ✇❤♦❧❡ ♣❛♣❡r ♦❢ ▼❛❧❧❡ ❛♥❞ ❲❡✐❣❡❧✳ ❋❡✇ ②❡❛rs ❛❣♦ ❥♦✐♥t❧② ✇✐t❤ ▼❛❧❧❡ ✇❡ st❛rt❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ r❡♠❛✐♥✐♥❣ ❝❛s❡s ❢♦r s✐♠♣❧❡ ❣r♦✉♣s G✳ ❚❤❡s❡ ❛r❡ t❤❡ s♣♦r❛❞✐❝ ❣r♦✉♣s✱ t❤❡ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣s ❛♥❞ t❤❡ ❣r♦✉♣s ♦❢ ▲✐❡ t②♣❡ ✐♥ ❝r♦ss ❝❤❛r❛❝t❡r✐s✐❝✱ t❤❛t ✐s✱ ✇✐t❤ p ❞✐st✐♥❝t ❢r♦♠ t❤❡ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐s✐t✐❝ ♦❢ G✳ ❚❤❡ ♣❛♣❡r ✐s st✐❧❧ ✉♥❞❡r ♣♦❧✐s❤✐♥❣✱ ❜✉t t❤❡ r❡s✉❧ts s❡❡♠ t♦ ❜❡ ❝♦♠♣❧❡t❡✳
SLIDE 6
❆♥ ❡❛s② ♣❛rt ♦❢ t❤❡ ✇♦r❦ ✇❛s t♦ ❞❡t❡r♠✐♥❡ ✐rr❡❞✉❝✐❜❧❡ ♣r♦❥❡❝t✐✈❡ FG✲♠♦❞✉❧❡ ♦❢ ❞✐♠❡♥s✐♦♥ |G|p✳❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ❧✐st✐♥❣ ♦r❞✐♥❛r② ✐rr❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡rs ♦❢ G ♦❢ ❞❡❣r❡❡ |G|p✳ ❚❤❡ ❞❡❣r❡❡s ♦❢ ✐rr❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡rs ♦❢ s✐♠♣❧❡ ❣r♦✉♣s ❛r❡ ✐♥ ♣r✐♥❝✐♣❧❡ ❦♥♦✇♥✱ s♦ ❡①tr❛❝t✐♥❣ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ ❞❡❣r❡❡ |G|p ✐s ❥✉st ❛ t❡❝❤♥✐❝❛❧ ✇♦r❦✳ ❚❤❡ ❧✐st ✐s ❛s ❢♦❧❧♦✇s✿ Pr♦♣♦s✐t✐♦♥ ✶✳ ▲❡t G ❜❡ ❛ ♥♦♥✲❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t G ❤❛s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡r χ ♦❢ ❞❡❣r❡❡ |G|p✳ ❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✭✶✮ G ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛ s✐♠♣❧❡ ❣r♦✉♣ ♦❢ ▲✐❡ t②♣❡ ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ p ❛♥❞ χ ✐s t❤❡ ❙t❡✐♥❜❡r❣ ❝❤❛r❛❝t❡r ♦❢ G❀ ✭✷✮ G = L2(q)✱ q ❡✈❡♥✱ ❛♥❞ p = χ(1) = q ± 1✱ ♦r G = L2(8)✱ p = 3 ❛♥❞ χ(1) = 9;
SLIDE 7
✭✸✮ G = L2(q)✱ q ♦❞❞✱ χ(1) = (q ± 1)/2 ✐s ❛ p✲ ♣♦✇❡r ❢♦r p > 2✱ ♦r p = 2 ❛♥❞ χ(1) = q ± 1 ✐s ❛ 2✲♣♦✇❡r❀ ✭✹✮ G = Ln(q)✱ q > 2✱ (n, q − 1) = 1✱ n ✐s ❛♥ ♦❞❞ ♣r✐♠❡ s✉❝❤ t❤❛t χ(1) = (qn−1)/(q−1) ✐s ❛ p✲♣♦✇❡r❀ ✭✺✮ G = Un(q)✱ n ✐s ❛♥ ♦❞❞ ♣r✐♠❡✱ (n, q+1) = 1✱ s✉❝❤ t❤❛t χ(1) = (qn + 1)/(q + 1) ✐s ❛ p✲ ♣♦✇❡r❀ ✭✻✮ G = S2n(q)✱ n > 1✱ q = rk ✇✐t❤ r ❛♥ ♦❞❞ ♣r✐♠❡✱ kn ✐s ❛ 2✲♣♦✇❡r s✉❝❤ t❤❛t χ(1) = (qn + 1)/2 ✐s ❛ p✲♣♦✇❡r❀ ✭✼✮ G = S2n(3)✱ n > 2 ✐s ❛ ♣r✐♠❡✱ s✉❝❤ t❤❛t χ(1) = (3n − 1)/2 ✐s ❛ p✲♣♦✇❡r❀
SLIDE 8
✭✽✮ G = Ap+1✱ χ(1) = p; ✭✾✮ G = S6(2)✱ χ(1) = 7; ✭✶✵✮ G ∈ {M11, M12} ❛♥❞ χ(1) = 11; ✭✶✶✮ G ∈ {M11, L3(3)} ❛♥❞ χ(1) = 16; ✭✶✷✮ G ∈ {M24, Co2, Co3} ❛♥❞ χ(1) = 23; ✭✶✸✮ G = 2F4(2)′ ❛♥❞ χ(1) = 27❀ ✭✶✹✮ G = U3(3) ∼ = G2(2)′ ❛♥❞ χ(1) = 32; ♦r ✭✶✺✮ G = G2(3) ❛♥❞ χ(1) = 64✳
SLIDE 9
❚❤✉s✱ t❤❡ ♣r♦❜❧❡♠ r❡❞✉❝❡s t♦ r❡❞✉❝✐❜❧❡ ♣r♦✲ ❥❡❝t✐✈❡ ♠♦❞✉❧❡s✳ ❲❡ st❛rt❡❞ ✇✐t❤ ❛♥♦t❤❡r ❡❛s② ❝❛s❡ ✇❤❡r❡ ❙②❧♦✇ p✲s✉❜❣r♦✉♣s ♦❢ G ❛r❡ ❝②❝❧✐❝✳ Pr♦♣♦s✐t✐♦♥ ✷✳ ▲❡t G ❜❡ ❛ ♥♦♥✲❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣ ✇✐t❤ ❝②❝❧✐❝ ❙②❧♦✇ p✲s✉❜❣r♦✉♣ ❙✳ ■❢ t❤❡r❡ ❡①✐sts ❛ r❡❞✉❝✐❜❧❡ ♣r♦❥❡❝t✐✈❡ FG✲♠♦❞✉❧❡ ♦❢ ❞✐✲ ♠❡♥s✐♦♥ |G|p t❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ (1) G = L2(q)✱ q > 4 ❡✈❡♥✱ |S| = q + 1❀ (2) G = L2(p)✱ |S| = p > 5❀ (3) G = Ln(q)✱ n ✐s ❛♥ ♦❞❞ ♣r✐♠❡✱ n |(q − 1)✱ |S| = (qn − 1)/(q − 1)❀ (4) G = Ap✱ |S| = p ≥ 5❀ (5) G = M11✱ |S| = 11❀ (6) G = M23✱ |S| = 23✳
SLIDE 10
❋✉rt❤❡r ❛♥❛❧②s✐s ❧❡❛❞s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✿ ❚❤❡♦r❡♠ ✶✳ ▲❡t G ❜❡ ❛ ♥♦♥✲❛❜❡❧✐❛♥ s✐♠♣❧❡ ❣r♦✉♣✳ ▲❡t M ❜❡ ❛ ♣r♦❥❡❝t✐✈❡ FG✲♠♦❞✉❧❡ ♦❢ ❞✐♠❡♥s✐♦♥ |G|p✳ ❚❤❡♥ ❡✐t❤❡r M ✐s ✐rr❡❞✉❝✐❜❧❡ ❛♥❞ (G, p) ✐s ❧✐st❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✶ ♦r ❙②❧♦✇ p✲s✉❜❣r♦✉♣s ♦❢ G ❛r❡ ❝②❝❧✐❝ ❛♥❞ (G, p) ❛r❡ ❛s ✐♥ Pr♦♣♦s✐t✐♦♥ ✷✱ ♦r G = PSL2(q)✱ q + 1 ✐s ❛ 2✲♣♦✇❡r ❛♥❞ p = 2✳ ❖♥❡ ❝♦♥❝❧✉❞❡s t❤❛t✱ ❢♦r s✐♠♣❧❡ ❣r♦✉♣s G✱ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ ♣r♦❥❡❝t✐✈❡ FG✲♠♦❞✉❧❡s ❛r❡ ❣r❡❛t❡r t❤❛♥ |G|p ✇✐t❤ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ❡①❝❡♣t✐♦♥s✳ ❚❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ r❡q✉✐r❡s q✉✐t❡ ❛ ❧♦t ♦❢ ✇♦r❦✳ ❖♥❡ ♦❢ t❤❡ r❡❛s♦♥ ❢♦r t❤✐s ✐s t❤❡ ❢❛❝t t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ❛ ♠♦❞✉❧❡ ✐s ♣r♦❥❡❝t✐✈❡ ✐s ♥♦t ❡❛s② t♦ ✉s❡✳
SLIDE 11
❆♥ ❡①♣❡r✐❡♥❝❡ ♦❢ ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ ❣r♦✉♣s ♦❢ ▲✐❡ t②♣❡ ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ p ❧❡❛❞❡❞ t♦ t❤❡ ❤✐♥t t❤❛t t❤❡ s❛♠❡ ♣❤❡♥♦♠❡♥♦♥ ❤♦❧❞s ♥♦t ♦♥❧② ❢♦r ♣r♦❥❡❝t✐✈❡ ❜✉t ❛❧s♦ ❢♦r ❛r❜✐tr❛r② ❝❤❛r❛❝t❡rs ✈❛♥✐s❤✐♥❣ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts✳ ✭❘❡❝❛❧❧ t❤❛t ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡s ❧✐❢t t♦ ❝❤❛r❛❝t❡r✐st✐❝ ✵ ❛♥❞ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ t❤❡♠ ✈❛♥✐s❤ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts✳✮ ❙♦ ■ ❛s❦❡❞✱ ✜rst ♠②s❡❧❢✱ t❤❡ q✉❡st✐♦♥ ✇❤❡t❤❡r ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❞❡t❡r♠✐♥❡ ✭❢♦r s✐♠♣❧❡ ❣r♦✉♣s G✮ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ ❞❡❣r❡❡ |G|p ✈❛♥✐s❤✐♥❣ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts✳ ❙♦♠❡ ♦❜s❡r✈❛t✐♦♥s ❛♣♣❡❛r❡❞ ❛❧r❡❛❞② ✐♥ ♠② ♣❛✲ ♣❡r ✐♥ ❏✳ ❆❧❣✳ ✷✵✶✸✳
SLIDE 12
■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ G ✐s ❛ ❣r♦✉♣ ♦❢ ▲✐❡ t②♣❡ ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ p t❤❡ q✉❡st✐♦♥ ✇❛s ❛♥s✇❡r❡❞ ✐♥ ❛ ❥♦✐♥t ♣❛♣❡r ✇✐t❤ ▼❛r❝♦ P❡❧❧❡❣r✐♥✐ ✭✷✵✶✻✮✳ ❙✉r♣r✐③✐♥❣❧②✱ ❢♦r ❡✈❡r② ❣r♦✉♣ ♦❢ ❇◆✲♣❛✐r r❛♥❦ ✶ ✇❡ ❞✐s❝♦✈❡r❡❞ ❛ r❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡r ♦❢ ❞❡❣r❡❡ |G|p ✈❛♥✐s❤✐♥❣ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts✳ ❲❡ ❡①♣❡❝t❡❞ t❤❛t ❡❛❝❤ s✉❝❤ ❝❤❛r❛❝t❡r ❝❛♥ ❜❡ ❛ ♠❡♠❜❡r ♦❢ ❛ s❡r✐❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r❛♥❦ n ♦❢ G✱ ❜✉t t❤❡ t❤✐♥❣s t✉r♥❡❞ ♦✉t t♦ ❜❡ ♦♣♣♦s✐t❡✳ ■♥ ❢❛❝t✱ ❢♦r ❣r♦✉♣s ♦❢ r❛♥❦ ❣r❡❛t❡r t❤❛♥ ✺ ♥♦ s✉❝❤ ❝❤❛r❛❝t❡r ❡①✐sts✳
SLIDE 13
■♥ t❤❡ ❝✉rr❡♥t ✇♦r❦ ✇✐t❤ ▼❛❧❧❡ ✇❡ ❛❧s♦ st✉❞② s✉❝❤ ❝❤❛r❛❝t❡rs✳ ❙♦ ✇❡ ❝♦♥s✐❞❡r s♣♦r❛❞✐❝ ❛♥❞ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣s ❛s ✇❡❧❧ ❛s ❣r♦✉♣s ♦❢ ▲✐❡ t②♣❡ ✇✐t❤ p ❞✐st✐♥❝t ❢r♦♠ t❤❡ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝✳ ❋♦r ❜r❡✈✐t②✱ ❛ ❝❤❛r❛❝t❡r ♦❢ ❞❡❣r❡❡ |G|p ✈❛♥✐s❤✐♥❣ ❛t ❛❧❧ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts ✐s ❝❛❧❧❡❞ ❙t❡✐♥❜❡r❣✲ ❧✐❦❡✳ ❚❤❡♦r❡♠ ✷✳ ▲❡t G = An✱ n > 5✱ ❜❡ ❛♥ ❛❧t❡r✲ ♥❛t✐♥❣ ❣r♦✉♣ ❛♥❞ p ❛ ♣r✐♠❡✳ ❚❤❡♥ G ❤❛s ♥♦ r❡❞✉❝✐❜❧❡ ❙t❡✐♥❜❡r❣✲❧✐❦❡ ❝❤❛r❛❝t❡r ✉♥❧❡ss G ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛ ❣r♦✉♣ ♦❢ ▲✐❡ t②♣❡ ♦r p = 2 ❛♥❞ n = 2k ♦r 2k + 1 ❢♦r s♦♠❡ k✳ ■♥ t❤❡ ❡①❝❡♣t✐♦♥❛❧ ❝❛s❡ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❛ ❙t❡✐♥❜❡r❣✲❧✐❦❡ ❝❤❛r❛❝t❡r✳ ■♥ ❢❛❝t✱ ❢♦r ❡✈❡r② n ❛♥❞ p = 2 ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ❛ ❝❤❛r❛❝t❡r ♦❢ ❞❡❣r❡❡ 2n−1 ✇❤✐❝❤ ✈❛♥✐s❤❡s ❛t t❤❡ ✷✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts✳ ❍♦✇❡✈❡r✱ 2n−1 = |G|2 ♦♥❧② ❢♦r n = 2k ♦r 2k + 1✳
SLIDE 14
◗✉❡st✐♦♥✳ ■s 2n−1 t❤❡ ♠✐♥✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛ ❝❤❛r❛❝t❡r ♦❢ An ✈❛♥✐s❤✐♥❣ ❛t t❤❡ ✷✲s✐♥❣✉❧❛r ❡❧✲ ❡♠❡♥ts❄ ❋♦r ♦t❤❡r ❝❛s❡s ♥♦ ❙t❡✐♥❜❡r❣✲❧✐❦❡ ❝❤❛r❛❝t❡r ❡①✐sts✳ ❚♦ ❜❡ ♣r❡❝✐s❡✱ ✇❡ ♣r♦✈❡ ❚❤❡♦r❡♠ ✸✳ ▲❡t G ❜❡ ❛ s♣♦r❛❞✐❝ ❣r♦✉♣ ♦r ❛ ❣r♦✉♣s ♦❢ ▲✐❡ t②♣❡✳ ❚❤❡♥ G ❤❛s ♥♦ r❡❞✉❝✐❜❧❡ ❙t❡✐♥❜❡r❣✲❧✐❦❡ ❝❤❛r❛❝t❡r✱ ✉♥❧❡ss G ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛ ❣r♦✉♣ ♦❢ ▲✐❡ t②♣❡ ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ p✱ ♦r G = PSL2(q)✱ q + 1 ✐s ❛ p✲♣♦✇❡r✱ ♦r ♦♥❡ ♦❢ t❤❡ ❝❛s❡s (3) − (6) ♦❢ Pr♦♣♦s✐t✐♦♥ ✷ ❤♦❧❞s✳ ◆♦t❡ t❤❛t t❤❡ ✇♦r❦ ✇✐t❤ ▼❛❧❧❡ ❝♦♥t❛✐♥s r❡s✉❧ts ❢♦r ❣r♦✉♣s ❝❧♦s❡❞ t♦ s✐♠♣❧❡ s✉❝❤ ❛s PGLn(q)✱ SLn(q)✱ GLn(q)✳
SLIDE 15
❖♥❡ ❝❛♥ ♥♦✇ st✉❞② t❤❡ ❛❜♦✈❡ ♣r♦❜❧❡♠s ❢♦r ♥♦♥✲s✐♠♣❧❡ ❣r♦✉♣s✳ ❋♦r s✐♠♣❧❡ ❣r♦✉♣s ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r❡st✐♥❣ Pr♦❜❧❡♠ ✸✳ ❉❡t❡r♠✐♥❡ t❤❡ ♠✐♥✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛ ♣r♦❥❡❝t✐✈❡ ❝❤❛r❛❝t❡r ♦❢ G ❢♦r ❡✈❡r② ♣r✐♠❡ ❞✐✈✐❞✐♥❣ |G|✳ Pr♦❜❧❡♠ ✹✳ ❉❡t❡r♠✐♥❡ t❤❡ ♠✐♥✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛ ❝❤❛r❛❝t❡r ♦❢ G ✈❛♥✐s❤✐♥❣ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡♠❡♥ts ✭✇❤❡♥ p ❞✐✈✐❞❡s |G|✮✳ ❙♦♠❡ ♦❜s❡r✈❛t✐♦♥s ❝♦♥❝❡r♥✐♥❣ t❤❡s❡ ♣r♦❜❧❡♠s ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ✇♦r❦s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳
SLIDE 16
■ ♠❡♥t✐♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦♥ ❣r♦✉♣s ♦❢ ▲✐❡ t②♣❡ ✭❩✱ ❏✳ ❆❧❣✳ ✷✵✶✸✮✿ ❚❤❡♦r❡♠ ✸✳ ▲❡t G = PSLn(pm)✱ n > 4✱ ❛♥❞ ❧❡t χ ❜❡ ❛ r❡❞✉❝✐❜❧❡ ♣r♦❥❡❝t✐✈❡ ❝❤❛r❛❝t❡r ❢♦r t❤❡ ♣r✐♠❡ p✳ ❚❤❡♥ χ(1) ≥ (n − 1) · |G|p✳ ❚❤✐s ❜♦✉♥❞ ✐s s❤❛r♣✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡ ♦❢ ❞✐♠❡♥s✐♦♥ n · |G|p ✐s ✇❡❧❧ ❦♥♦✇♥✱ ❛♥❞ ❢♦r q = 2 t❤❡r❡ ❡①✐sts s✉❝❤ ❛ ♠♦❞✉❧❡ ♦❢ ❞✐♠❡♥s✐♦♥ (n − 1) · |G|2✳ ■♥ ❛❞❞✐✲ t✐♦♥✱ t❤❡r❡ ❡①✐sts ❛ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ❞❡❣r❡❡ (n − 1) · |G|p ✈❛♥✐s❤✐♥❣ ❛t t❤❡ p✲s✐♥❣✉❧❛r ❡❧❡✲ ♠❡♥ts✳ ■ ❡①♣❡❝t t❤❛t t❤✐s ✐s ♣r♦❥❡❝t✐✈❡✳ ■♥ t❤❡ ❛❜♦✈❡ ♣❛♣❡r ❛ s✐♠✐❧❛r r❡s✉❧t ✐s ♣r♦✈❡❞ ❛❧s♦ ❢♦r ❣r♦✉♣s En(pm), n = 6, 7, 8✱ ✇❤❡r❡ t❤❡ ❜♦✉♥❞ ✐s s❤♦✇♥ t♦ ❜❡ n · |G|p✳
SLIDE 17 ❙♦♠❡ ❜✐❜❧✐♦❣r❛♣❤②
- ✳ ▼❛❧❧❡ ❛♥❞ ❚❤✳ ❲❡✐❣❡❧✱ ❋✐♥✐t❡ ❣r♦✉♣s ✇✐t❤ ♠✐♥✐♠❛❧
1✲P■▼✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳ ✶✷✻✭✷✵✵✽✮✱ ✸✶✺ ✲ ✸✸✷✳
- ✳ ▼❛❧❧❡ ❛♥❞ ❆✳ ❩❛❧❡ss❦✐✱ ■♥ ♣r❡♣❛r❛t✐♦♥
▼✳ P❡❧❧❡❣r✐♥✐ ❛♥❞ ❆✳ ❩❛❧❡ss❦✐✱ ❖♥ ❝❤❛r❛❝t❡rs ♦❢ ❈❤❡✈❛❧✲ ❧❡② ❣r♦✉♣s ✈❛♥✐s❤✐♥❣ ❛t t❤❡ ♥♦♥✲s❡♠✐s✐♠♣❧❡ ❡❧❡♠❡♥ts✱ ■♥t❡r♥✳ ❏✳ ❆❧❣❡❜r❛ ❛♥❞ ❈♦♠♣✉t✳ ▼❛t❤✳ ✷✻✭✷✵✶✻✮✱ ✼✽✾ ✲ ✽✹✶✳ ❆✳ ❩❛❧❡ss❦✐✱ ▲♦✇ ❞✐♠❡♥s✐♦♥❛❧ ♣r♦❥❡❝t✐✈❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡s ❢♦r ❈❤❡✈❛❧❧❡② ❣r♦✉♣s ✐♥ ❞❡✜♥✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝✱ ❏✳ ❆❧❣❡❜r❛ ✸✼✼✭✷✵✶✸✮✱ ✶✷✺ ✕ ✶✺✻✳
