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❖◆ 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 ✶

■♥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 ✳

❚❤✐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 ✳

◆♦t❡ t❤❛t ❡✈❡r② P■▼ Φ ❧✐❢ts t♦ ❝❤❛r❛❝t❡r✐st✐❝ ✵✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡r❡ ✐s ❛ ♣r♦❥❡❝t✐✈❡ ✐♥❞❡✲ ❝♦♠♣♦s❛❜❧❡ K p G ✲♠♦❞✉❧❡ Φ ✇❤♦s❡ r❡❞✉❝t✐♦♥ ♠♦❞✉❧♦ p ②✐❡❧❞s Φ ✳ ❍❡r❡ K p ✐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 ✲♠♦❞✉❧❡ 1 G ♦❝❝✉rs ✐♥ ❛ ✉♥✐q✉❡ P■▼ Φ 1 ✱ ✇❤✐❝❤ ✐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♦ ❜❡ ❞❡✈❡❧♦♣❡❞✳

◆❡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❡✳

❆♥ ❡❛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 = L 2 ( q ) ✱ q ❡✈❡♥✱ ❛♥❞ p = χ (1) = q ± 1 ✱ ♦r G = L 2 (8) ✱ p = 3 ❛♥❞ χ (1) = 9;

✭✸✮ G = L 2 ( q ) ✱ q ♦❞❞✱ χ (1) = ( q ± 1) / 2 ✐s ❛ p ✲ ♣♦✇❡r ❢♦r p > 2 ✱ ♦r p = 2 ❛♥❞ χ (1) = q ± 1 ✐s ❛ 2 ✲♣♦✇❡r❀ ✭✹✮ G = L n ( q ) ✱ q > 2 ✱ ( n, q − 1) = 1 ✱ n ✐s ❛♥ ♦❞❞ ♣r✐♠❡ s✉❝❤ t❤❛t χ (1) = ( q n − 1) / ( q − 1) ✐s ❛ p ✲♣♦✇❡r❀ ✭✺✮ G = U n ( q ) ✱ n ✐s ❛♥ ♦❞❞ ♣r✐♠❡✱ ( n, q +1) = 1 ✱ s✉❝❤ t❤❛t χ (1) = ( q n + 1) / ( q + 1) ✐s ❛ p ✲ ♣♦✇❡r❀ ✭✻✮ G = S 2 n ( q ) ✱ n > 1 ✱ q = r k ✇✐t❤ r ❛♥ ♦❞❞ ♣r✐♠❡✱ kn ✐s ❛ 2 ✲♣♦✇❡r s✉❝❤ t❤❛t χ (1) = ( q n + 1) / 2 ✐s ❛ p ✲♣♦✇❡r❀ ✭✼✮ G = S 2 n (3) ✱ n > 2 ✐s ❛ ♣r✐♠❡✱ s✉❝❤ t❤❛t χ (1) = (3 n − 1) / 2 ✐s ❛ p ✲♣♦✇❡r❀

✭✽✮ G = A p +1 ✱ χ (1) = p ; ✭✾✮ G = S 6 (2) ✱ χ (1) = 7; ✭✶✵✮ G ∈ { M 11 , M 12 } ❛♥❞ χ (1) = 11; ✭✶✶✮ G ∈ { M 11 , L 3 (3) } ❛♥❞ χ (1) = 16; ✭✶✷✮ G ∈ { M 24 , Co 2 , Co 3 } ❛♥❞ χ (1) = 23; ✭✶✸✮ G = 2 F 4 (2) ′ ❛♥❞ χ (1) = 27 ❀ ✭✶✹✮ G = U 3 (3) ∼ = G 2 (2) ′ ❛♥❞ χ (1) = 32; ♦r ✭✶✺✮ G = G 2 (3) ❛♥❞ χ (1) = 64 ✳