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r r r srs r r r srs s r r

  1. ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G , H ∈ G n ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ S n � G n ✳ ◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P ❄

  2. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G , H ∈ G n ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ S n � G n ✳ ◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P ❄ ◮ P ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

  3. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G , H ∈ G n ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ S n � G n ✳ ◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P ❄ ◮ P ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

  4. ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤ ✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳

  5. ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤ ✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳ ◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿

  6. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤ ✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳ ◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ◮ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

  7. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤ ✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳ ◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ◮ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

  8. ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

  9. ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧

  10. P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳

  11. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ ◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H ✳

  12. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ ◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H ✳ ◮ P 1 ≡ pol P 2!

  13. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s ♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ ◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H ✳ ◮ P 1 ≡ pol P 2!

  14. ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳ ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ❛♥❞ ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■

  15. ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ❛♥❞ ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■ ◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳

  16. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■ ◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳ ◮ ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G ❛♥❞ H ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✳

  17. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ◮ Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■ ◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳ ◮ ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G ❛♥❞ H ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✳

  18. ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ s✉❝❤ t❤❛t ✮✳ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s ◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮

  19. ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s ◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t ( x , y ) ∈ R ) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✮✳

  20. ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s ◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t ( x , y ) ∈ R ) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✮✳ ◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳

  21. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s ◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t ( x , y ) ∈ R ) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✮✳ ◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ◮ ❙●■✿ R ( G , H , φ ) ⇔ φ : G ≃ H ✳ ●✐✈❡♥ ( G , H ) s✉❝❤ t❤❛t ∃ φ : G ≃ H t♦ ✜♥❞ φ.

  22. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s ◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t ( x , y ) ∈ R ) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✮✳ ◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ( x , y ) ∈ R ) ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ◮ ❙●■✿ R ( G , H , φ ) ⇔ φ : G ≃ H ✳ ●✐✈❡♥ ( G , H ) s✉❝❤ t❤❛t ∃ φ : G ≃ H t♦ ✜♥❞ φ.

  23. Pr♦♦❢ ❆s❝❡♥t ✳ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳

  24. ❆s❝❡♥t ✳ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢

  25. ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳

  26. ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } .

  27. ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n

  28. ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n ◮ ❛♥❞ h 1 , . . . , h n ∈ V H s✉❝❤ t❤❛t✿

  29. ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n ◮ ❛♥❞ h 1 , . . . , h n ∈ V H s✉❝❤ t❤❛t✿ ◮ ✶✮ G i ≃ H i ❢♦r ❛❧❧ i ✱

  30. ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n ◮ ❛♥❞ h 1 , . . . , h n ∈ V H s✉❝❤ t❤❛t✿ ◮ ✶✮ G i ≃ H i ❢♦r ❛❧❧ i ✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s G i − 1 t♦ H i − 1 ❢♦r ❛❧❧ i = 1 , . . . , n ,

  31. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n ◮ ❛♥❞ h 1 , . . . , h n ∈ V H s✉❝❤ t❤❛t✿ ◮ ✶✮ G i ≃ H i ❢♦r ❛❧❧ i ✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s G i − 1 t♦ H i − 1 ❢♦r ❛❧❧ i = 1 , . . . , n , ◮ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ❢♦r ❛❧❧ i = 1 , . . . , n ✳

  32. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■ ◮ ❚❤❡♦r❡♠ ❙●■ ≤ pol ●■✳ ◮ Pr♦♦❢ ◮ ❆s❝❡♥t ✳ ◮ V G = { g 1 , . . . , g n } . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n ◮ ❛♥❞ h 1 , . . . , h n ∈ V H s✉❝❤ t❤❛t✿ ◮ ✶✮ G i ≃ H i ❢♦r ❛❧❧ i ✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s G i − 1 t♦ H i − 1 ❢♦r ❛❧❧ i = 1 , . . . , n , ◮ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ❢♦r ❛❧❧ i = 1 , . . . , n ✳

  33. ❚❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✦ ■♥❞❡❡❞✱ ❜② ✶✮ ✳ ❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t

  34. ■♥❞❡❡❞✱ ❜② ✶✮ ✳ ❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦

  35. ❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃ φ : G n → H n ✳

  36. t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃ φ : G n → H n ✳ ◮ ❇② ✷✮ φ t❛❦❡s g n t♦ h n ✳

  37. ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃ φ : G n → H n ✳ ◮ ❇② ✷✮ φ t❛❦❡s g n t♦ h n ✳ ◮ φ | G n − 1 t❛❦❡s G n − 1 t♦ H n − 1 ❜② ✷✮ ❛♥❞ t❛❦❡s g n − 1 t♦ h n − 1 ❜② ✸✮✳

  38. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃ φ : G n → H n ✳ ◮ ❇② ✷✮ φ t❛❦❡s g n t♦ h n ✳ ◮ φ | G n − 1 t❛❦❡s G n − 1 t♦ H n − 1 ❜② ✷✮ ❛♥❞ t❛❦❡s g n − 1 t♦ h n − 1 ❜② ✸✮✳ ◮ ❚❤❡♥ φ | G n − 1 t❛❦❡s G n − 2 t♦ H n − 2 ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t φ t❛❦❡s ❡✈❡r② g i t♦ h i ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

  39. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■ ◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃ φ : G n → H n ✳ ◮ ❇② ✷✮ φ t❛❦❡s g n t♦ h n ✳ ◮ φ | G n − 1 t❛❦❡s G n − 1 t♦ H n − 1 ❜② ✷✮ ❛♥❞ t❛❦❡s g n − 1 t♦ h n − 1 ❜② ✸✮✳ ◮ ❚❤❡♥ φ | G n − 1 t❛❦❡s G n − 2 t♦ H n − 2 ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t φ t❛❦❡s ❡✈❡r② g i t♦ h i ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

  40. ❋✐① ❛♥❞ ❝♦♥s✐❞❡r ❙✐♥❝❡ ✱ s✳t✳ ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳

  41. ❙✐♥❝❡ ✱ s✳t✳ ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 .

  42. ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) .

  43. ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳

  44. ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳ ◮ ❙❡t h 1 = h ✳

  45. ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳ ◮ ❙❡t h 1 = h ✳ ◮ g 1 , h 1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G 1 , H 1 ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ✳

  46. ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳ ◮ ❙❡t h 1 = h ✳ ◮ g 1 , h 1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G 1 , H 1 ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ✳ ◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G 1 ❛♥❞ H 1 t❛❦❡s G 0 t♦ H 0 ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1 ✳

  47. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳ ◮ ❙❡t h 1 = h ✳ ◮ g 1 , h 1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G 1 , H 1 ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ✳ ◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G 1 ❛♥❞ H 1 t❛❦❡s G 0 t♦ H 0 ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1 ✳ ◮ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ h 1 , . . . , h n ∈ H ❛♥❞ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n .

  48. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■ ❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r ◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k 1 ∈ K n +1 ❛♥❞ ❝♦♥s✐❞❡r G 1 = G ∪ g 1 = k 1 K n +1 , H 1 ( h ) = H ∪ h = k 1 K n +1 . ◮ ❙✐♥❝❡ G ≃ H ✱ ∃ h s✳t✳ G 1 ≃ H 1 = H 1 ( h ) . ◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs ( G 1 , H 1 ( h )) ✳ ◮ ❙❡t h 1 = h ✳ ◮ g 1 , h 1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G 1 , H 1 ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G i ❛♥❞ H i t❛❦❡s g i t♦ h i ✳ ◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G 1 ❛♥❞ H 1 t❛❦❡s G 0 t♦ H 0 ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1 ✳ ◮ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ h 1 , . . . , h n ∈ H ❛♥❞ G = G 0 < G 1 < · · · < G n ✱ H = H 0 < H 1 < · · · < H n .

  49. ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮ ❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■ ◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮ ■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳

  50. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■ ◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮ ■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳ ◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮ ❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳

  51. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■ ◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮ ■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳ ◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮ ❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳

  52. ■♥ ❞❡t❛✐❧✱ ✱ ✇❤❡r❡ ✳ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ ❢♦r ❛❧❧ ✱ ✐✳❡✳ ✳ ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

  53. ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ ❢♦r ❛❧❧ ✱ ✐✳❡✳ ✳ ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ◮ ■♥ ❞❡t❛✐❧✱ τ ( v ) = ( d 1 , ..., d d ( v ) ) ✱ ✇❤❡r❡ d 1 ≤ d 2 ≤ . . . ≤ d d ( v ) ✳

  54. ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ◮ ■♥ ❞❡t❛✐❧✱ τ ( v ) = ( d 1 , ..., d d ( v ) ) ✱ ✇❤❡r❡ d 1 ≤ d 2 ≤ . . . ≤ d d ( v ) ✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ ( φ ( v )) = τ ( v ) ❢♦r ❛❧❧ v ∈ V ( G ) ✱ ✐✳❡✳ τ ◦ φ = τ ✳

  55. ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ◮ ■♥ ❞❡t❛✐❧✱ τ ( v ) = ( d 1 , ..., d d ( v ) ) ✱ ✇❤❡r❡ d 1 ≤ d 2 ≤ . . . ≤ d d ( v ) ✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ ( φ ( v )) = τ ( v ) ❢♦r ❛❧❧ v ∈ V ( G ) ✱ ✐✳❡✳ τ ◦ φ = τ ✳ ◮ ❚❤❡ t②♣❡ τ G ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ( τ ( v 1 ) , . . . , τ ( v n )) , n = | V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳

  56. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ◮ ■♥ ❞❡t❛✐❧✱ τ ( v ) = ( d 1 , ..., d d ( v ) ) ✱ ✇❤❡r❡ d 1 ≤ d 2 ≤ . . . ≤ d d ( v ) ✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ ( φ ( v )) = τ ( v ) ❢♦r ❛❧❧ v ∈ V ( G ) ✱ ✐✳❡✳ τ ◦ φ = τ ✳ ◮ ❚❤❡ t②♣❡ τ G ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ( τ ( v 1 ) , . . . , τ ( v n )) , n = | V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ◮ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ G �→ τ G ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ τ G = τ H ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ✳

  57. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ ◮ ❚❤❡ t②♣❡ τ ( v ) ∈ N ∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢ ✈❡rt✐❝❡s ♦❢ N ( v ) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ◮ ■♥ ❞❡t❛✐❧✱ τ ( v ) = ( d 1 , ..., d d ( v ) ) ✱ ✇❤❡r❡ d 1 ≤ d 2 ≤ . . . ≤ d d ( v ) ✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ ( φ ( v )) = τ ( v ) ❢♦r ❛❧❧ v ∈ V ( G ) ✱ ✐✳❡✳ τ ◦ φ = τ ✳ ◮ ❚❤❡ t②♣❡ τ G ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ( τ ( v 1 ) , . . . , τ ( v n )) , n = | V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ◮ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ G �→ τ G ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ τ G = τ H ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ✳

  58. ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ ✐s ✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s s✉❝❤ t❤❛t ✳ ♣❛✉s❡ ■❢ ❛ ❣r❛♣❤ ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐♠♣❧✐❡s t❤❛t ❢♦r ❛❧❧ ✳ ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮ ▲❡t ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ r❡s♣❡❝t✐✈❡❧②✳ ■❢ ✐s ♦❜❧✐q✉❡ ❛♥❞ t❤❡♥ t❤❡ ♠❛♣ ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ♦♥t♦ ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❖❜❧✐q✉❡ ❣r❛♣❤s ◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱ ❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳

  59. ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮ ▲❡t ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ r❡s♣❡❝t✐✈❡❧②✳ ■❢ ✐s ♦❜❧✐q✉❡ ❛♥❞ t❤❡♥ t❤❡ ♠❛♣ ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ♦♥t♦ ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❖❜❧✐q✉❡ ❣r❛♣❤s ◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱ ❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳ ◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N ∗ ✐s ✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u , v ∈ V ( G ) s✉❝❤ t❤❛t τ ( v ) = τ ( w ) ✳ ♣❛✉s❡ ◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r τ G = ( τ ( v 1 ) , . . . , τ ( v n )) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ( τ ( v 1 ) , . . . , τ ( v n )) = ( τ ( w 1 ) , . . . , τ ( w n )) ✐♠♣❧✐❡s t❤❛t v i = w i ❢♦r ❛❧❧ i ✳

  60. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❖❜❧✐q✉❡ ❣r❛♣❤s ◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱ ❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳ ◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N ∗ ✐s ✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u , v ∈ V ( G ) s✉❝❤ t❤❛t τ ( v ) = τ ( w ) ✳ ♣❛✉s❡ ◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r τ G = ( τ ( v 1 ) , . . . , τ ( v n )) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ( τ ( v 1 ) , . . . , τ ( v n )) = ( τ ( w 1 ) , . . . , τ ( w n )) ✐♠♣❧✐❡s t❤❛t v i = w i ❢♦r ❛❧❧ i ✳ ◮ ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮ ▲❡t τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ G , H r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ♦❜❧✐q✉❡ ❛♥❞ G ≃ H t❤❡♥ t❤❡ ♠❛♣ g i �→ h i ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ G ♦♥t♦ H ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

  61. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❖❜❧✐q✉❡ ❣r❛♣❤s ◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱ ❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳ ◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N ∗ ✐s ✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u , v ∈ V ( G ) s✉❝❤ t❤❛t τ ( v ) = τ ( w ) ✳ ♣❛✉s❡ ◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r τ G = ( τ ( v 1 ) , . . . , τ ( v n )) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ( τ ( v 1 ) , . . . , τ ( v n )) = ( τ ( w 1 ) , . . . , τ ( w n )) ✐♠♣❧✐❡s t❤❛t v i = w i ❢♦r ❛❧❧ i ✳ ◮ ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮ ▲❡t τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ G , H r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ♦❜❧✐q✉❡ ❛♥❞ G ≃ H t❤❡♥ t❤❡ ♠❛♣ g i �→ h i ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ G ♦♥t♦ H ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

  62. ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ t♦ ❝♦♥str✉❝t s✉❝❤ t❤❛t ✳ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳

  63. ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳

  64. ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳ ◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ∈ G n ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) .

  65. ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳ ◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ∈ G n ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) . ◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳

  66. ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳ ◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ∈ G n ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) . ◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ◮ ❖t❤❡r✇✐s❡ G , H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ φ : g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳

  67. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳ ◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ∈ G n ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) . ◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ◮ ❖t❤❡r✇✐s❡ G , H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ φ : g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ◮ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s φ ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

  68. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A ◮ ■♥♣✉t s❡t✿ I = ∪ I n ✱ ✇❤❡r❡ I n = { ( G , H ) : G , H ∈ G n , G ≃ H } ✳ ◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ( G , H ) ∈ I n t♦ ❝♦♥str✉❝t π ∈ S n s✉❝❤ t❤❛t π G = H ✳ ◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G , H ∈ G n ✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τ G = ( τ ( g 1 ) , . . . , τ ( g n )) , τ H = ( τ ( h 1 ) , . . . , τ ( h n )) . ◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ◮ ❖t❤❡r✇✐s❡ G , H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ φ : g i �→ h i ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ◮ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s φ ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

  69. ❚❤❡♦r❡♠ ❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ t❡♥❞s t♦ ③❡r♦ ❛s t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts I n ∩ ( G o n × G o n ) ❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t I n ∩ ( S n × S n ) , ✇❤❡r❡ S n = G n − G o n ✳

  70. ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts I n ∩ ( G o n × G o n ) ❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t I n ∩ ( S n × S n ) , ✇❤❡r❡ S n = G n − G o n ✳ ◮ ❚❤❡♦r❡♠ ❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳

  71. ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts I n ∩ ( G o n × G o n ) ❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t I n ∩ ( S n × S n ) , ✇❤❡r❡ S n = G n − G o n ✳ ◮ ❚❤❡♦r❡♠ ❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ G n t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ |G o n | lim |G n | = 1 , n →∞ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ |S n | lim |G n | = 1 . n →∞

  72. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts I n ∩ ( G o n × G o n ) ❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t I n ∩ ( S n × S n ) , ✇❤❡r❡ S n = G n − G o n ✳ ◮ ❚❤❡♦r❡♠ ❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ G n t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ |G o n | lim |G n | = 1 , n →∞ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ |S n | lim |G n | = 1 . n →∞ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

  73. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts I n ∩ ( G o n × G o n ) ❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t I n ∩ ( S n × S n ) , ✇❤❡r❡ S n = G n − G o n ✳ ◮ ❚❤❡♦r❡♠ ❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ G n t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ |G o n | lim |G n | = 1 , n →∞ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ |S n | lim |G n | = 1 . n →∞ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

  74. ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ ❄ ❢♦r ❡✈❡r② ✳ ■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳

  75. ❢♦r ❡✈❡r② ✳ ■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄

  76. ■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄ � ≤ n ! ❢♦r ❡✈❡r② G ∈ G n ✳ ◮ � � pr − 1 � 1 G

  77. ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄ � ≤ n ! ❢♦r ❡✈❡r② G ∈ G n ✳ ◮ � � pr − 1 � 1 G ◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ � = n ! � pr − 1 � � 1 G

  78. ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄ � ≤ n ! ❢♦r ❡✈❡r② G ∈ G n ✳ ◮ � � pr − 1 � 1 G ◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ � = n ! � pr − 1 � � 1 G ◮ ■♥❞❡❡❞✱ Aut ( G ) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t S n G ❝♦♥s✐sts ♦❢ n ! ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ( G , S n G ) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr 1 ✳

  79. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄ � ≤ n ! ❢♦r ❡✈❡r② G ∈ G n ✳ ◮ � � pr − 1 � 1 G ◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ � = n ! � pr − 1 � � 1 G ◮ ■♥❞❡❡❞✱ Aut ( G ) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t S n G ❝♦♥s✐sts ♦❢ n ! ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ( G , S n G ) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr 1 ✳ ◮ ❈♦♥❝❧✉❞❡✿ pr 1 : I n ∩ ( G o n × G o n ) → G o n ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② n ! ✱ ❤❡♥❝❡ |I n ∩ ( G o n × G o n ) | = n ! |G o n | ❛♥❞ s♦ |I n | ≥ n ! |G o n | .

  80. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r pr 1 : I n → G n ✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr 1 ❄ � ≤ n ! ❢♦r ❡✈❡r② G ∈ G n ✳ ◮ � � pr − 1 � 1 G ◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ � = n ! � pr − 1 � � 1 G ◮ ■♥❞❡❡❞✱ Aut ( G ) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t S n G ❝♦♥s✐sts ♦❢ n ! ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ( G , S n G ) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr 1 ✳ ◮ ❈♦♥❝❧✉❞❡✿ pr 1 : I n ∩ ( G o n × G o n ) → G o n ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② n ! ✱ ❤❡♥❝❡ |I n ∩ ( G o n × G o n ) | = n ! |G o n | ❛♥❞ s♦ |I n | ≥ n ! |G o n | .

  81. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr 1 ♦✈❡r t❤❡ s❡t S n ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t |I n ∩ ( S n × S n ) | ≤ n ! |S n | .

  82. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr 1 ♦✈❡r t❤❡ s❡t S n ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t |I n ∩ ( S n × S n ) | ≤ n ! |S n | . ◮ |I n ∩ ( S n × S n ) | ≤ n ! |S n | n | = |S n | |S n | n | = n ! |G o |G o |I n | |G n | − |S n |

  83. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr 1 ♦✈❡r t❤❡ s❡t S n ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t |I n ∩ ( S n × S n ) | ≤ n ! |S n | . ◮ |I n ∩ ( S n × S n ) | ≤ n ! |S n | n | = |S n | |S n | n | = n ! |G o |G o |I n | |G n | − |S n | ◮   |S n | 1  → 0 , n → ∞ .  1 − |S n | |G n | |G n |

  84. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr 1 ♦✈❡r t❤❡ s❡t S n ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t |I n ∩ ( S n × S n ) | ≤ n ! |S n | . ◮ |I n ∩ ( S n × S n ) | ≤ n ! |S n | n | = |S n | |S n | n | = n ! |G o |G o |I n | |G n | − |S n | ◮   |S n | 1  → 0 , n → ∞ .  1 − |S n | |G n | |G n |

  85. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❉✐s❡❛s❡ ❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

  86. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❉✐s❡❛s❡ ❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

  87. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❉✐s❡❛s❡ ❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

  88. ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❉✐s❡❛s❡ ❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

  89. ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳ ❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦ ❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮ ◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳ ❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭ 10 6 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳

  90. ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦ ❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮ ◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳ ❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭ 10 6 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳ ◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳ ❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄

  91. ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡ ❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮ ◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳ ❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭ 10 6 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳ ◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳ ❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄ ◮ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦ ❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳

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