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  1. ●r♦✉♣s ✇❤♦s❡ ♥♦♥✲◆♦r♠❛❧ ❙✉❜❣r♦✉♣s ❛r❡ ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❉❛r✐♦ ❊s♣♦s✐t♦ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ◆❛♣♦❧✐ ❋❡❞❡r✐❝♦ ■■ ❨♦✉♥❣ ❘❡s❡❛r❝❤❡rs ✐♥ ❆❧❣❡❜r❛ ❈♦♥❢❡r❡♥❝❡ ✲ ✷✵✶✾ ◆❛♣♦❧✐ ✖ ❙❡♣t❡♠❜❡r✱ ✶✽t❤ ❉❛r✐♦ ❊s♣♦s✐t♦

  2. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀ H ≤ G , H �∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH ✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  3. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀ H ≤ G , H �∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH ✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  4. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ■♥ ✶✾✻✻✱ ●✳▼✳ ❘♦♠❛❧✐s ❛♥❞ ◆✳❋✳ ❙❡s❡❦✐♥ ✐♥tr♦❞✉❝❡❞ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✳ ❚❤❡② s❛② ❛ ❣r♦✉♣ G ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ✐❢✿ ∀ H ≤ G , H �∈ A ⇒ H ⊳ G ✇❤❡r❡ A ✐s t❤❡ ❝❧❛ss ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✳ ❲❡ ✇✐❧❧ ✉s❡ H t♦ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧❧② SH ✲❝❧♦s❡❞✱ ❜✉t ♥♦t ❝❧♦s❡❞ ❜② ❡①t❡♥s✐♦♥s ♦r ❡✈❡♥ ❞✐r❡❝t ♣r♦❞✉❝ts✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  5. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭ ✷✵✵✻ ✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  6. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭ ✷✵✵✻ ✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  7. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭ ✷✵✵✻ ✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  8. ▼❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣s ❘❡str✐❝t✐♥❣ ♦✉t ❛tt❡♥t✐♦♥ t♦ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ❣r♦✉♣s t♦ ❡①❝❧✉❞❡ ❚❛rs❦✐ ❣r♦✉♣s✱ t❤❡② ♣r♦✈❡❞ ❚❤❡♦r❡♠ ✶✳ ❬●✳▼✳ ❘♦♠❛❧✐s ✲ ◆✳❋✳ ❙❡s❡❦✐♥✱ ✶✾✻✻❪ ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② s♦❧✉❜❧❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥ ❣r♦✉♣✳ ❚❤❡♥ G ✐s s♦❧✉❜❧❡✱ t❤❡ ❞❡r✐✈❡❞ ❧❡♥❣t❤ ♦❢ G ✐s ❛t ♠♦st ✸ ❛♥❞ t❤❡ ❝♦♠♠✉t❛t♦r s✉❜❣r♦✉♣ G ′ ✐s ✜♥✐t❡ ✇✐t❤ ♣r✐♠❡✲♣♦✇❡r ♦r❞❡r✳ ❆ s✐♠♣❧✐✜❡❞ ♣r♦♦❢ ♦❢ t❤❡ r❡s✉❧t ✐♥ t❤❡ ❧❛r❣❡r ❝❧❛ss ♦❢ ❧♦❝❛❧❧② ❣r❛❞❡❞ ❣r♦✉♣s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛ ♣❛♣❡r ❜② ❋✳ ❉❡ ▼❛r✐ ❛♥❞ ❋✳ ❞❡ ●✐♦✈❛♥♥✐ ✭ ✷✵✵✻ ✮ ❛❜♦✉t ❣r♦✉♣s ✇✐t❤ ❢❡✇ ♥♦r♠❛❧✐③❡rs ♦❢ ♥♦♥✲❛❜❡❧✐❛♥ s✉❜❣r♦✉♣s✳ ❆ ❣r♦✉♣ G ✐s ❧♦❝❛❧❧② ❣r❛❞❡❞ ✐❢ ❛❧❧ ♦❢ ✐ts ♥♦♥✲tr✐✈✐❛❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣ ❤❛✈❡ ♣r♦♣❡r s✉❜❣r♦✉♣s ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❉❛r✐♦ ❊s♣♦s✐t♦

  9. ❚❤❡ ❝❧❛ss H ❝❛♥ ❛❧s♦ ❜❡ s❤♦✇♥ t♦ ❜❡ L ✲❝❧♦s❡❞✱ ✐✳❡✳✿ Pr♦♣♦s✐t✐♦♥ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ✇❤♦s❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ s✉❜❣r♦✉♣s ❛r❡ ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❚❤❡♥ G ✐ts❡❧❢ ✐s ♠❡t❛❤❛♠✐❧t♦♥✐❛♥✳ ❉❛r✐♦ ❊s♣♦s✐t♦

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