Pr rs r - - PowerPoint PPT Presentation

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p r r s

Pr rs r - - PowerPoint PPT Presentation

Sep 21, 2022 459 likes •1.66k views

Pr rs r Pr rs sst rts r

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SLIDE 1

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

◆♦✈❡♠❜❡r ✹✱ ✷✵✶✻

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 2

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t ✇❤✐❝❤ ❣❡♥❡r❛t❡s

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 3

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t X ⊆ G ✇❤✐❝❤ ❣❡♥❡r❛t❡s G

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 4

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t X ⊆ G ✇❤✐❝❤ ❣❡♥❡r❛t❡s G

g

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 5

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t X ⊆ G ✇❤✐❝❤ ❣❡♥❡r❛t❡s G

g x g · x

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 6

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t X ⊆ G ✇❤✐❝❤ ❣❡♥❡r❛t❡s G

g x g · x x2 gx2 gx3 x4 gx4 x3

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 7

❈❛②❧❡② ●r❛♣❤

❆ ❣r♦✉♣ G ❆ s✉❜s❡t X ⊆ G ✇❤✐❝❤ ❣❡♥❡r❛t❡s G

g x g · x x2 gx2 gx3 x4 gx4 x3

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 8

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✶}

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 9

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✶}

1 4 5 6 8 9 3 7 2 P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 10

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✶}

1 4 5 6 8 9 3 7 2 P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 11

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✷, ✺}✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 12

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✷, ✺}✿

7 2 1 3 4 5 6 8 9 P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 13

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✷, ✺}✿

7 2 1 3 4 5 6 8 9 P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 14

❊①❛♠♣❧❡s

G = Z✶✵✱ X = {✷, ✺}✿

7 2 1 3 4 5 6 8 9 P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 15

❊①❛♠♣❧❡s

G = S✹✱ X = {(✶✷), (✶✷✸✹)}✿

( ) (13)(24) (1234) (234) (14) (142) (1324) (123) (34) (132) (1243) (13) (12)(34) (14)(23) (24) (243) (1423) (12) (134) (124) (23) (1342) (143) (1432)

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 16

❊①❛♠♣❧❡s

G = S✹✱ X = {(✶✷), (✶✸), (✶✹)}✿

(13)(24) (1324) (243) (1432) (143) (14) (12) (123) (1234) (234) (1342) (13) (124) (23) (12)(34) (1423) (134) (24) (132) (1243) (14)(23) (34) (142) ( )

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 17

P❧❛♥❛r ●r❛♣❤

❉❡✜♥✐t✐♦♥ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ♣❧❛♥❛r ✐❢ ✐t ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ s✉❜s❡t ♦❢ t❤❡ ♣❧❛♥❡ ✭✇❤❡♥ t❤♦✉❣❤t ♦❢ ❛s ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡①✮✳ ❊q✉✐✈❛❧❡♥t❧②✱ ✐❢ ✐t ❝❛♥ ❜❡ ❞r❛✇♥ ✐♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ ♥♦ ✏❡①tr❛♥❡♦✉s ❝r♦ss✐♥❣s✑✖✐✳❡✳✱ s♦ t❤❛t t✇♦ ❡❞❣❡s ✐♥t❡rs❡❝t ♦♥❧② ❛t ❛ ❝♦♠♠♦♥ ✈❡rt❡①✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 18

P❧❛♥❛r ●r❛♣❤

❉❡✜♥✐t✐♦♥ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ♣❧❛♥❛r ✐❢ ✐t ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ s✉❜s❡t ♦❢ t❤❡ ♣❧❛♥❡ ✭✇❤❡♥ t❤♦✉❣❤t ♦❢ ❛s ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡①✮✳ ❊q✉✐✈❛❧❡♥t❧②✱ ✐❢ ✐t ❝❛♥ ❜❡ ❞r❛✇♥ ✐♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ ♥♦ ✏❡①tr❛♥❡♦✉s ❝r♦ss✐♥❣s✑✖✐✳❡✳✱ s♦ t❤❛t t✇♦ ❡❞❣❡s ✐♥t❡rs❡❝t ♦♥❧② ❛t ❛ ❝♦♠♠♦♥ ✈❡rt❡①✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 19

❊①❛♠♣❧❡s

✼ ✷ ✵ ✶ ✸ ✹ ✺ ✻ ✽ ✾

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 20

❊①❛♠♣❧❡s

✼ ✷ ✵ ✶ ✸ ✹ ✺ ✻ ✽ ✾ ✾ ✷ ✵ ✽ ✻ ✹ ✼ ✺ ✸ ✶ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 21

❊①❛♠♣❧❡s

✼ ✷ ✵ ✶ ✸ ✹ ✺ ✻ ✽ ✾ ✾ ✷ ✵ ✽ ✻ ✹ ✼ ✺ ✸ ✶ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-22

SLIDE 22

❊①❛♠♣❧❡s

( ) (13)(24) (1234) (234) (14) (142) (1324) (123) (34) (132) (1243) (13) (12)(34) (14)(23) (24) (243) (1423) (12) (134) (124) (23) (1342) (143) (1432)

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 23

❊①❛♠♣❧❡s

(13)(24) (1324) (243) (1432) (143) (14) (12) (123) (1234) (234) (1342) (13) (124) (23) (12)(34) (1423) (134) (24) (132) (1243) (14)(23) (34) (142) ( )

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-24

SLIDE 24

❊①❛♠♣❧❡s

(13)(24) (1324) (243) (1432) (143) (14) (12) (123) (1234) (234) (1342) (13) (124) (23) (12)(34) (1423) (134) (24) (132) (1243) (14)(23) (34) (142) ( ) (13)(24) (1324) (243) (1432) (143) (14) (12) (123) (1234) (234) (1342) (13) (124) (23) (12)(34) (1423) (134) (24) (132) (1243) (14)(23) (34) (142) ( ) (134) (234) (1234) (23) (12) (124) (142) (14) (34) (132) (243) (1432) (13)(24) (1342) (24) (1324) (143) (123)

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 25

❚❤❡ ◗✉❡st✐♦♥

❲❤✐❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞ ❣r♦✉♣s ❛❞♠✐t ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s❄ ■❞❡❛❧❧②✿ ❛ ✏❝❛t❛❧♦❣✉❡✑ ♦❢ ❞❡s❝r✐♣t✐♦♥s ✭s✉❝❤ ❛s ♣r❡s❡♥t❛t✐♦♥s✮ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ s✉❝❤ ❣r♦✉♣s✳ ❈❛❧❧ t❤❡♠ ♣❧❛♥❛r ❣r♦✉♣s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-26

SLIDE 26

❚❤❡ ◗✉❡st✐♦♥

❲❤✐❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞ ❣r♦✉♣s ❛❞♠✐t ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s❄ ■❞❡❛❧❧②✿ ❛ ✏❝❛t❛❧♦❣✉❡✑ ♦❢ ❞❡s❝r✐♣t✐♦♥s ✭s✉❝❤ ❛s ♣r❡s❡♥t❛t✐♦♥s✮ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ s✉❝❤ ❣r♦✉♣s✳ ❈❛❧❧ t❤❡♠ ♣❧❛♥❛r ❣r♦✉♣s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 27

❚❤❡ ◗✉❡st✐♦♥

❲❤✐❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞ ❣r♦✉♣s ❛❞♠✐t ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s❄ ■❞❡❛❧❧②✿ ❛ ✏❝❛t❛❧♦❣✉❡✑ ♦❢ ❞❡s❝r✐♣t✐♦♥s ✭s✉❝❤ ❛s ♣r❡s❡♥t❛t✐♦♥s✮ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ s✉❝❤ ❣r♦✉♣s✳ ❈❛❧❧ t❤❡♠ ♣❧❛♥❛r ❣r♦✉♣s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 28

❋✐♥✐t❡ ●r♦✉♣s

❋✐♥✐t❡ ♣❧❛♥❛r ❣r♦✉♣s ❂ ✜♥✐t❡ ❣r♦✉♣s ♦❢ ✐s♦♠❡tr✐❡s ♦❢ S✷ ✭▼❛s❝❤❦❡✱ ✶✽✾✻✮✿

✹ ✹ ✺

✳ ✳ ✳ ❛♥❞ ❛♥② ♦❢ t❤❡s❡

✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 29

❋✐♥✐t❡ ●r♦✉♣s

❋✐♥✐t❡ ♣❧❛♥❛r ❣r♦✉♣s ❂ ✜♥✐t❡ ❣r♦✉♣s ♦❢ ✐s♦♠❡tr✐❡s ♦❢ S✷ ✭▼❛s❝❤❦❡✱ ✶✽✾✻✮✿ Zn, Dn, A✹, S✹, A✺ ✳ ✳ ✳ ❛♥❞ ❛♥② ♦❢ t❤❡s❡

✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 30

❋✐♥✐t❡ ●r♦✉♣s

❋✐♥✐t❡ ♣❧❛♥❛r ❣r♦✉♣s ❂ ✜♥✐t❡ ❣r♦✉♣s ♦❢ ✐s♦♠❡tr✐❡s ♦❢ S✷ ✭▼❛s❝❤❦❡✱ ✶✽✾✻✮✿ Zn, Dn, A✹, S✹, A✺ ✳ ✳ ✳ ❛♥❞ ❛♥② ♦❢ t❤❡s❡ × Z✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 31

❇❛s✐❝ ❋❛❝ts

❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 32

❇❛s✐❝ ❋❛❝ts

❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

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SLIDE 33

❇❛s✐❝ ❋❛❝ts

❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-34

SLIDE 34

❇❛s✐❝ ❋❛❝ts

❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-35

SLIDE 35

❇❛s✐❝ ❋❛❝ts

P♦ss❡ss✐♥❣ ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤ ✐s ❛ ▼❛r❦♦✈ Pr♦♣❡rt②✿ t❤❡r❡ ✐s ♥♦ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s✱ ❢♦r ❛♥② ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ❣r♦✉♣ ❤❛s ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-36

SLIDE 36

❇❛s✐❝ ❋❛❝ts

P♦ss❡ss✐♥❣ ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤ ✐s ❛ ▼❛r❦♦✈ Pr♦♣❡rt②✿ t❤❡r❡ ✐s ♥♦ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s✱ ❢♦r ❛♥② ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ❣r♦✉♣ ❤❛s ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-37

SLIDE 37

■♥✜♥✐t❡ ❣r♦✉♣s

❊①❛♠♣❧❡s✿ Z✿

· · · · · ·

✷✿ ✸✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-38

SLIDE 38

■♥✜♥✐t❡ ❣r♦✉♣s

❊①❛♠♣❧❡s✿ Z✿

· · · · · ·

· · · · · · · · · · · · Z × Z✷✿

✸✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-39

SLIDE 39

■♥✜♥✐t❡ ❣r♦✉♣s

❊①❛♠♣❧❡s✿ Z✿

· · · · · ·

· · · · · · · · · · · · Z × Z✷✿

Z × Z✸✿

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-40

SLIDE 40

■♥✜♥✐t❡ ❣r♦✉♣s

Z × Z✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-41

SLIDE 41

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts

❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ ♦r ♦♥❡✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-42

SLIDE 42

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts

❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ Z ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ ♦r ♦♥❡✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-43

SLIDE 43

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts

❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ Z ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ ♦r ♦♥❡✿

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-44

SLIDE 44

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts

❚❤❡ s❡t ♦❢ ❱❆Ps ✭A✮ ✐s ❡ss❡♥t✐❛❧ ✐❢ ❛♥② t✇♦ ♦❢ ✐ts ♣♦✐♥ts ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ Γ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-45

SLIDE 45

❊ss❡♥t✐❛❧ ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-46

SLIDE 46

❊①❛♠♣❧❡

S✸ =

t, r | t✸ = r ✷ = (tr)✷ = ✶
✹

✸ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-47

SLIDE 47

❊①❛♠♣❧❡

S✸ =

t, r | t✸ = r ✷ = (tr)✷ = ✶
✹

✸ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-48

SLIDE 48

❊①❛♠♣❧❡

S✸ =

t, r | t✸ = r ✷ = (tr)✷ = ✶
A✹ =
t, g | t✸ = g ✷ = (tg)✸ = ✶
P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-49

SLIDE 49

❊①❛♠♣❧❡

✸ ✹ ✸ ✷ ✷ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-50

SLIDE 50

❊①❛♠♣❧❡

✸ ✹ ✸ ✷ ✷ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-51

SLIDE 51

❊①❛♠♣❧❡

✸ ✹ ✸ ✷ ✷ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-52

SLIDE 52

❊①❛♠♣❧❡

✸ ✹ ✸ ✷ ✷ ✷ ✸

✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-53

SLIDE 53

❊①❛♠♣❧❡

S✸ ∗t A✹ =

t, r, g | t✸ = r✷ = g✷ = (tr)✷ = (tg)✸ = ✶
P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-54

SLIDE 54

▼♦r❡ t❤❛♥ ♦♥❡ ❱❆P

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| > ✷✳ ❚❤❡♥ A ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ■♥ ❢❛❝t✱ ✐s ♣❡r❢❡❝t ✭♥♦♥❡♠♣t②✱ ❝❧♦s❡❞✱ ✇✐t❤ ♥♦ ✐s♦❧❛t❡❞ ♣♦✐♥ts✮ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-55

SLIDE 55

▼♦r❡ t❤❛♥ ♦♥❡ ❱❆P

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| > ✷✳ ❚❤❡♥ A ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ■♥ ❢❛❝t✱ A ✐s ♣❡r❢❡❝t ✭♥♦♥❡♠♣t②✱ ❝❧♦s❡❞✱ ✇✐t❤ ♥♦ ✐s♦❧❛t❡❞ ♣♦✐♥ts✮ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-56

SLIDE 56

▲♦✇ ❈♦♥♥❡❝t✐✈✐t②

< a, b, t | a✷ = b✷ = (ab)✷ = ✶, t−✶at = b >

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-57

SLIDE 57

▲♦✇ ❈♦♥♥❡❝t✐✈✐t②

< a, b, t | a✷ = b✷ = (ab)✷ = ✶, t−✶at = b >

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-58

SLIDE 58

▲♦✇ ❈♦♥♥❡❝t✐✈✐t②

< a, b, t | a✷ = b✷ = (ab)✷ = ✶, t−✶at = b >

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-59

SLIDE 59

▲♦✇ ❈♦♥♥❡❝t✐✈✐t②

< a, b, t | a✷ = b✷ = (ab)✷ = ✶, t−✶at = b >

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-60

SLIDE 60

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ ✵✱ ✐s ✜♥✐t❡ ■❢ ✶✱ ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-61

SLIDE 61

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ ✶✱ ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-62

SLIDE 62

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-63

SLIDE 63

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-64

SLIDE 64

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-65

SLIDE 65

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-66

SLIDE 66

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞

✷

✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡

✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-67

SLIDE 67

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞

✷

✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡

✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-68

SLIDE 68

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ = ⇒ U = S✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡

✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-69

SLIDE 69

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ = ⇒ U = S✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ U ≈ R✷

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-70

SLIDE 70

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ = ⇒ U = S✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ U ≈ R✷

U

U

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-71

SLIDE 71

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ = ⇒ U = S✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ U ≈ R✷

U

U

U

U

Γ

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-72

SLIDE 72

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ = ⇒ U = S✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ U ≈ R✷

U

U

U

U

Γ U

U

Γ

Γ

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-73

SLIDE 73

❚❤❡♦r❡♠

Γ ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❛ ❣r♦✉♣

G✳ ✐s ❡♠❜❡❞❞❡❞ ✐♥ ✇✐t❤ ♥♦ ❱❆Ps✱ s♦ ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-74

SLIDE 74

❚❤❡♦r❡♠

Γ ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❛ ❣r♦✉♣

G✳

Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥

U ✇✐t❤ ♥♦ ❱❆Ps✱ s♦ G ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-75

SLIDE 75

❊①❛♠♣❧❡

G = r, b, t | · · ·

✷ ✷ ✶ ✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-76

SLIDE 76

❊①❛♠♣❧❡

G = r, b, t | · · ·

✷ ✷ ✶ ✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-77

SLIDE 77

❊①❛♠♣❧❡

G = r, b, t | · · ·

G = r, b, t

✷ ✷ ✶ ✶

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-78

SLIDE 78

❊①❛♠♣❧❡

G = r, b, t | · · ·

G = r, b, t | r✷, b✷, rt−✶rt, bt−✶bt

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-79

SLIDE 79

❊①❛♠♣❧❡

G = r, g, t | · · ·

✷ ✷ ✷ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-80

SLIDE 80

❊①❛♠♣❧❡

G = r, g, t | · · ·

✷ ✷ ✷ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-81

SLIDE 81

❊①❛♠♣❧❡

G = r, g, t | · · ·

G = r, g, t

✷ ✷ ✷ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-82

SLIDE 82

❊①❛♠♣❧❡

G = r, g, t | · · ·

G = r, g, t | r✷, g✷, (rt)✷, (gt)✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-83

SLIDE 83

G = X | · · · Γ ⊆ S✷ ❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥

✷

❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-84

SLIDE 84

G = X | · · · Γ ⊆ S✷ ❚❤❡♦r❡♠

G = X | S✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥

S✷ − (Γ ∪ A) ❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-85

SLIDE 85

G = X | · · · Γ ⊆ S✷ ❚❤❡♦r❡♠

G = X | S✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥

S✷ − (Γ ∪ A) ❚❤❡♦r❡♠ G = X | S ∪ L ✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-86

SLIDE 86

G = X | · · · Γ ⊆ S✷ ❚❤❡♦r❡♠

G = X | S✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥

S✷ − (Γ ∪ A) ❚❤❡♦r❡♠ G = X | S ∪ L✱ ✇❤❡r❡ L ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ L ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-87

SLIDE 87

G = X | · · · Γ ⊆ S✷ ❚❤❡♦r❡♠

G = X | S✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥

S✷ − (Γ ∪ A) ❚❤❡♦r❡♠ G = X | S ∪ L✱ ✇❤❡r❡ L ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ L ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ Γ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ L

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-88

SLIDE 88

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, b, t | r✷, b✷, rt−✶rt, bt−✶bt

✸ ✷ ✷ ✶ ✶ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-89

SLIDE 89

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, b, t | r✷, b✷, rt−✶rt, bt−✶bt

L = {(rb)✸}

✷ ✷ ✶ ✶ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-90

SLIDE 90

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, b, t | r✷, b✷, rt−✶rt, bt−✶bt

L = {(rb)✸} G = a, b, t | r✷, b✷, rt−✶rt, bt−✶bt, (rb)✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-91

SLIDE 91

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, g, t | {r✷, g✷, (rt)✷, (gt)✸

✸ ✷ ✷ ✷ ✸ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-92

SLIDE 92

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, g, t | {r✷, g✷, (rt)✷, (gt)✸

L = {t✸}

✷ ✷ ✷ ✸ ✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-93

SLIDE 93

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞

G = r, g, t | {r✷, g✷, (rt)✷, (gt)✸

L = {t✸} G = r, g, t | r✷, g✷, (rt)✷, (gt)✸, t✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-94

SLIDE 94

❚❤❡ ◗✉❡st✐♦♥✱ r❡❢♦r♠✉❧❛t❡❞

❲❡ ❦♥♦✇ ✇❤❛t G ❝❛♥ ❜❡✳ ❋♦r ❛ ❣✐✈❡♥ ✱ ✇❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤❡ s❡t ❄

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-95

SLIDE 95

❚❤❡ ◗✉❡st✐♦♥✱ r❡❢♦r♠✉❧❛t❡❞

❲❡ ❦♥♦✇ ✇❤❛t G ❝❛♥ ❜❡✳ ❋♦r ❛ ❣✐✈❡♥ G✱ ✇❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤❡ s❡t L❄

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-96

SLIDE 96

❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps

❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐t❤ ✷ ❱❆Ps✳ ❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ ❛s ❛❜♦✈❡✳ ✱ ✇❤❡r❡ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-97

SLIDE 97

❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps

❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐t❤ ✷ ❱❆Ps✳

G ❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ X | S ❛s ❛❜♦✈❡✳

✱ ✇❤❡r❡ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-98

SLIDE 98

❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps

❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S✷ ✇✐t❤ ✷ ❱❆Ps✳

G ❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ X | S ❛s ❛❜♦✈❡✳

L = {ℓ}✱ ✇❤❡r❡ ℓ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-99

SLIDE 99

v

Γ

p

p ❤❛s ❧❛❜❡❧ ℓ❀ ❤❛s ❧❛❜❡❧ ❀ ❤❛s ❧❛❜❡❧ ❚❤❡ ♣❛t❤

✶

✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ ♦r t♦

✶✳ ✳ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-100

SLIDE 100

v

Γ

p v

Γ

p v′ q

p ❤❛s ❧❛❜❡❧ ℓ❀ q ❤❛s ❧❛❜❡❧ w❀ ❤❛s ❧❛❜❡❧ ❚❤❡ ♣❛t❤

✶

✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ ♦r t♦

✶✳ ✳ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-101

SLIDE 101

v

Γ

p v

Γ

p v′ q v

Γ

p v′ q p′

p ❤❛s ❧❛❜❡❧ ℓ❀ q ❤❛s ❧❛❜❡❧ w❀ p′ ❤❛s ❧❛❜❡❧ ℓ ❚❤❡ ♣❛t❤

✶

✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ ♦r t♦

✶✳ ✳ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-102

SLIDE 102

v

Γ

p v

Γ

p v′ q v

Γ

p v′ q p′ v

Γ

p v′ q p′

p ❤❛s ❧❛❜❡❧ ℓ❀ q ❤❛s ❧❛❜❡❧ w❀ p′ ❤❛s ❧❛❜❡❧ ℓ ❚❤❡ ♣❛t❤ q−✶p′q ✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ p ♦r t♦

✶✳ ✳ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-103

SLIDE 103

v

Γ

p v

Γ

p v′ q v

Γ

p v′ q p′ v

Γ

p v′ q p′ v

Γ

p v′ q p′

p ❤❛s ❧❛❜❡❧ ℓ❀ q ❤❛s ❧❛❜❡❧ w❀ p′ ❤❛s ❧❛❜❡❧ ℓ ❚❤❡ ♣❛t❤ q−✶p′q ✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ p ♦r t♦ p−✶✳ ✳ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-104

SLIDE 104

✳ ✳ ✳ s♦ t❤❡ ❣r♦✉♣ ❡❧❡♠❡♥t w−✶ℓw ∈ G ✐s ❡q✉❛❧ ❡✐t❤❡r t♦ ℓ ♦r t♦ ℓ−✶✳ ❙✐♥❝❡ t❤✐s ❤♦❧❞s ❢♦r ❡❛❝❤ ✇♦r❞ ✱ ✇❡ ❤❛✈❡✿ ❚❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ✐♥ ✐s ❡✐t❤❡r ❛❧❧ ♦❢ ✱ ♦r ✐t ❤❛s ✐♥❞❡① t✇♦ ✐♥ ✳ ❤❛s ✐♥✜♥✐t❡ ♦r❞❡r ✐♥ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-105

SLIDE 105

✳ ✳ ✳ s♦ t❤❡ ❣r♦✉♣ ❡❧❡♠❡♥t w−✶ℓw ∈ G ✐s ❡q✉❛❧ ❡✐t❤❡r t♦ ℓ ♦r t♦ ℓ−✶✳ ❙✐♥❝❡ t❤✐s ❤♦❧❞s ❢♦r ❡❛❝❤ ✇♦r❞ w✱ ✇❡ ❤❛✈❡✿ ❚❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ℓ ✐♥ G ✐s ❡✐t❤❡r ❛❧❧ ♦❢ G✱ ♦r ✐t ❤❛s ✐♥❞❡① t✇♦ ✐♥ G✳ ❤❛s ✐♥✜♥✐t❡ ♦r❞❡r ✐♥ ✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-106

SLIDE 106

✳ ✳ ✳ s♦ t❤❡ ❣r♦✉♣ ❡❧❡♠❡♥t w−✶ℓw ∈ G ✐s ❡q✉❛❧ ❡✐t❤❡r t♦ ℓ ♦r t♦ ℓ−✶✳ ❙✐♥❝❡ t❤✐s ❤♦❧❞s ❢♦r ❡❛❝❤ ✇♦r❞ w✱ ✇❡ ❤❛✈❡✿ ❚❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ℓ ✐♥ G ✐s ❡✐t❤❡r ❛❧❧ ♦❢ G✱ ♦r ✐t ❤❛s ✐♥❞❡① t✇♦ ✐♥ G✳ ℓ ❤❛s ✐♥✜♥✐t❡ ♦r❞❡r ✐♥ G✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-107

SLIDE 107

❙♦ G ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣ ✇✐t❤ ❛♥ ❡❧❡♠❡♥t ℓ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r ✇❤♦s❡ ❝❡♥tr❛❧✐③❡r ❤❛s ✐♥❞❡① ✶ ♦r ✷✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❛r❡ t❤❡ ✇❛❧❧♣❛♣❡r ❣r♦✉♣s ♣✶✱ ♣✷✱ ♣♠✱ ♣❣✱ ❝♠✱ ♣♠❣✱ ♣❣❣✱ ♣♠♠✱ ❛♥❞ ❝♠♠✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-108

SLIDE 108

❙♦ G ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣ ✇✐t❤ ❛♥ ❡❧❡♠❡♥t ℓ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r ✇❤♦s❡ ❝❡♥tr❛❧✐③❡r ❤❛s ✐♥❞❡① ✶ ♦r ✷✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜✐❧✐t✐❡s ❢♦r G ❛r❡ t❤❡ ✇❛❧❧♣❛♣❡r ❣r♦✉♣s ♣✶✱ ♣✷✱ ♣♠✱ ♣❣✱ ❝♠✱ ♣♠❣✱ ♣❣❣✱ ♣♠♠✱ ❛♥❞ ❝♠♠✳

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-109

SLIDE 109

❊①❛♠♣❧❡

♣♠❣ = b, r, g | b✷ = r✷ = g✷ = ✶, rgr = bgb

✹ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-110

SLIDE 110

❊①❛♠♣❧❡

♣♠❣ = b, r, g | b✷ = r✷ = g✷ = ✶, rgr = bgb ℓ = (bg)✹

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-111

SLIDE 111

❊①❛♠♣❧❡

♣♠❣ = b, r, g | b✷ = r✷ = g✷ = ✶, rgr = bgb ℓ = (bg)✹

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-112

SLIDE 112

❊①❛♠♣❧❡

♣♠❣ = b, r, g | b✷ = r✷ = g✷ = ✶, rgr = bgb ℓ = (bg)✹

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-113

SLIDE 113

❊①❛♠♣❧❡

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-114

SLIDE 114

❊①❛♠♣❧❡

♣❣ = b, r | b✷ = r✷

✶ ✸ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-115

SLIDE 115

❊①❛♠♣❧❡

♣❣ = b, r | b✷ = r✷ ℓ = (b−✶r)✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-116

SLIDE 116

❊①❛♠♣❧❡

♣❣ = b, r | b✷ = r✷ ℓ = (b−✶r)✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-117

SLIDE 117

❊①❛♠♣❧❡

♣❣ = b, r | b✷ = r✷ ℓ = (b−✶r)✸

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-118

SLIDE 118

❊①❛♠♣❧❡

P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

slide-119

SLIDE 119

♣✶✿ a, b | [a, b] = ✶✱ anbm = ✶ ❢♦r s♦♠❡ (m, n) = (✵, ✵) ♣✷✿ p, q, r, u | p✷ = q✷ = r✷ = u✷ = pqru = ✶✱ W = ✶ ✇❤❡r❡ W ✐s ❛♥② ✇♦r❞ ✐♥ p✱ q✱ r ❛♥❞ u ♦❢ ❡✈❡♥ ❧❡♥❣t❤✳ ♦r✱ ❢♦r s♦♠❡ ✐♥t❡❣❡r n = ✵✿ ♣♠✿

b, t, u | t✷ = u✷ = ✶✱ bt = tb✱ bu = ub✱ bn = ✶

b, t, u | t✷ = u✷ = ✶✱ bt = tb✱ bu = ub✱ (tu)n = ✶ ♣❣✿

p, q | p✷ = q✷✱ p✷n = ✶

p, q | p✷ = q✷✱ (p−✶q)n = ✶ ❝♠✿

p, t | t✷ = ✶✱ tp✷ = p✷t✱ p✷n = ✶

p, t | t✷ = ✶✱ tp✷ = p✷t✱ (p−✶tpt)n = ✶ ♣♠❣✿

t, r, s | t✷ = r✷ = s✷ = ✶✱ rtr = sts✱ (rs)n = ✶

t, r, s | t✷ = r✷ = s✷ = ✶✱ rtr = sts✱ (rt)✷n = ✶ ♣❣❣✿ p, r | (pr)✷ = (p−✶r)✷ = ✶✱ p✷n = ✶ ♣♠♠✿ a, b, c, d | a✷ = b✷ = c✷ = d✷ = (ac)✷ = (ad)✷ = (bc)✷ = (bd)✷ = ✶✱ (ab)n = ✶ ❝♠♠✿ a, c, t | a✷ = c✷ = t✷ = (ac)✷ = (atct)✷ = ✶✱ (at)✷n = ✶✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s