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1. S {2 {3} 4} R {{} 3 4 1} {} S,{} R,{,4,{3}} S,{{},1,3,4} R,RS,{} S,{} R,φ R,φ {{}} R E,{φ} S,φ R, φ {{3},4} {} S {} R { 4 {3}} S {{} 1 3 4 } R R S {} S {} R φ R φ {{}} R E {φ} S φ φ {{3} 4 } 2 { {}} {1 φ} {X Y Z} A{ {}} ρa{ φ {} {{}} { {}}} B{1 φ} ρb { φ {1} {φ} {1 φ}} C{X Y Z} ρc { φ {X} {Y} {Z} {X Y } {X Z } { Y Z } {X Y Z}} 3 A B 1A B ρa ρb 2ρA ρb ρa B 3ρA ρbρa B 4ρA-B ρa-ρb {φ} ρa ρb ρ A B 1 x ρa x A A B x B x ρb ρa ρb x A {x} A {x} ρa ρa ρb {x} ρb x B A B 2 X ρa ρb X ρa X ρb X A X B X A B X ρa B ρa ρb ρ A B 3 ρa ρb ρ A B X ρa ρb X ρa X ρb X A X B X A B X ρ A B ρa ρb ρ A B ρ A B ρa ρb 1

Y ρa B Y A B Y A Y B Y ρa Y ρb Y ρa ρb ρ A B ρa ρb ρa ρb ρ A B A{1} B{} ρa{ φ {1}} ρb{ φ {}} ρa ρb { φ {1} {}} ρ A B{ φ {1} {} {1 }} {1 } ρ A B {1 } ρa ρb ρa ρb ρ A B 4 x xφ x ρ A B x ρ A -ρ B φ} x φ x ρ A B x A B x A B x ρ A ρ B x ρa-ρb ρa-b ρa-ρb {φ} 4. A,B,C 1 A B B C A C 2 A B B C A C 3 A B B C A C 4 A B B C A C 1 2 3 4 5. 24 13 5 10 9 2 4 A B C D A 13 B 5 C 10 D 9 A B 2 A C A D C D 4 B C B D A B C A B D B C D A B C D 0 A B C D 24 A B C D 2437-14 A C D A C D 1 1 x 1 x 2 x 3 x 4 x 1 A - B C D A A - B A C A A D B A C A A D x 1 4 2

x 2 3 x 3 3 x 4 2 6. A B A B A A B A 2. A m B n A B A{ } B{1 2} A B A B 2 mn A B φ {,1} {,2} {,1} {,2} {,1,,2} {,1,,1} {,1,,2} {,1,,2} {,2,,1} {,2,,2} {,1,,2,,1} {,1,,2,,2} {,1,,1,,2} {,2,,1,,2} {,1,,2,,1,,2} A B 7.R S A 1 RS S R R R R S R S R S R S RS S R x y RS y x RS A y R x S x S y R x y S R RS S R S R RS x y S R A x S y R y R x S y x RS x y RS S R RS R R x y R y x R x y R R R R R x y R y x R x y R R R R R R S R S x y R S y x R S y x R y x S x y R x y S x y R S R S R S R S R S x y R S x y R x y S y x R y x S y x R S x y R S R S R S R S R S R S R S x y R S y x R S y x R y x S x y R x y S x y R S R S R S R S R S x y R S x y R x y S y x R y x S y x R S x y R S R S R S R S R S 8. R A R {x, y x A y A n>0 xr n y} R R R R 3

1 R R x y R x R y n1 x R n y x y R R R 2 R x y z A xr y yr z m nm>0 n>0 xr m y yr n z xr m R n z xr mn z xr z R 1.2.4 x y R R A x R y R m nm>0 n>0 x R m R n y xr m R n y x R mn y xr y R R R R 3 R P P R P R P x y R nn>0 x R n y n1 x R y x y R x y P n>1 1 2 n-2 n-1 x R 1 1 R 2 2 R 3 n-2 R n-1 n-1 R y P R x P 1 1 P 2 2 P 3 n-2 P n-1 n-1 P y P x P y x y P R P Tn R R 2 R 3 R 4 R n>0 R R T P P R P T P T1 R P nk Tk R R 2 R 3 R 4 R P k1 Tk1 R R 2 R 3 R 4 R R Tk R Tk P R P x y R A x R, y R R Tk P R P x P, y P P x y P R P Tk1 P T P R P 9. R R R R R R R R 10. A R R A 8. A R R R R R R R R R R R R R R R R 11. R A 1 A R 2 R Rc Rc R 4

A R R A R R 2 R R c A R R c R R Rc 2 R c R R 12. A A R R R A R R S RS RS S R RS SR RS x y A x y RS A x R y S R S y S x R y x SR SR RS y x RS RS RS RS SR RS SR x y RS RS y x RS A y R x S R S x S y R x y SR RS SR SR RS x y SR A x S y R R S y R x S y x RS RS x y RS SR RS RS SR 14. R R R R R R R ρ A B σ B C τ C D A τσρ τσρ τσ ρ τσρ τσρ τσ ρ τσρ τσρ σ c c τ c c τσ τσ τ 5

στ σ τ σ c τσ στ σ M N M A B σ A B σ A σ B σ A B σ A σ B y σa B N y x σxy y σ A B x A B x A x B y σa y σb y σ A σ B σ A B σ A σ B M{1 2 3 4} A{1 2 3} B{2 3 4} σ 1 2 3 c 4 σ A B σ {2 3}{ c} σ A σ B { c} { c}{ c} 17. σ M N A N M σ A A σ A A B N σ A B σ A σ B σ A B σ A σ B x σ A B σx A B σx A σx B x σ A x σ B x σ A σ B σ A B σ A σ B σ A σ B σ A B x σ A σ B x σ A x σ B σx A σx B σx A B x σ A B σ A σ B σ A B σ A B σ A σ B 1. P Q R 1 c d P R Q Q R c P d P Q 2 Q R P 6

R Q c Q R R Q d R Q c d 2. 1 2 3 n 2 x n y n z n 4 1 2 3 x n y n z n n 2 x n y n z n n 2 4 3. P Q 1 R S 0 P Q R P Q R S P Q R P Q R S c P Q R Q P R S d P Q R P Q S G P Q R P Q R S T I G 1 1 0 1 1 0 0 0 01 G P Q R P Q R S T I G 1 1 0 1 1 0 0 1 01 c G P Q R Q P R S P Q R Q P P Q R S P Q R Q P P Q R S T I G 1 1 0 1 1 1 1 0 0 1 11 d G P Q R P Q S P Q R P Q S P Q R P Q S P Q R P Q S Q S P Q R P P Q R P Q S Q S P Q R P T I G 1 1 0 1 1 0 1 0 7

1 1 0 1 1 11 4. 1P P Q Q 2 P Q P Q 3 P Q Q R P R 4 P Q P Q P Q 1 2 3 4 5. P Q P Q P Q P Q P Q P Q I I P Q I P P Q I P P P Q I P Q Q P Q P Q P Q I P Q I P Q P Q P Q P Q P Q P Q P Q P Q 6. P Q R P Q R P P Q P c Q P P Q d P Q P Q 1 GP Q R P Q R P Q R P Q R P Q R P P Q R Q R P P P Q R P Q R Q R P P P Q R P Q P R Q R P P P Q R P Q Q P R R P Q R P Q R P Q R G P Q R G 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 1 G 2 G P P Q P P P Q P P P T P Q G 0 0 1 8

0 1 1 1 0 1 1 1 1 G 3 GQ P P Q Q P P Q P Q P Q F P Q G 0 0 0 0 1 0 1 0 0 1 1 0 G 4 G P Q P Q P Q P Q Q P P Q P Q Q P P Q P Q P Q P Q G 0 0 0 0 1 1 1 0 1 1 1 1 G 7. 1 P Q R Q R P RR 2 P Q P P P Q 3 P Q RP Q P R 4 P Q R QP R Q 1 P Q R Q R P R P Q R Q P Q R R P R P Q R Q P R R P R P Q R Q R P R P Q R P R P R Q R P R R P P Q R R 2 P Q P P Q P 9

P Q P P P Q T P P Q P P Q T 3 P Q R P Q P R P Q R 4 P Q R Q P R Q P R Q P R Q 8 S{G 1 G n } S G 1 G n G S G G G 1 G n G G G m G 1 m 1 S m n0 n m 2 m 1 H 1 S>H G>H I GG 1 G n 1 G 1 1 H 1 S>H 2 H S H H G G H m k G m k 0 I I H 0 I m k 1 G G I 1 G I 1 G>H 9. P, P P P>P P Q P>Q Q>P P Q Q P P Q Q P P Q PQ c P Q R P>Q Q>R P Q R I I P I Q 10

I Q I R I P R P>R 10. S G S G S G 1 G n G 1 G n G 1 G n G 1 G n >G G 11. G 1 G n {G 1 G n } G {G 1 G n G} R R R {G 1 G n G}>R R G 1 G n G>R R R R G 1 G n G G 1 G n G G 1 G n G G 1 G n >G {G 1 G n } G {G 1 G n }>G G 1 G n G G 1 G n G G 1 G n G G 1 G n G R R R G 1 G n G > R R {G 1 G n G} R R 12. 1P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q Q T P Q P Q P Q Q Q P Q 2P Q P P Q P P Q P P P Q T P Q P 3 P Q R P Q P R P Q R P Q P R P Q R P Q R T 4 P Q P P Q 11

: P P Q P Q P Q P Q 5 P Q Q P Q P Q Q P Q Q P Q Q P Q Q Q >P Q 6 P P Q P P R Q R Q R R Q P P R P P Q P P Q P P R P Q 7 Q P P R P P R Q R Q Q R R P P Q P P Q P P R P P 13. {C D C D H H A B A B R S} R S 1 C D 1 2 C D H 1 3 H 2 1,2 4 H A B 1 5 A B 2 3,4 6 A B R S 1 7 R S 2 5,6 14. {P Q Q R P M M} R P Q 1 P M 1 2 M P 2 1 3 M 1 4 P 2 2,3 5 P Q 1 6 P Q 2 5 7 Q 2 4,6 8 Q R 1 9 R 2 7,8 10 R P Q 2 5,9 15. { P Q Q R R S} P S 1 P Q 1 2 P 3 3Q 2 1,2 4 Q R 1 5R 2 3,4 6 R S 1 12

7S 2 5,6 8 P S 3 2,7 16. P Q P Q R P Q P R P Q P Q R P Q P R P Q P Q R P Q P R P Q P Q P R P Q P R P Q R P Q R T 17. 6 P 1 P n n n P 1 P n P 1 P n 7 G P 1 P n G G P 1 P n G G 1 G G, G P 1 P n G G i G i P 1 P n G i P j1, P jk G i G i G i P j1 P j1 P jk P jk M i1 M i2 k G G i G G P 1 P n G H G H P 1 P n G H M i G H I M i 0 G 0 I M i 1 I H 1 G H 18. G G G G 1 G 2 G n G G G i i1,2, n G i G i I 13

1 0 G i 1 G i G i I 1 0 G i I 0 G i G 19. P P Q P Q P Q P P Q P P Q P P P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P P Q Q P Q P Q 20. 1 P P Q Q P 2 P P Q Q R 1 P P Q Q P P P Q Q P P Q P P Q Q Q P P Q P Q Q P P Q P P Q P P P Q 2 P P Q Q R P P Q Q R P P Q R P Q R P Q Q R R Q P P R R R P P Q Q P Q R P Q R P Q R P Q R Q P R Q P R Q P R Q P R R P Q R P Q R P Q R P Q P Q R P Q R P Q R P Q R Q P R Q P R R P Q 14

P Q R P Q R P Q R P Q R P Q R P Q R P Q R 1. { c} 1 xrx xsx 2 xpx Qx 3 x Px xpx 1 xrx xsx R R Rc S S Sc 2 xpx Qx P Q P Q Pc Qc 3 x Px xpx P P P P P P 2 1 xpx xqx xpx Qy 2 x ypx Qy zrz 3 Az x ybx y 4 x Ax ybx y 5 xfx ygx y z zhx y z 1 Qy y Px Qx Px x x Px 2 x Qx x Px 2 x y z x y Px Qy zrz z Rz 3 z x y x y Bx y 4 x x y x Ax y Bx y 5 x y z x y z x Fx y Gx y z z Hx y z 3. I D{ } 15

P P P P 1 0 0 1 I 1 x ypx y 1 2 x ypx y 0 3 x ypx y 0 4 y P y 1 5 x ypx y Py x 1 6 xpx x 1 4. G xp x xpx 1 I D G I 1 G P P G I 1 2 D{ } D I G I 0 P P 1 0 5. I D{3 2} f3 f2 P3 3 P3 2 P2 3 P2 2 3 2 2 3 1 1 0 0 I 1 f P f 0 2 x ypy x 1 3 x ypx y Pfx fy 0 6. G 1 xpx Qx G 2 Q P G 1 G 2 G G 1 G 2 I G P I G P I P G 2 Q P Q G 1 G I G 7. 1 xpx yqx y x Px y Qx y x y Px Qx y 2 x ypx y zqz Rx x ypx y zqz Rx x ypx y zqz Rx x ypx y z Qz Rx x y Px y z Qz Rx x y z Px y Qz Rx 16

3 x y zpx y z uqx u vqy v x y z Px y z uqx u vqy v x y z Px y z u Qx u vqy v x y z Px y z u v Qx u Qy v x y z u vpx y z Qx u Qy v 8. Skolem 1 xpx y zqy z xpx y zqy z xpx y zqy z xpx y zqy z x y Px z Qy z x y z Px Qy z fx y z Skolem : x ypx Qy fx y 2 x Ex 0 yey gx zez gx Ey z x Ex 0 yey gx z Ez gx Ey z xex 0 yey gx z Ez gx Ey z x y z Ex 0 Ey gx Ez gx Ey z x y z Ex 0 Ey gx Ex 0 Ez gx Ey z fx y Skolem : Skolem : x zex 0 E fx gx Ex 0 Ez gx E fx z 9. x ymx y G Mx y x y f Mx y G xmx fx G x ymx y G 1 xmx fx I G 1, I x D fx D Mx fx I G I G G 1 G I G x 0 y 0 D 2 Mx 0 y 0 I I fx fx 0 y 0 I G 1 1. GP L PG LG m n n C 2 2 m C m m 2 GP,L u v u v, uv m PG LG C m2 GP,L PG 17

m LG n C m 2 C m 2 m 2 2. G M A G MM A 2 G PG m LG n M m n M n m 1 BMM m m d ii i ' j ji ; i1,2, m j 1 M M ij ji ij 0 1 d ii j n 1 j n i d ii M i 1 d ii v i i1,2,,n n n ' ji 2 ij j 1 j 1 ij 2 CA 2 C C ii j n 1 ij ji ; i1,2, m A mxm ij ji ij 0 1 C ii i v i i1,2,,n n j 1 n 2 ij j 1 C ii A i 1 ij 3. G PG LG m n δ G δ 2n/m mδ d G v m 1 v P G d G v 2n mδ 2n m v P G m>0 δ 2n/m 2 4. GP L PG LG m n n> C m 1 G G G G 1 G 2 PG 1 m 1 PG 2 m 2 1 m 1 m 2 <m m 1 m 2 m n 2 2 C C m 1 m 1-1/2 m 2 m 2-1/2 m1 m 2 1/2m 2 1 m 2 2 - m 1 - m 2 1/2m 2 1 m-m 1 2 - m 1/2m 2 -m-2mm 1 2m 1 1/2m 2 -m-2m 1 m-m 1 1/2m 2 -m-2m 1 m 2 m 1-1 m 2-1 0 m 1 m 2 - m 1 m 2 1 0 m 1 m 2 m-1 n 1/2m 2 -m-2m 1 m 2 1/2m 2 -m-2m-1 1/2m 2-3m2 18

1/2m-1m-2 C 2 m 1 2 n> C m 1 5. GP L l 1 v 1, v 2, v n l 2 v 1, v 2, v m l 1 l 2 l 1 v 1 l 2 v 1 G l 3 : x 1, x 2,,x k x 1 v 1 x k v 1 x j l 3 l 2 v h l 4 x 1, x 2,,x j x i l 4 l 1 v g l 5 x i, x i1,,x j l 5 1 l 5 x i x j l 1 l 2 v 1, v 2, x i x i, v g1, v n, v 1, v 2, x j x j, v h1, v m, l 6 v 1, v 2, x i, x i1,,x j, v h-1, v 1, l 5 1 l 6 >min{ l 1, l 2 }, 6. c 1 c 2 c 6 c i c j i j 0 50 40 25 10 50 0 15 20 25 15 0 10 20 40 20 10 0 10 25 25 20 10 0 55 10 25 25 55 0 C 1 C 2,,C 6 LC 1 lc 2 lc 3 lc 4 lc 5 lc 6 S 0 {C 1 } 50 40 25 10 {C 1, C 6 } C 1 C 6 35 35 25 {C 1, C 6, C 5 } C 1 C 5 35 45 35 {C 1, C 6, C 5, C 2, C 4 } C 1 C 6 C 2 C 1 C 6 C 4 C 1 C 5 C 4 45 {C 1, C 6, C 5, C 2, C 4, C 3 } C 1 C 5 C 3 C 1 C 6 C 4 C 3 19

C 1 C 2 C 3 C 4 C 5 C 6 C 35 45 35 25 10 0 C 1 C 6 C 2 C 1 C 5 C 3 C 1 C 5 C 4 C 1 C 5 C 1 C 6 C 1 C 6 C 1 C 6 C 4 C 4 C 3 C 1 C 5 C 4 C 3 C 2 35 0 15 20 30 25 C 2 C 6 C 1 C 2 C 3 C 2 C 4 C 2 C 4 C 5 C 2 C 6 C 3 45 15 0 10 20 35 C 3 C 4 C 6 C 3 C 2 C 3 C 4 C 3 C 4 C 5 C 3 C 4 C 6 C 1 C 3 C 5 C 3 C 5 C 1 C 3 C 4 C 5 C 1 C 4 35 25 10 0 10 25 C 4 C 5 C 1 C 4 C 2 C 4 C 3 C 4 C 5 C 4 C 6 C 4 C 6 C 1 C 5 25 30 20 10 0 35 C 5 C 1 C 5 C 4 C 2 C 5 C 4 C 3 C 5 C 4 C 5 C 4 C 6 C 5 C 3 C 5 C 1 C 6 C 6 10 25 35 25 35 0 C 6 C 1 C 6 C 2 C 6 C 4 C 3 C 6 C 4 C 6 C 4 C 5 C 6 C 1 C 5 7. G G G 1 G 2 G G u v G u v u v G uv G G u v G uv u v G G 8. G G 1 2 G G G 1 2 PG n k3 kn-1 G 1 2 20

k3 kn-1 G kn v 1 G 1 v 2 v 1 v 1 v 1 v 2 G G v 2 1 0 G 2 G v 2 3 G v 1 v 1 v 2 G G 9. 10. G e 1 e n fine n inite 1 G e 1 e n fine n inite 1 e 1 e n G e 1 e n k kmin{i i n fine i inite j 1 i,j e 1 e n G G 11. G G 1 2 G G G 1 G 2 G G r r 12. n n-2 n n 4 n nn 3 n G 21

G Hmilton u, v G u,v v k u,v u v v, v k u n-3 v k u u v v k v dudv2n-2nn-4 n-1 u,v G dudv n-22>n-1 G dudv n-1 3 G Hmilton n n 4 u,v dudv2n-2nn-4 n u,v dudv n-22n dudv n CG G Hmilton 13 G n-1 n PG Hmilton G G G 1, G 2 G n 1, n 2 n 1 n 2 <n v 1, v 2 PG 1, PG 2 d G1 v 1 n 1-1 d G2 v 2 n 2-1 d G1 v 1 d G2 v 2 n 1-1n 2-1 n 1 n 2-2 n-2<n-1 G Hmilton Lv 1, v 2,, v l G l-1 L 1 ln L G Hmilton 2 l<n G v 1, v 2,, v l i v 1 v l v 1, v 2,, v l G ii v 1 v l v 1 v i1 v 2,v i2,,v ik v i1,v i2,,v ik L k 2 k1 v 1 v l dv l l-2 dv 1 dv l 1l-2l-1<n-1 v l v i1-1,v i2-1,, v ik-1 v l l-1-1-k-1l-k-1 dv 1 dv l kl-k-1l-1<n-1 v l v ij-1 2 j k v ij-1, v ij v 1, v ij,,v l, v ij-1,,v 2, v 1 v k v 1 v 2 v x v ij-1 v ij v l 3 l<n G G v x v k v k-1, v k v k-1, v k-2,,v 1, v ij,,v l-1, v l, v ij-1,, v k, v x 1 2 3 L G Hmilton 14. 4.4.10 Hmilton 1 2 6 5 22

7 0 8 9 3 4 4.4.10 G Hmilton T{16,27,38,49,50} G-T G Hmilton T G T Hmilton G Hmilton T 4 C 27 38 49 50 16 Hmilton C 12 15 68 69 23 45 3 4 C 34 C {34,49,96,68,83} C G Hmilton 15. G 6 12 G Hmilton n Hmilton 6 12 G Hmilton G 1/2n-1n-22,G Hmilton n G Hmilton 1/2n-1n-21 6 11 Hmilton G PGn LGm m 1/2n-1n-22 Hmilton n Hmilton 1/2n-1n-21 Hmilton 4.4.5 m,n m,n m,n Hmilton n Hmilton m,n m,n 1,n n fmc 2 n-2m C 2 m C 1 n-2m C 1 m C 1 1 m C m C v u u v fm m,n fm1/2n-2mn-2m-11/2mm-1n-2mmm 2 3/2m 2 -n-1/2m1/2n 2 -n f m3m-n-1/2 1 m 1/3n-1/2 fm 1/3n-1/2 m n-1/2 f1 fn-1/2 fm [ n-1/2] m n m,n 1,n f11/2n-1n-21 n Hmilton 1/2n-1n-21 16. 23

6-1-1 X X 6-1-2 Y 6-1-3 2 2 c d 6-1-1 c d 6-1-2 d c c d d c c d 6-1-3 17 2n2 n 0 6-2-1 2n1 2n3 2n x y z 1 5 3 6 4 4 2 7 5 7 5 8 6 6 4 9 7 9 7 10 8 8 6 11 9 1 2 0 3 3 6-2-1 4 2 5 5 6 4 7 1. W 1 W 2 W 3 6 Z 6 W 1 { 0 3 } W 2 { 0 2 24

4 } W 3 {1 3 5 } W 1 W 2 W 3 2. S{2 n n N} S S 3. N * x * y x y * 4. * S x y S x * y y * x x y * S x y z x*y*zx*y*z x*x*xx*x*x x*xx 5. * S S x y x y x * x y z S x*yzx*yx*yx*z yz*xy*xy*xz*x * 6. G G G * G G * * c d c d c d G G G G * G G G * G G G * G c d e f G,c d e f G G *c d*e f *c e d f c e d f c e d f c d*e f *c d*e f G G G * G 1 G G * 1 1 *1 1 1 1 1 1* 1 1 G G * G 1 G 1 1 G G * 1 G G * G G * 1 1 * G 1 1 * G G * 7. y y y 25

y G G G{ 1 2 n } G 1 2 n i j i j i j G i i x y G 8. 6.2.2 1 2 1 G 1 1 2-1 -1 1 G G 0 0 1 0-1 0 1 G 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 G 1 9. G{ c} c c c c c c c G G G G G G G c 10. 1 2 3 2 3 4 1 4 2 3 1 2 3 2 3 4 1 4 2 3 1 3 2 4 11. 1 2 n M M 1 2 1 3 1 n 1 2 2 3 n-1 n 26

M r s 1 r 1 s 1 r r s 1 1 r 1 22 3 r-2 r-1 r-1 r r-2 r-1 2 31 2 12. σ τ τ στσ -1 σ τ σ 1 2 3 τ 1 2 3 4 στσ -1 2 3 1 4 σ τ 1 2 2 3 3 1 στσ -1 τ 11 12 1r 21 2s m1 mt 11 1r 21 2s m1 mt σ 11 1r 21 2s m1 mt jk jk 1 σ jk τ στσ -1 jk jk1 σ jk 1 jk 1 12. n n n n n A 4 1 1 2 3 1 3 2 1 2 4 1 4 2 1 3 4 1 4 3 2 3 4 2 4 3 1 2 3 4 1 3 2 4 1 4 2 3 1 n! 2 14. 1 1 2 3 4 1 3 2 4 1 2 2 3 klein klein H Klein σ S 4 τ H στσ 1 σ τ H στσ 1 H σhσ 1 H 15. 1 1 2 1 3 2 3 1 2 3 1 3 2 1 1 2 3 1 3 2 1 1 2 1 1 3 1 2 3 1 16. G G G G G H 1 H H H -1-1 H H H g G h H h ghg -1 ghg -1 H ghg -1 H 27

17. H G N G HN H N HN G 1 HN NH HN h 1 n 1 h 2 n 2-1 h 1 n 1 h 2 n 2-1 h 1 n 1 n -1-1 2 h 2 h 1 n 1 h 2 n 2 h 1 h 2 n 1 n 2 HN HN G 18. H G H H H G G H G G H H H H G 19. G H G H m>0 m m<0 m -m H -m>0 m H H m H S Stmr 0 r<m r S-tm S m -t H m 0 r<m r0 S m t H S m H m 20. G n r n d r G r n e G e e r m m r m 1 n d m e rm e n rm n r n r n m d 1 m 2 d d d d 1 2 m d n r d n G n r d1 r n1 ϕ n n d rn d r d r d 21. G G G P G G r r r P P r 1, rp G G 22. 1 G G 2-1 2 G 2 G Ael 1 1 G 1 2-1 1-1 2-1 -1 2 2 1-1 G G 2 G 28

-1-1 -1 G Ael 23. G 6 G G G{1, 2 3 4 5 } 9 5 G 8 G G {1} < 3 >{1 3 } < 2 >{1 2 4 } G 24. K H G H K G K H H K k h K H k h -1 h -1 k -1 H K H K G k h -1-1 H K K H H K h k H K h k -1 H K h 1 H k 1 K h k -1 h 1 k 1 h kh 1 k 1-1 k 1 h 1 K H H K K H K H H K 25. σ G G τ G G τσ G G τσ σ -1 τ -1 1 1 G τσ G G G G ττστσστστστστσ τσ τσ G G 1 τσ G σ -1 τ -1 1 τσσ -1 τ -1 1 I1 1 τσ σ -1 τ -1 1 26. σ G G H σ N G H σ H H G G H G H ' ' H G g G g G σgg ghg -1 H σ ghg -1 σgσhσg -1 σhh g H g -1 H H G ' G G' H G τ G ~ ' H 1 H H ' ' G σ G G τσ G H ' σ -1 τ -1 1 σ -1 H H N σ -1 H H G G' τσ H. H H ' G G / N 27. H N G H N H H / N N G σ G G/N σ H G/N G/N G N N 1,N 2,, i Gi1,2,, H G/N N H i, H G/N 29

G / N H/N H/N G/N G/H H / N HN H 28. N G H G N H N N G σ G G/N G σ H H G σ -1 H σ -1 σhhn σ HN H HN σ H σ HN H N H HN/N σ σ H H H~H N H H N H N H H/H N HN/N H/N N 29. G 6.3 G G -1 G 30. G G, x x -1 x G G G G G 1-1 g 1 g 2 G g 1 g 2 g 1 g 2-1 g 1-1 g 2-1 g 1 g 2 G 31. G 1 x y g G y x g y xgx -1 y xgx -1 y 1 yx gyx -1 yx g 1-1 x x-1 32. G N G G/N 33. f x x -1 G G x G 30

34. p m p p m 1 p m p m k 1 pk p m p p m 1 35. xy, x-y0 0 x-y0 xy 36. R 1 R 1 2 3 R 0 37. L R 1 R 0 2 R 31

38. R 39. K K R R 0 K σ σ σ R K σ R K, σσ C- 0, σ C 0 C K C 1 CC 1 1 σ1σcc -1 σcσc -1 σc -1 0 x K σxσx 1σx σ10 0 R 0 K 40. I/12I { 0,1, 2,3, 4,5, 6, 7,8,9,10, 11 }; { 0,2,4,6,8, 10 }; { 0,3,6, 9 };{ 0,4, 8 }; { 0, 6 };{ 0 } 41. F Fx,y 0 0 N 1 f,g N, f-g 0 f-g N 2 f N g Fx,y g f 0 g f N N N Nfx,yFx,y x N y N r,s Fx,y xfx,y r yfx,y S 32

r1 rx fx,yx fx,y1 y fx,y1 Nfx,y Fx,yFx,y fx,yx yfx,y S yx S S x S Fx,y N 42. A B N R P A, B P AB P AB I 5I I/5I{ 0,1, 2,3, 4 } A 0 {0 5 10 } B2{2 7 12 } 5 AB P AB 43. σ R R N R σ -1 N R, σ -1 N σ N,σ N [σ-σ] N σ-[σ-σ] N - σ -1 N σ-1n x R σxσ σx σ N,σx R, σσx N x σ -1 N x σ -1 N σ -1 N R 44. A B R 0 A 0 B 0 A B A B x y A B x y A A R x-y A x-y B x-y A B x A B r R x A r R A R xr A rx A xr B rx B xr A B rx A B 1. S S S S A B A BA A BB. 2. L, L S{x x L x } L,, S, L c d S, c,d L c,d c*d c*d c d c d S * 33

S * L 3. D S 1 S {1 2 3 4 6 12} 2 S {1 2 3 4 6 8 10 12} 3 S {1 2 3 4 5 6 7 8 9 10} 4 S {2 3 6 12 24 36} 1 4 4. L,,c L, c *c* *c * c c *c *,*c,, cc * *c * c*c * *c * c. 5 L,,c d,,,c,d L, c *d c d *c**d*c**d**c*d*c *c *d, c d c d c d d, c d. 6 L * c*d c* d; * *c c* * c*c * c*d [* c]*[* d]] c* c d* d [ c* d]*[ d* c] c* d * *c c* [* * c c* [** c] c* [** c c]*[** c ] [ c** c c]]* *[* c ] [ c* * c* *[* c] c* * c* c * c*c 7. L L L L{1,2,,n} L *, 1*2* *nl 1 2 nl i L i L, 8. S{ } I D S D n I I S g:n 2n * n 1,n 2 I,n 1 *n 2 d. 2n 1,2n 2 S,2n 1 *2n 2 2n 1 2n 2 34

2d gn 1 *n 2 gd2d2n 1 *2n 2 gn 1 *gn 2. gn 1 n 2 gn 1 gn 2 g I D S D 9. 4 L * I 4 S 6 D 4 L{ 1, 2, 3, 4 } L 4, 1, 1 2, 3 4 2, 3 1 2 3 2, 3 I 4 2 2, 3 S 6 D. 10. E O E O E D O D D E O g:m m-1 m E * m 1,m 2 E m 1 m 2 gm 1 *m 2 gm 1 m 1-1 m 1-1*m 2-1gm 1 *gm 2 gm 1 m 2 gm 2 m 2-1m 1-1 m 2-1gm 1 gm 2 E O E D O D 6 5 12 11 E D O D * 1 * 2 g6* 1 12g65 g6* 2 g125* 2 111 g E D O D 11. L * S g L S g g g-1 g L S g-1 S L, L,, S g g,*c, c c,c g gc,g c gc gc c c c c g-1 g-1 c,g-1c gg-1 c gc c g c c g c c gc g*gcc g g g g g. g 12. L * g L L, L g*g*g e L, gee g*g*g m g1*2* *mg1*g2* *gm L 1,g12, 21, e1., 3g2 : j gjj1 j n, e j gji 1 i j., ei*i1* *j gegi*i1* *j 35

gi*gi1* *gj i1*i2* *j*i e 13 L L 0 1 0 0 0 1 11 1 0 14 L c c c c c, c c L c c c c c c c c c c c,c c c c. 15. B - 0 1 B B, 36

37 0 000 0 0 0 0 100 1 1 1 1 0 10 c c c c c c c c 0 0 0 0 10 0 00 c 0 0

38 0, 0 c c c0 c c c c c 0 0,. 16. 1 2 0 3 1 4 0 17. 1 2 3 4 5 1 D S S {1 2 3 4 6 8 10 12} S {1 2 3 4 5 6 7 8 9 10} 2 x 1 x 2 1 1 1 1 x 1 x 2 1 2

1 2 3 0 3 0 0 0 c d 3 4 c 39