A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x A (1)

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1 A µ : A A A µx, y x y x y z x y z A x, y, z x y y x A x, y A e x e e x x A x e A e x A xy yx e y x x x y y x A 1 A R x y xy + x + y R x, y, z, : xyz xy+x+yz xy+x+yz+xy+x+y+z xyz+y+z+x+yz+y+z xyz+y+z xyz : xy xy+x+y yx+y+x yx : 0 x : x+1 x 1 2 A N x y x y : : N x, y, z, gcdx, y, z : d, gcdx, y : d 1, gcdd 1, z : d 2, ie k 1, k 2, l 1, l 2 Z st d 1 k 1 x + k 2 y, d 2 l 1 d 1 + l 2 z d 1

2 1 8 d 2 l 1 k 1 x + l 1 k 2 y + l 2 z, ie d 2 d, m 1, m 2, m 3, Z st d m 1 x + m 2 y + m 3 z, m, Z st md 1 m 1 x + m 2 y md 1 d md 1 + m 3 z, ie d d 2, d d 2 gcdx, y, z gcdx, y, z, gcdx, y, z gcdx, y, z : gcda,ba b, gcdc,b1 c, : 3 A N 2 a, b, c, d A a, b c, d ad + bc, bd A a, b, c, d, e, f, : a, b c, d e, f ad + bc, bd e, f ad + bcf + bde, bdf adf + bcf + de, bdf a, b cf + de, df a, b c, d e, f : a, b c, d ab + dc, bd cb + da, db c, d a, b : ab + bc a, bd b, d 1, bc 0, : 4 X A X A 2 X x, y A x y x y : x y z x y z x y z x y z x y z : x y x y y x y x : A : A, A 5 A x y x y A A x 1, y 1, x 2, y 2, x 3, y 3, : x 1, y 1 x 2, y 2 x 3, y 3 x1+x2 2, y1+y2 2 x 3, y 3 x1+x2+2x3 4, y1+y2+2y3 4, x 1, y 1 x 2, y 2 x 3, y 3 x 1, y 1 x2+x3 2, y2+y3 2x1+x2+x3 4, 2y1+y2+y3 x2+x1 2, y2+y1 4,, : x 1, y 1 x 2, y 2 x1+x2 2 x 2, y 2 x 1, y 1 : : 2, y1+y Y a z, A Y you are stupid, but i am not A A ξ α 1 α 2 α n ξ α 1α 2 α m n m α i α i 1 i n ξ ξ α 1 α 2 α n α 1α 2 α m A a, b, c, : a b c a bc abc ab c a b c : a b ab, ba b a : : 7 n A M n n C x y 2 2 : a b e f i j ae + bg af + bh i j c d g h k l ce + dg cf + dh k l iae + bg + kaf + bh jae + bg + laf + bh ice + dg + kcf + dh jce + dg + lcf + dh a b ei + fk ej + fl a b e f i j c d gi + hk gi + hl c d g h k l :

3 1 9 : 0 1 : det A 0 A, 8 n A M n n R x y xy yx :, , : xy yx yx xy : e x A, x e xe ex e x ex xe x x xe ex xe x xe x, 2x 0, ie x 0 : 9 A X x y X n A f, g, h, : f g hx f ghx fghx fg hx f g hx :, A {1, 2, 3}, f : 1 1, 2 3, 3 2, g : 1 3, 2 2, 3 1 f g : 1 2, 2 3, 3 1 g f : 1 3, 2 1, 3 2 : id : A 113 x 1 x 2 x n n x n 1 0 n 2 i> 0 n a n x 1 x 2 x n [ ] 114

4 1 10 e e e e e e e e e e [ ] 115 x x x y z e x y y x e x z z x e y y e y x z y x z e z z x x [ ] 116 A e x e x x A e x x e X {0, 1} x y : max{0, x y} 0 X 0 0 0, 0 1 0, 1 0 1, X 117 e x A x y e y x x Z x + y 1 y < 2 x y : xy y 2 1 Y Z, 2 0 1,

5 G G G G 1 x y z x y z x, y, z G 2 e G such that x e e x x x G 3 x G, x 1 G such that x x 1 x 1 x e G G x y y x x, y G + 0 x x G G G G n G {a 1, a 2,, a n } a i, a j 1 i, j n a i a j a k a k e x e e x x x e G x 0 e, x 1 x, x 2 x x,, x n+1 x n x G x G x n G x 121 G 1 x y x z y z x 1 x y x 1 x z x 1 x y x 1 x z y z 2 x x x x e x 1 x x x 1 x x 1 x x e 3 a, b G a x b x : x a 1 b : a x b, a x b x a 1 b a 1 a x a 1 a x 4 a G a G {a x x G} G a {x a x G} a G G G a G b, b a a 1 b a a 1 b, G a G, G G a

6 a 1 a 2 a n 1 a 1 n a 1 2 a 1 1 n 2, a 1 2 a 1 1 a 1 a 2 a 1 a 2 a 1 2 a 1 1 a 1 a 2 a k 1 a k 1 a 1 k a 1 a 2 a k 1 1 a 1 k a 1 k 1 a 1 1 e n k , 4, 5, 6 3, 5 4, 6 og 3, e x y e e x y x x y e y y e x og 4, e x y z e e x y z x x e z y y y z e x z z y x e e x y z e e x y z x x y z e y y z e x z z e x y, og 5, e x y z u e e x y z u x x y z u e y y z u e x z z u e x y u u e x y z og 6, e x x 2 x 3 x 4 x 5 e e x x 2 x 3 x 4 x 5 x x x 2 x 3 x 4 x 5 e x 2 x 2 x 3 x 4 x 5 e x x 3 x 3 x 4 x 5 e x x 2 x 4 x 4 x 5 e x x 2 x 3 x 5 x 5 e x x 2 x 3 x 4,

7 1 13 e x y y 2 xy xy 2 e e x y y 2 xy xy 2 x x e xy xy 2 y y 2 y y xy 2 y 2 e x xy y 2 y 2 xy e y xy 2 x xy xy y 2 xy 2 x e y xy 2 xy 2 y x xy y 2 e e x x 2 x 3 x 4 x 5 x 6 e e x x 2 x 3 x 4 x 5 x 6 x x x 2 x 3 x 4 x 5 x 6 e x 2 x 2 x 3 x 4 x 5 x 6 e x x 3 x 3 x 4 x 5 x 6 e x x 2 x 4 x 4 x 5 x 6 e x x 2 x 3 x 5 x 5 x 6 e x x 2 x 3 x 4 x 6 x 6 e x x 2 x 3 x 4 x n n e x x 2 x n 1 e e x x 2 x n 1 x x x 2 x 3 e x 2 x 2 x 3 x 4 x x n 1 x n 1 e x x n x G a G x n e n G x, x n x m, n > m m, x n m e k Z, k qn + r,0 r n, x k x r G b G n n x m e m x m e m {x, x 2, x 3,, x m 1, x m e}, x n e n m c G G Z x G ϕ : G Z, x i i ϕx i x j ϕx i+j i + j ϕx i + ϕx j ϕx i ϕx j i j x i x j

8 a Z Z a, b, c, a + b Z, a + b + c a + b + c a + b + c, 0, a b R {0} R {0} a, b, c, ab R {0}, abc abc abc, 1, 1 a c n GL n C GL n C A, B, C, detab detadetb, AB GL n C, ABC ABC I n, A 1 d n S n σ, τ, ρ S n, σ τ ρ σ τ ρ, ε, σ S n σ e M G f, g, h G, f g h f g h, 1 M, f G f 126 G n x G x m e n m x G, x G n, x m e n [ ] 127 n Z x, y x y n n a b mod n Z Z/nZ x [x] Z/nZ {[0], [1], [2],, [n 1]} Z/nZ [x] + [y] [x + y] Z/nZ [0] Z/nZ [1] n n Z/nZ

9 1 15 ix y n x y x x 0, n 0, x x, x y, y x x x x y, y z x z ii[x] x 1, [y] y 1, x 1 + x 2 [x + y] x 1 x + nk, y 1 y + nk x 1 + y 1 x + y + nk + k iii,iv[x] + [y] + [z] [x + y] + [z] [x + y + z] [x] + [y + z] [x] + [y] + [z] [x] + [y] [x + y] [y + x] [y] + [x] [x] + [0] [x + 0] [0 + x] [0] + [x] [x], [0] [x] + [ x] [x x] [0] [ x + x] [ x] + [x], [ x] v,vi G x ϕ : G Z/nZ, x [1] ϕx i ϕx j [i] [j] x i x j, G Z/nZ iv,

10 G H G H G G H G G e e H x, y H x y 1 H H G G x, y x y x 1 y H G/H ah {ah h H} G/H {ah a G} ah a H x y xh yh G x, y x y x y 1 H H\G Ha {ha h H} H\G {Ha a G} Ha a H x y Hx Hy G/H H [G : H] G H [G : H] G H H [G : H] G G x x n e n x order ordx G ordx G 131 G H a G R H Z b G x H H {x n n Z} c G GLn, C {X n detx 0} H SLn, C {X GLn, C detx 1} d G GLn, C {X n detx 0} T {X GLn, C X } { } t 0 e G SL2, R H 0 t 1 t 0 R f n G S n H A n a 1 G 0 H 2 m, n H m + n m n H H G

11 1 17 b 1 G 1 x 0 H 2 x m, x n H x m x n x m n H H G c 1 G E H 2 A, B H det AB 1 H H G d 1 G E H 2 A, B H AB 1 H H G e 1 G E H 2 A, B H AB 1 H H G f 1 G ε H 2 σ, τ H στ 1 H H G [ ] 132 G H G/H H\G a G/H {r + H r G} r + H r + H r r H G/H [0, 1 S 1 H\G {H+r r G} H+r H+r r r H H\G [0, 1 S 1 b G/H {gh g G}, gh g H g 1 g H H \ G {Hg g G}, Hg Hg g g 1 H c G/H {AH A G} AH BH B 1 A H G/H C H \ G {HA A G} HA HB AB 1 H H \ G C d G/H {AH A G}, AH BH B 1 A H H \ G {HA A G}, HA HB AB 1 H e G/H {AH A G}, AH BH B 1 A H H \ G {HA A G}, HA HB AB 1 H f G/H {σh σ G} σh τh τ 1 σ H G/H 1 Z/2Z H \G {Hσ σ G} Hσ Hτ στ 1 H H \G 1 Z/2Z

12 n pq p, q G Z/nZ G H x n pq, p q x p, {0} H p : {0, x, 2x,, p 1x} H, H p H x q, G G,H p,h q,{0} G n G n x G x, 1 m < n, x m e x n e, 1 l < k m, x l x k, G x [ ] 135 G G G x, G x G H G, y H y G x, n y x n n, x n H r, x r H G, H x r x n H n, n mr +l, 0 l r 1, x n x mr+l x mr x l H, x l H, l 0, H x r, H [ ] 136 G n n m m G G : {x, x 2, x 3,, x n 1, x n e} n md, H : {x d, x 2d,, x m 1d, x md e} K m K y, y m e ie y H K H 137 G H G/H H\G ϕ : G/H H \ G gh Hg g2 1 g 1 H, g1 1 g 2 g2 1 g 1 1 H, ϕ well-defined ϕ ϕ, G/H H \ G [ ] 138 Q H, Q Q H, [Q : H] n < y Q/H, Q/H, ny 0 x Q, z Q, st x nz, x Q, nx H, nz x H 139 G H H K [G : K] [G : H][H : K] [G : K] og/ok og/oh oh/ok [G : H][H : K]

13 G G x /x n e G x i x n e G x n a G x b G x, y /x 2 e, y 2 e, xy yx c G x, y /x 3 e, y 2 e, yx x 2 y a ϕ : Z G x n x n, well-defined, ϕ, G, G H i x i i 0, 1, 2, b G x, y /x 2 e, y 2 e, xy yx {e, x, y, xy} Z/2Z Z/2Z, G 4, G G,{e, x},{e, y},{e, xy},{e} 5 c G x, y /x 3 e, y 2 e, yx x 2 y {e, x, x 2, y, xy, x 2 y} S 3, G 6, G G,{e, x, x 2 },{e, y},{e, xy},{e, x 2 y},{e} 6 [ ] 1311 n Z/nZ n ϕn ϕ a ϕn {i 1 i < n, i, n 1} i Z/nZ, gcdn, i 1 i, Z/nZ k 1 n + k 2 i 1, k 1, k 2 Z, k 2 s 1 mod n, < s > 1 b p ϕp r p r 1 p 1 p r p r c n m n ϕm n p i1 1 pi2 2 pit t, m p j1 1 pj2 2 pjt t, j 1 i 1, j 2 i 2,, j t i t n p i1 1 pi2 2 pit t j 1,j 2,,j t i 1,i 2,,i t ϕpj1 1 pj2 2 pjt t gcdn, m 1,

14 1 20 ϕnm ϕnϕm, j 1,j 2,,j t i 1,i 2,,i t ϕp j 1 1 ϕp j 2 2 ϕp j t t p j1 1 pj1 1 j 1,j 2,,j t i 1,i 2,,i t 1 p j2 p j1 1 pj1 1 j 1,j 2,,j t 1 i 1,i 2,,i t 1 j 1,j 2,,j t 1 i 1,i 2,,i t 1 p i 1 1 p i 2 2 p i t t 2 pj2 1 1 p j2 2 p jt t 2 pj2 1 p jt 1 t 2 p j t 1 p j 1 1 p j p j 2 2 p j p j t 1 t 1 pj t 1 1 t 1 p jt t j t i 1 t 1 pj t 1 1 t 1 p i t t p jt 1 t

15 G G f : G G f f a x, y G fx y fx fy f b fe e c fx 1 fx 1 d fx y 1 fx fy 1 f : G G f G G G G f : G G Imf {fx G x G} Imf f Image Imf G f : G G f Imf G f : G G Kerf {x G fx e } Kerf f Kernel Kerf G f : G G f Kerf {e} 141 f f fxy 1 fx fy 1 fe fee fefe, e fe fe fxx 1 fxfx 1, fx 1 fx 1 fxy 1 fxfy 1 fxfy a f : GLn, C GL1, C fa deta GLn, C A, B, fab detab detadetb fafb b R C C {0} GL1, C f : R C fx e 2πix R x, y, fx + y e 2πix+y e 2πix e 2πiy fxfy Kerf {x R e 2πix 1}, e 2πix cos2πx + isin2πx, Kerf Z

16 1 22 c f : Z Z/2Z 0 x fx 1 x fx + y 0 x, y, 1 Kerf 2Z fx 0 x, 1 x fy 0 y, 1 y 0 x, y fx + fy 1 d n S n n f : S n Z/2Z fσ sgnσ sgnσ σ 0 x sgnσ 1 x,,, f, , , , , f, Kerf { } A n [ ] f : G G a f G G G x, y, fxy fxfy fyfx fyx f, xy yx b f G G G x, y, x fa, y fb xy fafb fab fba fbfa yx 145 G G f : G G f G G

17 1 23 G G G G f : G G 146 f : G G f 1 : G G G a, fa a, f 1 a, b, ab f 1 a, f 1 b, f 1 a, f 1 b, a b fa fb a, b,, 147 f : G G a f Kerf {e} b Kerf {e} G Imf a f, f Kerf {e} Kerf {e} fx fy, e fxfy 1 fxfy 1 fxy 1, xy 1 e, x y, f b f : G Imf f, f [ ] 148 G f : G G fx x 1 f f G x G, x 1 G, fx 1 x 1 1 x, f fx fy, x 1 y 1 x y, f f f, fxy fxfy, xy 1 x 1 y 1 xy yx, yx xy G G, x, y G, fxy xy 1 y 1 x 1 x 1 y 1 fxfy, f [ ] 149 G f : G G fx x 2 G fxy xy 2 x 2 y 2 fxfy, xyxy x 2 y 2, yx xy fxy xy 2 xyxy x 2 y 2 fxfy 1410 Q f : Q Q f1 1 f Q a, b, gcda, b 1 1 f1 fb1/b bf1/b, fa/b af1/b a/b 1411 Z Q

18 1 24 q Q q q/2 q 1412 Q + {x Q x > 0} Q + Z Q Q +, Z, + f : Z Q + f1 a fi a i, i Z, f, Q +, Q, + f : Q Q +, fa 2 fa/2 ± G a G φ a : G G φ a x axa 1 φ a G φ a a G : φ a xy axa 1 axa 1 aya 1 φ a xφ b y : φ a x φ b y axa 1 aya 1 x y : φ a a 1 xa x

19 G N N G normal subgroup a xn Nx b xnx 1 N c xnx 1 N for any x G for any x G for any x G N G G H NH HN G G f : G G Kerf G Imf G N G G/N G/N G N G/N xn yn xyn well-defind xn x N, yn y N xyn x y N N G/N N en, xn x 1 N 151 GL2, R { } a c a N 1 ab 0, c R 0 b / N b N 2 { c N 3 { a 0 0 b a b 0 1 ab } a 0, b R } / N 2 / N 3

20 1 26 d N 4 { 1 b 0 1 } b R / N H G 2 H G g G, g H ghg 1 H g / H, gh H Hg, H gh H Hg H gh G H Hg, gh Hg ghg 1 H, H [ ] 153 N H G N H {e} N H x N y H, xyx 1 H yx 1 y 1 N, xyx 1 y 1 xyx 1 y 1 xyx 1 y 1 N H {e}, xy yx [ ] 154 G n H i i 1, 2,, n N n i1 H i G N a, b, ab 1 H i, i 1, 2,, n N a,g x, xax 1 H i, i 1, 2,, n 155 N G H G a H N H b NH {n h n N, h H} G c H G NH G a x H N h H, N G, hxh 1 N H G, hxh 1 H, hxh 1 H N, H N H b n 1, n 2 N h 1, h 2 H, n 1 h 1 n 2 h 2 n 1 h 1 n 2 h 1 1 h 1h 2 NH e e e NH, nh NH, enh nh nhe nh NH, h 1 n 1 h 1 n 1 hh 1 NH, nhh 1 n 1 e h 1 n 1 nh, NH G c nh NH g G, gnhg 1 gng 1 ghg 1 NH, NH G [ ] S 4

21 1 27 S 4 H n, n 24 n 1, H {ε}, n 2, H Z/2Z 9 H S 4 n 3, H 3-4 H S 4 n 4, H 4 3, {ε, 1, 2, 3, 4, 1, 23, 4} 3, H S 4, 4 V {ε, 1, 23, 4, 1, 32, 4, 1, 42, 3}, n 6, H S 3 4 H S 4 n 8, H 2-3 H S 4 n 12, H A 4, n 24, H S 4, 157 n S n A n A n, i, j 1 i, j, A n S n σ σa n σ 1, 158 G ZG {a ab ba for any b G} ZG G center a ZG G b ZG G H G c H ZG G G/H G d G/ZG G a x ZG g G, gxg 1 gg 1 x x ZG, ZG G b x H g G, H ZG gxg 1 gg 1 x x H, H G c G/H a i H, x, y G, h, h H, x a i h, y a j h xy a i ha j h a i a j hh a j a i h h a j h a i h yx, G d x, y G, z, z ZG, xyzz xzyz yz xz yxz z yxzz zz 1 xy yx, G [ ] 159 G a, b [a, b] aba 1 b 1 a, b G G [G, G] G a [G, G] G

22 1 28 b G/[G, G] c H G G [G, G] H G/H d G/H G H [G, G] H a [a, b], [x, y] [G, G], [a, b][x, y] [G, G] e [e, e] [G, G], [a, b] [G, G], [a, b][e, e] [a, b] [e, e][a, b] [a, b] [a, b] [G, G], [b, a] [G, G], [a, b][b, a] [e, e] e [b, a][a, b] [e, e] e, [G, G] G, [a, b] [G, G] g G, g[a, b]g 1 gaba 1 b 1 g 1 gaba 1 g 1 b 1 bgb 1 g 1 [ga, b][b, g] [G, G], [G, G] G b [a, b] e ab ba, c π : G/[G, G] G/H, G/[G, G], G/H d [x, y] [G, G], xhyh yhxh xyh yxh, [x, y] xyx 1 y 1 H 1510 S n ZS n [S n, S n ] ZS n {ε}, ε [S n, S n ] A n [ ]

23 f : G G H Kerf H G f f : G/H G [x] G/H f[x] fx f : G G G/Kerf Imf H K G K HK {h k h H, k K} G H K H K HK H/H K HK/K K H G G H/K G/K G/K G/K/H/K H a, b H, fab fafb e ab H H G x G y H, fxyx 1 fxfyfx 1 fxe fx 1 e, H G 2 f : G/H G well-defined [x] [y] G/H, x yh y 1 x H, fy 1 fx fy 1 x e fx fy 3 f : G/H G [x], [y] G/H fx fy, y 1 x H [x] [y] 4 f : G/H G f 5 f : G/H G [x], [y] G/H, f[x][y] f[xy] fxy fxfyf[x]f[y], f : G/H G [ ] f : G G, G/Kerf Imf G Imf, 2 H K G, K HK {h k h H, k K} G, H K H, K HK, H/H K HK/K ϕ : H H/K; h hk, Kerϕ H K Imϕ HK, 1

24 K H G G, H/K G/K, G/H G/K/H/K ψ : G/K G/H; gk gh, Kerψ H/K, [ ] 163 n S n A n S n /A n Z/2Z 1 C Z/2Z, C 2 S n /A n C2 sgn : S n C 2 Kersgn A n A n S n, S n /A n C2 [ ] 164 SLn, C GLn, C GLn, C/SLn, C C C {0} det : GLn, C C Kerdet SLn, C SLn, C GLn, CS n, GLn, C/SLn, C C [ ] 165 S {z C z 1} S C C {0} C /S : C R + : {a R a > 0 } Ker S S C, C /S R + [ ] 166 n R n R n H {x 1,, x n R n a 1 x a n x n 0} R n R n /H R 1 a 1,, a n 0,, 0 a a 1,, a n, a, b a b, a, : R n R 1 Ker a, H H R n, R n /H R 1 [ ] 167 n, m Z Z nz {nz z Z} mz a nz mz n m b d n m c n m nz + mz dz, nz mz cz n m an + bm a, b Z c dz/mz nz/cz

25 1 31 a nz mz, n n 1 nz mz, l Z, n ml, n m, n m, n mz, nz mz b nz + mz dz nz mz cz c b, nz/cz nz/nz mz nz + mz/mz dz/mz [ ] 168 S 4 V {1, 1234, 1324, 1423} S 4 S 4 /V S 3 σ S 4, σ1σ 1 1 S 4, σ1234σ S 4, σ1324σ S 4, σ1423σ S 4, V S 4 f : S 3 S 4 /V, S 3 S 4 S 4 S 4 /V, Kerf V S 3 1, f, S 4 /V 6 S 3, f, S 4 /V S 3 [ ] 169 G H K [G : H] [G : K] G HK f : H K G; h, k hk HK fh 1, k 1 fh 2, k 2 h 1 k 1 h 2 k 2 x : h 1 2 h 1 k 2 k1 1 H K h 2 h 1 x 1, k 2 xk 1, x H K, hk HK, f 1 hk {hx 1, xk x H K}, H H H K HK H K, [G : H] [G : K], [G : H K] [G : H][H : H K] [G : K][K : H K], [G : H] [K : H K] [G : H], [G : H] [K : H K] G, [G : H K] [G : K][G : H], H K G H G K H K, G H K HK, G HK [ ] 1610 a G 1 G 2 G 1 G 2 {g 1, g 2 g 1 G 1, g 2 G 2 } g 1, g 2 h 1, h 2 g 1 h 1, g 2 h 2 G 1 G 2 G 1 G 2 G 1 G 2 e 1, e 2 g 1, g 2 1, g 1 2 b G 1 {g 1, e 2 g 1 G 1 } G 1 G 2 G 1 G 2 /G 1 G 2 G 1 G 2 /G 2 G 1 c G G 1 G 2 G ZG [G, G] ZG ZG 1 ZG 2, [G, G] [G 1, G 1 ] [G 2, G 2 ] d F G 1 G 2 G 1 G 2 F G 1 G 2 π : G 1 G 2 G 1 G 2 Kerπ a g 1, g 2, h 1, h 2, f 1, f 2 G 1 G 2, g 1, g 2 h 1, h 2 f 1, f 2 g 1 h 1, g 2 h 2 f 1, f 2 g 1 h 1 f 1, g 2 h 2 f 2 g 1 h 1 f 1, g 2 h 2 f 2 g 1, g 2 h 1 f 1, h 2 f 2 g 1

26 1 32 g 1, g 2 h 1, h 2 f 1, f 2 e 1, e 2 G 1 G 2, g 1, g 2 G 1 G 2, g 1, g 2 e 1, e 2 g 1, g 2 e 1, e 2 g 1, g 2 g 1, g 2 G 1 G 2, g1 1, g 1 2 G 1 G 2, g 1, g 2 g1 1, g 1 2 e 1, e 2 g1 1, g 1 2 g 1, g 2 G 1 G 2 b f : G 1 G 2 G 2 ; g 1, g 2 g 2, Kerf G 1, G 1 G 2 /G 1 G 2 c, d π : G 1 G 2 G 1 G 2 ; g 1, g 2 g 1 g 2 πg 1, g 2 πh 1, h 2 g 1 g 2 h 1 h 2 g 1 g 2 h 1 g2 1 g 2h 2 G 1 G 2,, Kerπ G 1 G 2 [ ] 1611 H K G a G HK d π H K b HK h k h H, k K c a b e a H, b K a b e d H K {e} a b b c a b e e e, a b e c d a H K, a a 1 e, a a 1 e d a 1610, [ ] 1612 G H K H K G H K p q Z/pqZ Z/pZ Z/qZ H K, H K 1 H K {e} 1611, HK H K G HK, G HK, G H K [ ]

A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x A (1)

7 2 2.1 A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x 1 2.1.1 A (1) A = R x y = xy + x + y (2) A = N x y = x y (3) A =