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 = N 2 (a, b), (c, d) A (a, b) (c, d) = (ad + bc, bd) (4) X A X A = 2 X ) x, y A x y = x y (5) A x y = x y

2 8 (6) 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) (7) n A = M n n (C) x y (8) n A = M n n (R) x y = xy yx (9) A X x y X n 2.1.2 (1) (2) A 2.1.3 x 1 x 2 x n 2.1.4 2.1.5 x x 2.1.6 A e x e = x x A ( e x = x ) e 2.1.7 e x A x y = e y x x

2 9 2.2 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) (2) 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 2.2.1 G (1) x y = x z y = z (2) x x = x x = e (3) a, b G a x = b x (4) a G a G = {a x x G} G a = {x a x G} a G = G = G a (5) (a 1 a 2 a n ) 1 = a 1 n a 1 2 a 1 1 2.2.2 (1) 3, 4, 5, 6 3, 5 4, 6

2 10 (2) 7 7 2.2.3 n n 2.2.4 x G (a) G x n = e n (b) G n n x m = e m (c) G G Z 2.2.5 (a) Z (b) R {0} (c) n GL n (C) (d) n S n (e) M G 2.2.6 G n x G x m = e n m 2.2.7 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

2 11 2.3 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) ord(x) G ord(x) G 2.3.1 G H (a) G = R H = Z (b) G x H H = {x n x Z} (c) G = GL(n, C) = {X n det(x) 0} H = SL(n, C) = {X GL(n, C) det(x) = 1} (d) G = GL(n, C) = {X n det(x) 0} T = {X GL(n, C) X } {( ) } t 0 (e) G = SL(2, R) H = 0 t 1 t 0 R (f) n G = S n H = A n 2.3.2 G H G/H H\G

2 12 2.3.3 n = pq p, q G = Z/nZ 2.3.4 G n G n x G x 2.3.5 G G 2.3.6 G n n m m G 2.3.7 G H G/H H\G 2.3.8 Q 2.3.9 G H H K [G : K] = [G : H][H : K] 2.3.10 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) 2.3.11 n Z/nZ n ϕ(n) ϕ (a) ϕ(n) {i 1 i < n, (i, n) = 1} (b) p ϕ(p r ) = p r 1 (p 1) (c) n = m n ϕ(m)

2 13 2.4 G G f : G G f f (a) x, y G f(x y) = f(x) f(y) f (b) f(e) = e (c) f(x 1 ) = f(x) 1 (d) f(x y 1 ) = f(x) f(y) 1 f : G G f G G G = G f : G G Im(f) = {f(x) G x G} Im(f) f (Image) Im(f) G f : G G f Im(f) = G f : G G Ker(f) = {x G f(x) = e } Ker(f) f (Kernel) Ker(f) G f : G G f Ker(f) = {e} 2.4.1 f f f(x y 1 ) = f(x) f(y) 1 2.4.2 (a) f : GL(n, C) GL(1, C) f(a) = det(a) (b) R C = C {0} = GL(1, C) f : R C f(x) = e 2πix (c) f : Z Z/2Z 0 x f(x) = 1 x

2 14 (d) n S n n f : S n Z/2Z f(σ) = sgn(σ) sgn(σ) σ 0 x sgn(σ) = 1 x 2.4.3 2.4.4 f : G G (a) f G G (b) f G G 2.4.5 G G f : G G f G G 2.4.6 f : G G f 1 : G G 2.4.7 f : G G (a) f Ker(f) = {e} (b) Ker(f) = {e} G Im(f) 2.4.8 G f : G G f(x) = x 1 f f G 2.4.9 G f : G G f(x) = x 2 G 2.4.10 Q f : Q Q f(1) = 1 f 2.4.11 Z Q 2.4.12 Q + = {x Q x > 0} Q + Z Q 2.4.13 G a G φ a : G G φ a (x) = axa 1 φ a G φ a a G

2 15 2.5 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 Ker(f) G Im(f) 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 2.5.1 GL(2, R) {( ) } a c (a) N 1 = ab 0, c R 0 b {( ) } a 0 (b) N 2 = 0 b ab 0 {( ) } a b (c) N 3 = 0 1 a 0, b R {( ) } 1 b (d) N 4 = 0 1 b R 2.5.2 H G 2 H G 2.5.3 N H G N H = {e} N H 2.5.4 G n H i (i = 1, 2,..., n) N = n i=1 H i G

2 16 2.5.5 N G H G (a) H N H (b) NH = {n h n N, h H} G (c) H G NH G 2.5.6 4 S 4 2.5.7 n S n A n 2.5.8 G Z(G) = {a ab = ba for any b G} Z(G) G (center) (a) Z(G) G (b) Z(G) G H G (c) H Z(G) G G/H G (d) G/Z(G) G 2.5.9 G a, b [a, b] = aba 1 b 1 a, b G G [G, G] G (a) [G, G] G (b) G/[G, G] (c) H G G [G, G] H G/H (d) G/H G H [G, G] H 2.5.10 S n Z(S n ) [S n, S n ]

2 17 2.6 f : G G H = Ker(f) H G f f : G/H G [x] G/H f([x]) = f(x) f : G G G/Ker(f) = Im(f) 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) 2.6.1 2.6.2 2.6.3 n S n A n S n /A n = Z/2Z 2.6.4 SL(n, C) GL(n, C) GL(n, C)/SL(n, C) C = C {0} 2.6.5 S = {z C z = 1} S C = C {0} C /S 2.6.6 n R n R n H = {(x 1,..., x n ) R n a 1 x 1 + + a n x n = 0} R n R n /H = R 1 (a 1,..., a n ) (0,..., 0) 2.6.7 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

2 18 2.6.8 S 4 V = {(1), (12)(34), (13)(24), (14)(23)} S 4 S 4 /V = S 3 2.6.9 G H K [G : H] [G : K] G = HK 2.6.10 (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 Z(G) [G, G] Z(G) = Z(G 1 ) Z(G 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 Ker(π) (g 1 2.6.11 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} 2.6.12 G H K H K G = H K p q Z/(pq)Z = Z/pZ Z/qZ