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1 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1

2 Armstrong culture Web A 37 N 0 Z = {0, ±1, ±2,... }, R =, C =. 2

3 1 1.1 n 2 a n a = qn + r q, r 0 r < n r a n (remainder) a r r = r a a n a a mod n a a mod n a a n a, a a a n a a mod n (congruence) 1.1 ( ). a a, b b = a ± b a ± b, ab a b m a l... a 2 a 1 m a 1 + a a l mod = 28 1 mod [r] = {a Z; a r}, r = 0, 1,..., n 1 3

4 n (congruence class) [a] = a + Zn a [a] (representative) [1] [ n + 1] [1] 1 n Z n Z n n Z n n n = 7 a a mod mod 7 n = 2 [a] + [b] = [a + b] 3. (well-definedness) ([a] + [b]) + [c] = [a] + ([b] + [c]), [a] + [b] = [b] + [a]. 0 [0] = Zn [a] + [0] = [a] = [0] + [a] k [a] k k[a] = [ka] 4

5 (well-defined ) [a] = [ a] [a] + [ a] = [ a] + [a] = [0] [a] 4 (Optional). p m, n (m + n) p m p + n p mod p. m p m mod p 1.2 x, y, z f(x, y, z) x y, f(y, x, z) x y, y z, z x, f(y, z, x). (symmetric expression) x + y + z, x 2 y + y 2 z + z 2 x + xy 2 + yz 2 + zx 2, ring (elementary symmetric polynomial) (t x)(t y)(t z) t (t x)(t y)(t z) = t 3 (x + y + z)t 2 + (xy + yz + zx)t xyz. x, y, z t 3 + at 2 + bt + c = 0 ( Viete x + y + z = a, xy + yz + zx = b, xyz = c n f(x 1,..., x n ) ( ) x 1,..., x n n σ(1) σ(2)... σ(n) σ = f(x σ(1),..., x σ(n) ) 5

6 σ f(x 1,..., x n ) = f(x σ(1),..., x σ(n) ) x 1,..., x n s i (x 1,..., x n ) (i = 1, 2,..., n) (t x 1 )(t x 2 )... (t x n ) = t n s 1 (x 1,..., x n )t n 1 + s 2 (x 1,..., x n )t n ( 1) n s n (x 1,..., x n ) x 2 y + y 2 z + z 2 x + xy 2 + yz 2 + zx 2 = (x + y + z)(xy + yz + zx) 3xyz x 2 + y 2 + z 2 = (x + y + z) 2 2(xy + yz + zx) 5. x 3 + y 3 + z 3 3xyz x, y, z Lagrange Galois Lagrange f(x 1,..., x n ) σ f(x σ(1),..., x σ(n) ) Lagrange G(f) = {σ; f(x σ(1),..., x σ(n) ) = f(x 1,..., x n )} 6. Lagrange n! G(f) (i) n = 3, f = xy, f = (x y)(y z)(z x) (ii) f = x x n 6

7 G(f) f f, g G(f) G(g) g f 7. xy + z 2 x 2 y + y 2 z + z 2 x n = 4 x 1 + x 2 x 1 + x 2 + x K K K K K K ABC A, B, C ABC {( ) A B C, A B C ( ) ( ) A B C A B C, B C A C A B ( ) A B C, B A C ( A B C A C B ), {( ) ( )} A B C A B C,. A B C B A C {( )} A B C. A B C ( )} A B C. C B A 8. 9 (Optional). 7

8 f(x + T ) = f(x) = = f( x) = ±f(x). z = x + iy z = x iy 2 σ, τ στ (στ)(i) = σ(τ(i)), 1 i n. σ σ 1 σ n S n S n 1 1 n σ S n σ 1 S n σσ 1 = σ 1 σ = 1 Z Z n A. Cayley 1878 G a, b G a b G G G (binary operation) (i) (associativity law) a (b c) = (a b) c 8

9 (ii) a G e a = a e = a e G ( unit element (iii) a G a a = a a = e a G (a inverse element G (group) ab a 1 1 G 1 G X X X X (transformation) (inverse transformation) (identity transformation) 1 X [ ] X S(X) S(X) S(X) X (general transformation group) X n X = {1, 2,..., n} S(X) S n n (the symmetric group of degree n) g G 2.1. G 1 G g g g g 1 Proof. 10. (ab) 1 = b 1 a 1 9

10 2.2. a 1 a 2... a n (a 1 a 2 )(a 3 a 4 ) = (a 1 (a 2 a 3 ))a 4. Proof. 11. a, b, c, d abcd 12. R a b = a + b (Optional). a 1... a n (... ((a 1 a 2 )a 3 )... )a n n 2.3. G (commutativity law) ab = ba, a, b G (commutative group) (non-commutative group) (abelian group) (additive group) 0 (zero element) a (multiplicative group) Z n, Z, R, C C n, T, R +, C 10

11 C n = {z C; z n = 1}, T = {z C; z = 1}, C = C \ {0}, R + = 2.4. G (finite group) (infinite group) Z n Z (discrete group) (continuous group) G(f) G(f) = S n G(f) S n σ, τ G(f) σ 1, στ G(f) G(f) S n 2.5. G H G H H G (subgroup) a, b H ab H (i) ah = ha (a H) H h H (ii) a H H a aa = h = a a 2.6. (i) nz Z R, (ii) C n T, (iii) R + C G H (i) a, b H = ab H, (ii) a H = a 1 H h = 1 a H H a G a 1 11

12 Proof. 14 (Optional). a, b H = ab 1 H. G H G G S(X) X (transformation group) 2.8. K K 2.9. n n (the dihedral group of degree n) D n n GL(n, C) C n GL(n, R) GL(n, C) (general linear group) GL(n, C) (matrix group) SL(n, C) = {g GL(n, C); det(g) = 1} U(n) = {g GL(n, C); g g = gg = I} O(n) = {g GL(n, R); t g g = g t g = I} (special linear group), (unitary group), (orthogonal group). SU(n) = U(n) SL(n, C), SO(n) = O(n) SL(n, R). 15 (Optional). SO(n) O(n), SU(n) U(n) 12

13 16. SL(n, Z) = {g SL(n, C); g } SL(n, R) (i) N = {0, 1,... } Z. (ii) C C (iii) GL(n, R) G G ϕ : G G ϕ(ab) = ϕ(a)ϕ(b), a, b G (homomorphism) ab G ϕ(a)ϕ(b) G (isomorphism) ϕ : G G ϕ(1 G ) = 1 G, ϕ(g) 1 = ϕ(g 1 ), g G 18. G, G 19. ϕ : G G {g G; ϕ(g) = 1 G } = {1 G } 13

14 3.3. ϕ : G G (i) G H ϕ(h) G (ii) G H ϕ 1 (H ) G (iii) ϕ ϕ 1 : G G (Optional). G ϕ(g) G G 3.4. G, G G G (isomorphic) G = G = = 22. [ ] 23. m n S m S n 3.5. D n X π(g) π : D n S(X) D n S n 24. n = 3 D 3 S 3 n = 4 D 4 S 4 Z Z n, Z C n Z n = Cn SO(2) = T R T R t e at R + det : GL(C) C sgn : S n {±1} = C 2 C z z R + 14

15 25. σ S n n T (σ) T (σ) : e i e σ(i) T : S n O(n) 26 (Optional). R R 27 (Optional). R C R GL(n, C) 28 (Optional). X, Y S(X) = S(Y ) G {H i } i I i I G H i Proof. G S S S S (the subgroup generated by S) G = S G S S = {s ɛ 1 1 s ɛ s ɛ k k ; s j S, ɛ j {±1}} n n n 1 C 2π/n π/n π 15

16 2n n D n = 2n 2π/n r s r k a πk/n srs = r 1, r n = 1, s 2 = 1 D n = r, s D n = {r k, r k s; 0 k < n} 29 (Optional). s, rs a G a a (cyclic group) G a a... a (n ) if n 1, a n = 1 (G ) if n = 0, a 1... a 1 ( n ) if n 1 a = {a k ; k Z} 4.3. (a m ) n = a mn, a m a n = a m+n, m, n Z 30. a m 1 a m 1 {a k } k Z ak = a l a k l = 1 a k l = 1 k l 0 k l a k a l 16

17 Z k a k G a Z m 1 a m = 1 a m = 1 m 1 n n 1, a, a 2,..., a n 1 a k = a l (0 k < l < n a l k = 1 1 l k < n n a n n 1 Z n [k] a k a Z n G a a n 4.4. G a a n = 1 n 1 a (order) a m = 1 m 1 a a a a 31. m a n b ab = ba m n ab mn 32. ( cos θ ) sin θ sin θ cos θ (singly generated) Z Z n G = Z n n n = Zn = {0, ±n, ±2n,... } 17

18 4.5. Z Zn Zm Zn m n Proof. H {k H; k > 0} n k H n 4.6. Proof. Z G 33. m, n Z Zm, Zn 34 (Optional). T = SO(2) C n O(2) C n D n 4.7. G, G G G (a, a )(b, b ) = (ab, a b ) G G (the direct product of G and G ) G, G G G G G 4.8. Z 2 Z 2 C 2 H Z 4 Z 2 Z Z m Z n m n Z m Z n = Zmn 4.9 ( ). 18

19 5 G S(X) π : G GT (X) λ : G X X λ(g, x) = π(g)(x) λ(a, λ(b, x)) = λ(ab, x) λ(1, x) = x 5.1. G X λ : G X X G X G X λ G X (g, x) λ(g, x) gx (i) a(bx) = (ab)x (ii) 1x = x (point) λ : G X X g G π(g) : X X π(g) : x λ(g, x) π(a) π(b) = π(ab) π(1) = 1 X π(g) S(X) π : G S(X) 5.2. X G S(X) 19

20 37. ρ : X G X 5.3. G X = G G X G X G (a, x, b) axb X G (regular action) 5.4. G X Y X Y Y X G Y X (gf)(x) = f(g 1 x), g G, x X 38. G {g 1, g 2,..., g n } g G {gg 1, g 2,..., gg n } g 1,..., g n 5.5. n x 1,..., x n indeterminate f(x 1,..., x n ) σ S n (σf)(x 1,..., x n ) = f(x σ(1),..., x σ(n) ) (τ(σf))(x 1,..., x n ) = (σf)(x τ(1),..., x τ(n) ) = f(x στ(1),..., x στ(n) ) = ((τσ)f)(x 1,..., x n ) Z R 39 (Optional). X X {0, 1} X 2 X X 20

21 Gx = {gx X; g G} G x (orbit) G g 1,..., g n (n = G ) Gx = {g 1 x, g 2 x,..., g n x} g 1,..., g n x 5.6. SO(2) R 2 R 40. S n {1, 2,..., n} 41. S n 2 X (stabilizer) G X x G(x) = {g G; gx = x} 42. G(x) G. G x Gx 5.7. G = S n, X n Lagrange 5.8. (orbit space) G\X X/G G\X = {Gx; x X}. Gx, Gy X 21

22 5.9. Gx, Gy Gx Gy = Gx = Gy. Proof. z Gx Gy z = ax = by gx = ga 1 by Gy Gx Gy Gy Gx Gx = Gy ( ). X σ = X = ( ) Gx G\X Gx X = {1, 4, 5} {3} {2, 6} C = σ I {O i } i I X O i X x y Gx = Gy O i O i = Gx i x i X {x i } i I (representative set) G X X gx = x for all g G G(x) = G x X (fixed point) x G (invariant) O(n) R n R n O(n) S n 1 S n 1 G(x) = {1} 22

23 x (free point) (free action) G H G H H G H x X G g gx Gx G Gx H Gx H G H H H (left coset) (left space) H (right coset) (right coset space) Hg = {hg; h H} 43. a, b G ah = bh a 1 b H 44 (Optional). D n s. D. Hilbert notation G S a, b G G as = {as; s S}, Sb = {sb; s S}, S 1 = {s 1 ; s S} 23

24 a(bs) = (ab)s, (Sa)b = S(ab), (as)b = a(sb), (as) 1 = S 1 a 1, (Sb) 1 = b 1 S G gh = H G = m g i H, i=1 G = G/H H m = G/H 5.16 (Lagrange). G H G, H H G G a G a G = G G G. Lagarange A 4 12 A ( ). G X X x Gx G/G(x) gg x gx Gx G/G(x) Gx G G(x) Gx 5.19 ( ). Gx Gx Gx = 24 G G(x)

25 f(x 1, x 2, x 3 ) = x 1 x x 2 x x 3 x 2 1 {( ) ( ) ( )} G(f) = ( ) ( ) G(f), G(f) Gf f = x 1 x x 2 x x 3 x 2 1, (12)f = x 2 1x 2 + x 2 2x 3 + x 2 3x 1 G(f) f, (12)G(f) (12)f 47. f(x 1, x 2, x 3 ) = x 1 Langange 48 (Cauchy). G G p G p Z p X = {(g 1,..., g p ) G n ; g 1... g p = 1} 49 (Sylow). G G p p m p m G /p m G p m (Sylow ) G p m X G (i) X p (ii) p (iii) Sylow 25

26 6 G G µ(g, g ) = gg g 1 = g.g (adjoint action) (conjugacy class) a G C(a) C(a) = {gag 1 ; g G}. G(x) C(a) a = GL(n, C) Jordan GL(2, C) ( ) ( ) λ 0 ν 1,. 0 µ 0 ν 50 (Optional). D n G S G g.s = gsg 1. S gsg 1 H, K H K (conjugate) H = gkg 1 g G H K 6.2. G X x y X G(x) G(y) G(gx) = gg(x)g 1 (invariant subgroup) g G, ghg 1 = H g G, gh = Hg ( ) (normal subgroup) 26

27 G G / S 3 {( ) N =, ( ) 1 2 3, ( 1 2 )} {( ) ( )} H =, ϕ : G G ϕ 1 (1) = ker ϕ G G H ϕ 1 (H ) G 53 (Optional). H G ϕ(h) 54 (Optional). (i) ghg 1 H g H H (ii) H, K H K N gn = Ng g N\G = G/N G/N (an)(bn) = (ab)n G/N G N (quotient group) 27

28 55. (well-defined) G g gn G/N G G/N (the canonical homomorphism) 6.6. G = Z, N = Zn G/N Z n G = V N = W G/N V/W 6.7 ( ). ϕ : G G N = ker ϕ G/N gn ϕ(g) ϕ(g) G/N ϕ(g) G G 6.8. C /R + = T 56. R/Z = T 57. O(n)/SO(n) = Z 2 58 (Optional). G H N HN = {hn; h H, n N} G H N H HN/N = H/H N 59 (Optional). N G/N = 2 G H { H/(H N) Z 2 if H N, = {1} if H N. G = S n, N = A n S n σ A n C(σ) (i) A n (ii) A n 28

29 60 (Challenging). SU(2)/{±1} = SO(3) 7 n S n i j {1, 2,..., n} i j (transposition) (ij) S n = n(n ) 1)/2 (i i + 1) n ( ). S n {(1 2), (2 3),..., (n 1 n)} Proof. σ i = (i i + 1) n σ S n σ(n) = i < n σ n 1... σ i σ n S n 1 ( ) σ = σ sgn(σ) n e 1,..., e n sgn(σ) = det(e σ(1),..., e σ(n) ) sgn(σ) = 1 (even permutation) sgn(σ) = 1 (odd permutation) 1 S n {±1} S n n (the alternating group of degree n) A n 62. A n A n m j 1, j 2,..., j m ( n 2 j 1 j 2,..., j m 1 j m, j m j 1 29

30 (j 1 j 2... j m ) (cyclic permutation) m m σ σ σ (cycle decomposition) n σ l λ l λ 1 σ n = n lλ l l=1 λ 1, λ 2,..., λ n λ = 1 λ 1 2 λ 2... n λ n σ (type) n λ λ l = σ = (2) (3) (6) (1 9) (7 11) ( ) 1 λ 1 2 λ 2... l ( λ l l (Young diagram) n

31 7.3. σ, τ S n 7.4. m (i 1 i 2... i m ) S n n σ σ(i 1 i 2... i m )σ 1 = (σ(i 1 ) σ(i 2 )... σ(i m )) Proof. σ(i i... i m ), (σ(i 1 )... σ(i m ))σ S n n p(n) n (the partition number of n) n p(n) (Hardy-Ramanujan) p(n) 1 4 3n eπ 2n/3. 63 (Optional). p(n)x n = n 0 p(0) = 1 1 l 1 (1 xl ) σ S n C(σ) λ l l l λ l (i 1 i 2... i l ) l λ l l λ l λ l λ l! 31

32 l l lλ l λ l! C(σ) = n! l lλ l λl! 64 (Optional). 1 n n = 3, 4, 5 n = (1), (3), 3 1 (2). n = (1), (6), 2 2 (3), (8), 4 1 (6). n = (1), (10), (15), (20), (20), (30), 5 1 (24). 65. S n (n = 3, 4, 5) 7.5. S 4 4 = = S 5 60 = A 5 66 (Optional). A 4, A 5 A 4 4 A 5 f(x 1,..., x n ) σ S n, σf = f (symmetric polynomial) σ, σf = sgn(σ)f (alternating polynomial) 32

33 (difference product) (x 1, x 2,..., x n ) = (x i x j ) 1 i<j n Vandermonde x 1 x 2... x n = ( 1) n(n 1)/2 (x 1, x 2,..., x n ). x n 1 1 x n xn n 1 67 (Optional). Vandermonde 7.6. σ S n σf sgn(σ)σf σ S n. (z 1,..., z n ) C n σ(z 1,..., z n ) = (z σ 1 (1),..., z σ 1 (n)) 68 (Optional). S n m 8 G X g G X g X g = {x X; gx = x} X 1 = X 33

34 69. π : G S(X) ker π = {g G; X g = X} 8.1. S n X = {1, 2,..., n} n = 5, ( ) σ = X σ = {2, 4} 70. O(3) X = R 3 T if det(t ) = 1, X T = if T = I, otherwise. 8.2 (Burnside). G X 1 X g G g G X g = {x X; gx = x} g O G(x) = G x O Proof. Y = {(g, x) G X; gx = x} g X g g G 34

35 x G(x) = G(x) x X O G\X x O G(x) = G x O O 71 (Optional). m X = {1, 2,..., m} n Y X Y Z = Y X X S m Z σ = ( m) S m C C X Z X m Y Y m X Z C m m C Z C m = 6 langleσ l X X σl l = 1, 5 l = 2, 4 l = 3 n if l = 1 or l = 5, n 2 if l = 2 or l = 4, Z σl = n 3 if l = 3, n 6 if l = 0 2n + 2n 2 + n 3 + n

36 8.3. l m (l, m) C\Z = 1 m m l=1 n (l,m) 72. σ l o(l) m/(l, m) Z σl = n m/o(l) = n (l,m) m m p C = {1, σ, σ 2,..., σ p 1 } p 1 l < p C = σ l Z σl { Z σl n if 1 l p 1, = n p if l = 0. C\Z n p + (p 1)n p = n + np n p ( ) m τ = m m τ 2 = 1 τστ = σ 1 36

37 73. σ l τ (l = 1,..., m 1) m σ G G = {1, σ,..., σ m 1, τ, στ,..., σ m 1 τ} G = 2m G g = σ l τ Z g m m = 2k + 1 m l Z g = n k+1 G\Z = n k m m l=1 n (l,m) m = 2k τ, σ 2 τ,..., σ 2m 2 τ Z g = n k {στ, σ 3 τ,..., σ 2k 1 τ Z g = n k+1 G\Z = nk (n + 1) m m l=1 n (l,m) 8.4. m = 6 G\Z = 1 12 (n6 + 3n 4 + 4n 3 + 2n 2 + 2n). n = (Challenge). n A 37

38 76. A = {1, 2, 1, 2, 1, 1}. 77. B = {1, {1}, {1, 2}, {1, {2}}}. { } 78. C = {, { }, {{ }}, {{{ }}}}. X X (power set) 2 X 79. X 2 X = 2 X 2 n = n nc k. k=0 X, Y X = Y X Y Z (X Y ) Z X Y Z 38

39 Maclane, coherence theorem (... (f 1 f 2 ) f n ) = f 1 f 2 f n f 1 f = f 2 f = f 1 = f 2 f(a) = {f(a) Y ; a A}, f 1 (B) = {a X; f(a) B} a A a A (disjoint union) A B X = i I X i X = i I X i, X i X j = if i j q : X I X x y q(x) = q(y) 39

40 X = {X i ; i I} X (quotient set) I X x x X q : X I x = q 1 (q(x)) X X X Y X Y f : X Y x y = f(x) = f(y) f : X Y f(x) = f(x) X Y x f(x) (well-defined) f : X Y 40

ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University

ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,