ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
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1 ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University hsuzuki@icu.ac.jp 0 0
2 1 1.1 G G1 G a, b, c (a b) c = a (b c) G2 G e e a = a e = a G a G3 G a a b = b a = e G b G4 G a, b a b = b a G1 (semigroup) G1 G2 (monoid) G1 G2 G3 (group) G4 (abelian group) G f : G G G ((a, b) a b) 1.1 (Z, +) (Q, +) (R, +) (C, +) (Z, ) (Q, ) (R, ) (C, ) 0 # 0 (Q #, ) (R #, ) (C #, ) 1.2 Mat(n, R) n GL(n, R) n (Mat(n, R), +) (Mat(n, R), ) (GL(n, R), ) 1. (G, ) a 1, a 2,..., a n G a 1 a 2 a n = ( ((a 1 a 2 ) a 3 ) ) a n ((a 1 a 2 ) a 3 ) a 4 = (a 1 a 2 ) (a 3 a 4 ) = a 1 (a 2 (a 3 a 4 )) = a 1 ((a 2 a 3 ) a 4 ) = (a 1 (a 2 a 3 )) a
3 2. (G, ) e, e G a e = e a = a a e = e a = a a G e = e e = e. 3. (G, ) u G u v = v u = 1 v G u v u u v w u v = v e = v (u w) = (v u) w = e w = w.. a b ab e 1 u u 1 x n x x x = x n x 1 n x 1 x 1 x 1 = x n n x n 1.1 G x x 2 = 1 G a, b G G G4 ab = ab(ba) 2 = abbaba = ba. 1.2 G U(G) G G U(G) a, b U(G) abb 1 a 1 = 1 ab ab U(G) 1 1 = 1 1 U(G) a 1 a = aa 1 = 1 a 1 a a 1 U(G) U(G) G 1.3 Z U(Z) = {±1} U(Q) = Q # U(R) = R # U(C) = C # U(Mat(n, R)) = GL(n, R) 1 2
4 2 2.1 G H G H G H G 2.1 G (1) H G (2) (i) = H G (ii) a, b H ab H (iii) a H a 1 H (1) (2) (2) (1) (ii) (i) a H (iii) (ii) a 1 H 1 = aa 1 HG H H G A, B G AB = {ab a A, b B}, A 1 = {a 1 a A} B = {b} AB = Ab BA = ba H G = H G, HH H, H 1 H. G G {1} G {1} 1 S G < S >= {a n 1 1 a n 2 2 a n r r a 1, a 2,..., a r S, n 1, n 2,..., n r Z} S G a < a >= {a n n Z} a a G (order) G o(a) = < a > a 2.2 < a > S = {n N a n = 1} (1) S = min S = n o(a) = < a > = n (i) a m = 1 n m (ii) < a >= {1, a, a 2,..., a n 1 } 2 1
5 (2) S < a >..., a 2, a 1, 1, a, a 2,... (1) m = nq + r q, r Z 0 r < n a n = 1 a m = a nq+r = (a n ) q a r = a r r < n a m = 1 r = 0 n m. < a > {1, a, a 2,..., a n 1 } < a > a i = a j a i j = 1 n i j, 0 i, j n 1 i = j 1, a,..., a n 1 o(a) = n (2) a i = a j i > j a i j = 1 i j N S = a i < a > 2.1 (Z, +) a i ia (Z n, +) Z n = { 0, 1,..., n 1} ī + j = i + j n Z n n 2.2 X X X X X σ X X a X σ a σ σ, τ X X σ τ : a (a σ ) τ σ τ X X X X S X = U(X X ) X X X X = {1, 2,..., n} S X S n n σ S n σ = ( ) ( i = i σ 1 2 n 1 σ 2 σ n σ (i 1, i 2,..., i r ) i σ j = i j+1 (j = 1,..., r 1) i σ r = i 1 i i σ = i r ) 2 2
6 3 3.1 A (i) a a (ii) a b b a (iii) a b, b c a c (equiavelence relation) C a = {b A a b} a a 3.1 (1) a C a (2) b C a C a = C b (3) C a C b C a C b = (1) (i) (2) (ii) b C a a C b C a C b c C a a c, a b b c c C b (3) c C a C b (2) C a = C c = C b (3) {C λ λ Λ} A = λ Λ C λ (disjoint union) a λ C λ {a λ λ Λ} H G r a, b G a r b (modh) ab 1 H. r a {b G a r b (modh)} = Ha Ha = Hb ab 1 H G H {Ha i i I} H\G G G H G = i I Ha i = Ha 1 + Ha Ha n {a i i I} H\G H\G H G (index) G : H 3 1
7 l a, b G a l b (modh) a 1 b H. l a {b G a l (modh)} = Ha H\G = {Ha i i I} Ha a 1 H {a 1 i H i I} = G/H H\G = G/H = G : H. 3.2 (Lagrange ) G H G G = G : H H. H G a G H = Ha r a : H Ha (h ha) G = n Ha i i=1 H G n = G : H Ha i Ha j = (i j) n n n G = Ha i = Ha i = H = n H = G : H H. i=1 i=1 3.3 G a G o(a) G a G = 1 i=1 o(a) = < a > o(a) G 3.4 G G G = 1 1 a G o(a) 1 o(a) G o(a) = G < a > = G G =< a > 3 2
8 3.1 S 3 = {1, (12), (13), (23), (123), (132)} 1, 2, 2, 2, 3, 3 S 3 = 6 S 3 1, 2, 3, S < (123) >=< (132) >= {1, (123), (132)} < (12) >, < (13) >, < (23) > 3 < (123) > S 3 H K G a, b G a b (mod(h, K)) b = hak for some h H, k K a HaK (H, K) G G (H, K) {Ha i K i I} H\G/K G = i I Ha i K G (H, K) 3.5 H, K G a G K K a 1 Ha K = j J (K a 1 Ha)k j {Hak j j J} HaK H G HaK = H K : K a 1 Ha. G = i I Ha ik {a i i I} G : H = i I K : K a 1 Ha i. k, k K Hak = Hak akk 1 a 1 H kk 1 K a 1 Ha (K a 1 Ha)k = (K a 1 Ha)k 3.6 HK = H K : K H = H K / K H. a = 1 3 3
9 4 G (cyclic group) G a G =< a >= {a n n bz} a G 4.1 G =< a > (1) 1 H G h = min{i N a i H} H =< a h > (2) G = n = ml < a l > G m (1) 1 a i H a i = (a i ) 1 H {i N a i H} = h = min{i N a i H} a h H < a h >= {(a h ) i i Z} H. a i H i = hq + r, q, r Z, 0 r < h a r = a i hq = (a i )(a h ) q HH H. h r = 0 a i = a hq < a h > (2) G =< a >= {1, a, a 2,..., a n 1 } 2.2 n n = ml m (a l ) m = < a l >= {1, a l, a 2l,..., a (m 1)l }, (a ml = 1) G m H G H = m h (1) H =< a h > a n = 1 H n h (1) H = n/h = mh = l 4.2 (Z, +) 1 a 1, a 2,..., a r Z d < a 1, a 2,..., a r >=< d >. a 1 x 1 +a 2 x 2 + +a r x r = d x 1, x 2,..., x r x 1, x 2,..., x r H =< a 1, a 2,..., a r >= {a 1 x 1 + a 2 x a r x r x 1, x 2,..., x r Z} Z 4.1 H =< c >= {cx x Z} a i H c a i c d c = a 1 x 1 + a 2 x a r x r d c c = ±d < c >=< d > m, n (m, n) 4 1
10 4.3 G =< a > n < a r >=< a (n,r) >, o(a r ) = n/(n, r). d = (n, r) d r < a r > < a d > 4.2 d = nx + ry x, y Z a d = (a n ) x (a r ) y = (a r ) y < a r >. < a d > < a r > 4.1 o(a d ) = n/d 4.1 (Z n, +) Z n = { 0, 1,..., n 1} ā + b = a + b n Z 6 < 0 >= {0}, < 1 >=< 5 >= Z 6, < 2 >=< 4 >= { 0, 2, 4}, < 3 >= { 0, 3}. 4.2 (Z n, ) ā b = a b n. Z n = U(Z n, ) Z n φ(n) φ (Euler function) ā Z n ā b = 1 for some b Z n ab + qn = 1 for some b, q Z (a, n) = 1 Z 6 = { 1, 5} φ(6) = 2 Z 5 = { 1, 2, 3, 4} φ(5) = 4 p φ(p) = p 1 Z p Z n 3.3 (a, 6) = 1 a 2 1 (mod6) (a, 5) = 1 a 4 1 (mod5) (a, n) = 1 a φ(n) 1 (modn) n 4 2
11 5 5.1 G N G a G a 1 Na = N N G N G Na = an G a N an Na 5.1 G N N G (an)(bn) = abn for all a, b G. ( ) anbn = abnn = abn H G HH = H ( ) a 1 a a b a a a 1 b a 1 Na a 1 NaN = a 1 an = N = a 1 ana 1 a a 1 ana 1 Na = a 1 Na N G an, bn G/N 5.1 anbn = abn G/N. G/N G/N G N (factor group) 1 G/N = N = 1N, (an) 1 = a 1 N. 5.1 Z nz Z a r b (mod nz) a b (mod n). Z/nZ = {nz, 1 + nz,..., n 1 + nz} (a + nz) + (b + bz) = (a + b) + nz = a + b + nz Z/nZ Z n 5.2 G 2 a H Ha = H = ah a H ah H Ha G : H = 2 G H G = H + Ha = H + ah ah = G H = Ha G a ah = Ha H G 5 1
12 5.2 S n : A n = 2 Exercise 3.5 ) A n S n S n /A n = {A n, (12)A n } 2 Z 2 ({±1}, ) 5.2 G S G 1. N G (S) = {x G x 1 Sx = S} S (normalizer) 2. C G (S) = {x G x 1 sx = s, for all s S} S (centralizer) 3. Z(G) = C G (G) G 4. S = {a} N G (S) = C G (S) C G (a) H G H G N G (H) = G. 5.3 G = S 3 H =< (123) >= {1, (123), (132)} = A 3 C G (H) = H N G (H) = G (12) 1 (123)(12) = (132) S G x G x 1 Sx = {x 1 sx s S} S x S x (conjugate) S, T G T = S x x G T S T G S G P(G) = 2 G G a G b ( {a} G {b}) a b a G b b = g 1 ag for some g G. N G a N g 1 ag g 1 Ng = N N G 5.4 S 3 {1}, {(12), (13), (23)}, {(123), (132)} S 3 1, A 3, S 3 τ = ( ( i τ = i τ ) ( ) i, σ = i σ ( ) i τ 1 τ = i ( )( )( ) ( ) i τ 1 τ i i i τ στ = =. i i σ i τ i στ ), σ = (123)(45) τ 1 στ = (512)(43). (Exersise 5.8 ) 5 2
13 6 G G f : G G f(ab) = f(a)f(b) (for all a, b G) G G G G f G G 6.1 ω = ( 1 + 3)/2 ω 1 3 Z Z/3Z + 3Z 1 + 3Z 2 + 3Z 3Z 3Z 1 + 3Z 2 + 3Z 1 + 3Z 1 + 3Z 2 + 3Z 3Z 2 + 3Z 2 + 3Z 3Z 1 + 3Z ({1, ω, ω 2 }, ) + 1 ω ω ω ω 2 ω ω ω 2 1 ω 2 ω 2 1 ω 6.2 G =< a >= {a n n Z} Z f : Z < a >= {a n n Z} (n a n ) 2.2 f(i + j) = a i+j = a i a j = f(i)f(j) Z + G 6.3 G =< a >= {1, a, a 2,..., a n 1 } n f : Z/nZ < a >= {1, a, a 2,..., a n 1 } (i + nz a i ) f i + nz = j + nz i j nz n i j a i = a j. i + nz j + nz a i = a j Z/nZ < a > f f ( ) well-defined f((i + nz) + (j + nz)) = f(i + j + nz) = a i+j = a i a j = f(i + nz)f(j + nz) f G n Z/nZ f(i+j +nz) = a i+j i+nz a i well-defined i + nz i 0, 1,..., n 1 f(i + j + nz) = a i+j 6 1
14 6.4 R R + f : R R + (a e a ). f(a + b) = e a+b = e a e b = f(a)f(b) f g : R + R (a log a) fg = id R + gf = id R f : X Y g : Y X gf = id X f g 6.1 G G f : G G f(ab) = f(a)f(b) (for all a, b G) f (homomorphism) 6.1 f : G G (1) f(1 G ) = 1 G (2) G a f(a 1 ) = f(a) 1 (3) Imf = {f(a) G} G f G (4) Kerf = f 1 (1 G ) = {a G f(a) = 1 G } G f G (1) f(1) = f(1 1) = f(1)f(1) f(1) 1 1 = f(1) (2) 1 = f(1) = f(aa 1 ) = f(a)f(a 1 ) f(a) 1 f(a) 1 = f(a 1 ) (3) f(a)f(b) = f(ab) Imf f(a) 1 = f(a 1 ) Imf Im G (4) a, b Kerf f(a) = f(b) = 1 f(ab) = f(a)f(b) = 1 f(a 1 ) = f(a) 1 = 1 ab Kerf a 1 Kerf Kerf G x G f(x 1 ax) = f(x 1 )f(a)f(x) = f(x 1 )f(x) = f(1) = 1 x 1 ax Kerf Kerf G 6.5 N G f : G G/N (a an) (cannonical homomorphism) f f(ab) = abn = anbn = f(a)f(b) 6 2
15 7 7.1 G G f : G G G/Kerf Imf. K = Kerf 6.1 K G a, b G f(a) = f(b) f(a)f(b) 1 = f(ab 1 ) = 1 ab 1 K Ka = Kb f(ka) = f(a) f well-defined Ka a f f(ka) = f(kb) Ka = Kb f f(kakb) = f(kab) = f(ab) = f(a)f(b) = f(ka) f(kb). Im f = Imf f : G/K Imf G/K Imf 7.2 (1) H G, N G NH/N H/H N. (2) f : G G H G H = f 1 (H ) H G G/H G /H (1) f : H G/N (h Nh) f Imf = f(h) = NH/N, Kerf = H N 7.1 H/H N NH/N (2) g : G f G G /H, a f(a) H f(a) Kerg = {a G H f(a) = H } = {a G f(a) H } = f 1 (H ) 7.1 G/H G /H 7.3 H, N G N H G/H (G/N)/(H/N) f : G G/N f 1 (H/N) = H 7.2 (2) 7 1
16 7.1 G =< a > f : Z G =< a >= {a n n Z} (i a i ) f(i + j) = a i+j = a i a j = f(i)f(j). 7.1 Z/Kerf < a > Kerf Z 4.1 Kerf Kerf nz n = 0 G < a > Z/{0} Z n 0 G n Z/nZ 7.2 f : R C (a e 2πai ) Imf = {z C z = 1} Kerf = {a R e 2πai = 1} = Z R/Z S GL(n, R) n n {( ) } (( ) ) A O A O f : G = A GL(r, R), C GL(s, R) GL(r, R) A. B C B C {( Kerf = I B ) } O C GL(s, R) G. C I r 7.1 G/Kerf GL(r, R) G N G H N f : G G/N (a an) f(h) = {hn h N} = HN/N G/N H G/N H = f 1 ( H) = {h G hn H} N H G f(h) = H/N = H S(G, N) = {H G N H} G N S(G/N, 1) = { H H G/N} G/N S(G, N) H f(h) = H/N S(G/N, 1) S(G/N, 1) H f 1 ( H) S(G, N) 7.4 Z Z/12Z S(Z, 12Z) = {nz 12Z nz, n 12} = {12Z, 6Z, 4Z, 3Z, 2Z, Z} S(Z/12Z, 0) = {mz/12z n 12} = {0, 6Z/12Z, 4Z/12Z, 3Z/12Z, 2Z/12Z, Z/12Z} 7 2
17 8 8.1 G X f : X G X, ((α, a) X G f(α, a) = α a ) α 1 = α, α ab = (α a ) b G X X G- X G-α, β X α G β α a = β for some a G G G- (G-orbit) α G- {α a a G} α G α G G- X = α G transitive G α = {a G α a = α} G α G β = α a x G α G β = a 1 G α a β a 1 xa = α aa 1 xa = α xa = α a = β G β a 1 G α a α = β a 1 G α ag β a 1 a 1 G α a G β G α α G 8.1 α G = G : G α a, b G α a = α b α ab 1 = α ab 1 G α G α a = G α b φ : G α \G α G (G α a α a ) X G-a G σ(a) : X X (α α a ) σ(a) X σ : G S X (a σ(a)) σ Kerσ Ker(X, G) G X σ : G S X X G X ((α, a) α σ(a) ) X G- G G GL(n, C) G 1 8 1
18 8.1 H S X σ : H S X X H X, ((α, h) α h ) X H S n {1, 2,..., n} H S X X H S n n 8.2 H G G H\G f : H\G G H\G, ((Hx, a) Hxa) (Hx) 1 = Hx1 = Hx, (Hx) ab = Hxab = (Hxa) b = (Hxa) b = ((Hx) a ) b Ker(H\G, G) = {a G Hxa = Hx for all x G} = x G x x Hx G H = 1 Ker(G, G) = 1 σ : G S G G S G 8.3 X = G G G G ((x, a) a 1 xa = x a ) G G- G x = {a G x a = x = {a G a 1 xa = x} = C G (x) x G x 8.1 x G = G : C G (x) G Ker(G, G) = {a G x a = x, x G for all x G} = Z(G) G x G = 1 G : C G (a) = 1 x Z(G). 8 2
19 9 9.1 G G = E 1 E 2 E m I 1 I 2 I r, E i Z ei e i > 1 i = 1, 2,..., m e i e i+1 i = 1, 2,..., m 1 I j Z j = 1, 2,..., r 1 (e 1, e 2,..., e m ; r) G 9.2 G =< x 1, x 2,..., x n > y 1 = x a 1 x a2 x a n a i Z (i = 1, 2,..., n) (a 1, a 2,..., a n ) = 1 G =< y 1, y 2,..., y n > y 2,..., y n m = a 1 + a a n m = 1 i a i = ±1 y 1 = x ±1 i m > 1 (a 1, a 2,..., a n ) = 1 a i a j > 0 a i, a j a i ɛa j < a i ɛ = ±1 y 1 = x a 1 1 x a i ɛa j i (x j x ɛ i) aj x a n n b k = a k (k i) b i = a i ɛa j z l = x l (l j) z j = x j x ɛ i (b 1, b 2,..., b n ) = (a 1,..., a i 1, a i ɛa j, a i+1,..., a n ) = (a 1,..., a n ) = 1 G =< x 1,..., x n >=< x 1,..., x j 1, x j x ɛ i, x j+1,..., x n >=< z 1, z 2,..., z n >. y 1 = z b 1 1 z b n n b b n < m y 2,..., y n G =< y 1,..., y n > Proof of Theorem 9.1 G x 1,..., x n 1. G =< x 1,..., x n > n 2. G =< x 1,..., x i 1, y i,..., y n > x i y i 9 1
20 (x 1 1 x i 1 x i x 1,..., x i 1 y i ) < x i >= E i E i = e i G = E 1 E n e i e i+1 φ : E 1 E n G, ((x a 1 1,..., x an n ) xx a 1 1 x an n ) x a i i x a j j = 1 0 < a k e k d = (a i,..., a j ) a k = db k (b i,..., b j ) = y i = x b i i x b j j, y i+1,..., y n < x i,..., x n >=< y i,..., y n > y i y d i = x db i i x db j j = x a i i x a j j = 1. y i d x i e i o(y i ) d a i e i. e i = d = a i x a i i = 1 φ x e i i xe i+1 i+1 = 1 d = (e i, e i+1 ) e i = d e i e i+1 e 1,..., e m e m+1 G = E 1 E m I 1 I r E i Z ei I j Z e i e i+1 1 < e i G = E 1 E m I 1 I r = E 1 E m I 1 I r E i =< x i > I j =< z j > I j =< w j > E i =< y i > e i = E i e i = E i T (G) G T (G) = E 1 E m = E 1 E m p e 1 p e i+1 T (G) p = {x T (G) x p = 1} =< x e 1/p 1,..., x em/p m >=< y e i+1 /p i+1,..., y e m /p m > T (G) p = p m = p m i m m m m m = m p e 1 T (G) p = {x p x T (G)} =< x p 1,..., x p m >=< y p 1,..., y p m > 9 2
21 e 1 /p,..., e m /p e 1/p,..., e m/p e i = e i e = e m G e = {x e x G} =< z e 1,..., z e r >=< w e 1,..., w e r > G 2e = {x 2e x G} =< z1 2e,..., zr 2e >=< w1 2e,..., wr 2e > G e /G 2e 2 r = 2 r r = r 9.1 G = e 1 e 2 e r e 1 e 2, e 2 e 3... e r 1 e r e r 12(12), (2, 6) G Z Z 2 Z 6 9 3
22 10 G 10.1 p G p G p ( G = p r ) G = p n g, (p, g ) = 1 p n G P G p- (Sylow-p-group) P G P = p n 1. p p 2. p p 10.1 p G G p G p G 1 G p G : H p Z(G) Z(G) 1 Exercise p p r G G p r G p G G 1 Case 1. G H ( G : H, p) = 1 p r G = G : H H p r H H P p r P G Case 2. G p G : H 10.2 p Z(G) Z(G) 10.1 Z(G) p x A =< x > A = < x > = o(x) = p A Z(G) A G G/A p r G = G : A A = p G/A p r 1 G/A G/A p r 1 P 1 π : G G/A P = π 1 (P 1 ) A P P/A P 1 P = p r Syl p (G) G p 10 1
23 10.4 G p (1) H G p G p P H (2) P, Q G p Q = g 1 P g G g (3) Syl p (G) 1 (mod p) p kp + 1 (k Z) (1) P Syl p (G) H G p (P, H) G G = P a 1 H + P a 2 H + + P a r H 3.5 G : P = r i=1 H : H a 1 i P a i P G p p p ( H : H a 1 i P a i, p) = 1 H p H = H : H a 1 i P a i H a 1 i P a i H : H a 1 i a 1 i P a i = 1 H = H a 1 i P a i P G a 1 i P a i H a 1 i P a i P a i G p H G p (2) Q = H (1) G g = a i Q g 1 P g Q = P = g 1 P g Q = g 1 P g (3) Syl p (G) G Syl p (G), ((P, a) a 1 P a) a 1 P a Syl p (G) Syl p (G) G-(2) P Syl p (G) N 8.1 N = {a G P a = P } = {a G a 1 P a = P } = N G (P ) Syl p (G) = P G = G : N = G : N G (P ) G (N, P ) b 1 = G = Nb 1 P + Nb 2 P + + Nb r P G : N = r i=1 P : P b 1 i Nb i 10 2
24 P N G (P ) = N G : N G : P G : N p p P : P b 1 i Nb i = 1 P = P b 1 i Nb i P b 1 i Nb i b i P b 1 i N b i P b 1 i Syl p (N) b i P b 1 i = n 1 P n = P n N b i N = N G (P ) Nb i P = N1P (= N) b i = b 1 b i P : P b 1 i Nb i p P : P b 1 1 Nb 1 = P : P N = P : P = 1 Syl p (G) = G : N G (P ) = kp
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