Genel matemtaik - soyut cebir n> 5 A n G N G f : G! G=N G H G=N =H N G f : G! H G= (f) =H K = (f) G - f H G=K @ @ @ R r g - f f(g) Kg @ @ @ R r g2 G r(Kg) = f(g) r : G=K! H r r X = Kg X = Kg 0 r(X) = f(g) r(X) = f(g 0 ) f(g) =f(g 0 ) X =Kg =Kg 0 =) k 1 ;k 2 2K k 1 g =k 2 g 0 =)g =k 1 1 k 2 g 0 =k 3 g 0 =)f(g) =f(k 3 g 0 ) =f(k 3 )f(g 0 ) =e H f(g 0 ) =f(g 0 ) k 3 2 (f)) r r(XY ) =r(X)r(Y ) X;Y 2G=K X =Kg Y =Kg 0 r(XY ) =r(KgKg 0 ) =r(Kgg 0 ) K G=f(gg 0 ) =f(g)f(g 0 ) =r(Kg)r(Kg 0 ) =r(X)r(Y ) =)r r h2H X2G=K r(X) =h h2H f g 0 2G h =f(g 0 ) X =Kg 0 r(X) =r(Kg 0 ) =f(g) =h r r r G=K (r) =fKg r X2 (r) =)X =Kg =)r(X) =e H =)r(Kg) = e H =)f(g) =e H =)g2 (f) =K =)X =Kg =K =) (r) =fKg G=K =H f : G ! H G= (f) = f(G) f f(G) =H G = (Z; +);H = (Qnf 0g;:) f :G! H x2G f(x) = ( 1; x 1; x f : G ! f(G) f x;y2Z x y =)f(x +y) = 1 f(x)f(y) = 1 1 = 1 x y =)f(x +y) = 1 f(x)f(y) = ( 1) ( 1) = 1 x y =)f(x +y) = 1 f(x)f(y) = 1 =) (f) =fx2Zjf(x) = 1g =fx2Zjx g = 2Z (f) f (f); (f) + 1g G= (f) =f (f); (f) + 1g: G - f f(G) =f 1; 1g G= (f) : f(G)! G= (f) ( 1) = (f) + 1; (1) = (f) G= (f) =f(G)G K f : G! K jGj jf(G)j jGj G - f f(G) K G= (f) @ @ @ R G= (f) =f(G) G G= (f) jG= (f)j =jf(G)j jGj j (f)j =jf(G)j =) jGj jf(G)j =j (f)j =)jf(G)j jGj: G =GL n (R) G G 0 = (Rnf 0g;:) f :G! G 0 ; f(X) = det(X) f (f) G= (f) GL n (R)=SL n (R) =Rnf 0g X;Y 2 G =) f(XY ) = det(XY ) = det(X) det(Y ) = f(X)f(Y ) =) (f) = fX2Gjf(X) = det(X) = 1g = SL n (R) G= (f) = f (f):Y jY 2Gg f G - f G 0 =f(G) G= (f) @ @ @ R ( (f) X) = f(X) ( (f) X (f) Y ) = ( (f) X Y ) =f(XY ) =f(X)f(Y ) = ( (f) X) ( (f) Y ) 8 y2G 0 9 (f) X2G= (f) ( (f)X) =y f X2G f(X) =y (X) = (f)X2G= (f) ( (f)X) =f(X) =y ( (f)X) = ( (f)Y ) =)f(X) =f(Y ) =)f(X)f(Y ) 1 = 1 =)f(XY 1 ) = 1 =)XY 1 2 (f) =) (f)XY 1 = (f) =) (f)X = (f)Y G =f a;b j a;b :R! R;8x2R a;b (x) =ax +b;a6= 0g 3;5 (x) = 3x + 5 f :G! G;f( a;b ) = a;0 f G= (f) =f(G) f [ a;b c;d ](x) = a;b (cx+d) =a(cx+d)+b =acx+(ad+b) a;b c;d = ac;ad+b f( a;b c;d ) =f( ac;ad+b ) = ac;0 = a;0 c;0 =f( a;b ) f( c;d ) =) f (f) G ( 1;0 )(x) =x (f) =f a;b jf( a;b ) = a;0 = 1;0 g =f 1;b jb2Rg G= (f) =f (f) a;b j a;b 2Gg G - f f(G) G= (f) @ @ @ R ( (f) a;b ) =f( a;b ) f : G! H T 6 H S =fx2Gjf(x)2Tg S6G (f) S T H S G S6G x;y2S f(xy 1 ) =f(x)f(y 1 ) =f(x) |{z} 2T f(y) 1 | {z } 2T 2T =)f(xy 1 )2T =)xy 1 2S =)S6G(f) S k2 (f) =)f(k) =e H 2T T6H) =)k2S =) (f) S: T H =)S G T H s2S g2G gsg 1 2S f(gsg 1 ) =f(g)f(s)f(g 1 ) =f(g) |{z} h2H f(s) |{z} 2T f(g) 1 | {z } h 1 2T (T H ) =)f(gsg 1 )2T =)gsg 1 2S =)S G y (f) &% ’$ ’ & $ % S G &% ’$ ’ & $ % T H - f G L H L6G; (f) L L 0 =fh2Hj l2L f(l) =hg L 0 6H y (f) &% ’$ ’ & $ % L G &% ’$ ’ & $ % L 0 H - f x q - L L 0 x2GnL f(x)2 L 0 l2 L f(x) = f(l) =) f(x)f(l) 1 = e H =) f(xl 1 ) = e H =) xl 1 2 (f) L =)xl 1 =l 1 2L =)x =l 1 l2L x2LH G (f) G (f) H f : G! H S f=S : S! T S= (f) = T G H6G N G H\N H HN6G H= (H\N) =HN=N H\N H n2 H\N h2 H hnh 1 2 H\N hnh 1 2 H N G h2G hnh 1 2N hnh 1 2H\N =)H\N H HN6G x =h 1 n 1 ;y =h 2 n 2 2HN xy 1 = (h 1 n 1 )(h 2 n 2 ) 1 =h 1 n 1 n 2 1 h 2 1 =h 1 n 1 h 2 1 h 2 n 2 1 h 2 1 | {z } 2N =h 1 n 1 h 2 1 | {z } 2Nh 2 1 n 3 =h 1 h 2 1 n 4 n 3 (N G Nh 2 1 =h 2 1 N) =h 3 n 5 2HN =)HN6G f : H! HN=N h2 H f(h) = Nh f Nhn2HN=N N G Nhn =hNn =hN =Nh =f(h) f f h 1 ;h 2 2H f(h 1 h 2 ) =Nh 1 h 2 =Nh 1 Nh 2 (N G ) =f(h 1 )f(h 2 ) =)f H= (f) =HN=N(f) =H\N HN=N N (f) =fx2Hjf(x) =Ng =fx2HjNx =Ng =fx2Hjx2Ng (Nx =N()x2N) =H\N H=(H\N) =HN=N f : G! H T H S = fx2Gjf(x)2Tg G=S =H=T G=S = G= (f) = S= (f) fG - H H=T @ @ @ R g - f f(g) Th Tf(g) @ @ @ R T H H! H=T (h) =Th :G! H=T (g) =Tf(g) Th2 H=T f 9g 0 2 G f(g 0 ) = h (g 0 ) = Tf(g 0 ) = Th g 1 ;g 2 2G (g 1 g 2 ) =Tf(g 1 g 2 ) =Tf(g 1 )f(g 2 ) =Tf(g 1 )Tf(g 2 ) (T H ) = (g 1 ) (g 2 ) =) =) G= ( ) =H=T ( ) =S k 2 ( ) =) (k) = T =) Tf(k) = T =) f(k)2 T =) k 2 S =) ( ) S s 2 S =) f(s) 2 T (s) = Tf(s) = T s 2 ( ) =) S ( )S = ( ) G=S =H=T G= (f) =H S= (f) =T f(S) =T G=S = G= (f) = S= (f) G I(G);G I(G) O(G) G=Z(G) =I(G) Z(G) :G I(G) = t a ja2G;t a :G! G;t a (x) =axa 1 I(G)6O(G) t a ;t b 2I(G) t b t b 1 t a (t b ) 1 (x) =t a (t b ) 1 (x) =t a [t b 1(x)] =at b 1(x)a 1 =a(b 1 xb)a 1 = (ab 1 )x(ba 1 ) =cxc 1 (c =ab 1 ) =t c (x)2I(G) =)I(G)6O(G) I(G) O(G) f2O(G) t a 2I(G) ft a f 1 2I(G) G=Z(G) =I(G) f :G! I(G);f(g) =t g f f t g 2I(G) f(g) =t g f f t g 1 g 2 (x) =g 1 g 2 xg 2 1 g 1 1 =t g 1 (g 2 xg 2 1 ) =t g 1 t g 2 (x) = [t g 1 t g 2 ](x) t g 1 g 2 =t g 1 t g 2 f(g 1 g 2 ) =t g 1 g 2 =t g 1 t g 2 =f(g 1 )f(g 2 ) f =) G= (f) =I(G) (f) =Z(G) I k2 (f) =)f(k) =I =)t k =I =)t k (x) =I(x) =)kxk 1 =x =)k2Z(G) =) (f) Z(G) z2Z(G) =)8x2G;zxz 1 =x =)t z (x) =x =I(x) =)t z =I =)z2 (f) =)Z(G) (f) Z(G) = (f) G=Z(G) =I(G) (Z(G) g) =t g :Z 12 ! Z 3 (1) = 2 K = ( ) Z 12 =K :Z 12 =K! Z 3 (1) = 2 =) (2) = (1) + (1) = 2 + 2 = 1 =) (3) = (2) + (1) = 2 + 1 = 0; (4) = (2) + (2) = 1 + 1 = 2; (5) = (4) + (1) = 2 + 2 = 1; (6) = (5) + (1) = 1 + 2 = 0; (7) = (6) + (1) = 0 + 2 = 2: (8) = 1; (9) = 0; (10) = 2; (11) = 1 (0) = 0 K = ( ) =fx2Z 12 j (x) = 0g =f 0; 3; 6; 9g K + 1 =f 1; 4; 7; 10g;K + 2 =f 2; 5; 8; 11g K + 3 =K Z 12 =K =fK;K + 1;K + 2gZ 12 - Z 3 Z 12 =K @ @ @ R (K + 0) = (0) = 0; (K + 1) = (1) = 2; (K + 2) = (2) = 1 Z 24 H =< 4 >6 Z 24 N =< 6 > Z 24 H\N H (H +N)6Z 24 H=(H\N) = (H +N)=N H =< 4 >=f 0; 4; 8; 12; 16; 20g N =< 6 >=f 0; 6; 12; 18g H\N =f 0; 12g (H\ N)6H H =)(H\N) H H +N =fh +njh2H;n2Ng =f 0; 6; 12; 18; 4; 10; 16; 22; 8; 14; 20; 2g H +N6Z 24 (H +N)=N = fN +mjm2H +Ng = fN;N + 4;N + 8g f : H ! (H +N)=N; f(h) = N +h f f(0) = f(12) = N f(4) =f(16) =N + 4 f(8) =f(20) =N + 8 (f) =fh2Hjf(h) =Ng =f 0; 12g (f) =H\N H= (f) = (H +N)=N H=(H\N) = (H +N)=N: G =Z 12 ;H =Z 4 ;f : G! H;f(1) = 1 f T =< 2> Z 4 S =fx :f(x)2Tg G=S =H=T H=T =Z 4 =< 2>=f< 2>;< 2> +1g :G! H=T =Z 4 =< 2> (g) =T +f(g) f f(0) =f(4) =f(8) = 0; f(1) =f(5) =f(9) = 1; f(2) =f(6) =f(10) = 2; f(3) =f(7) =f(11) = 3: (0) =T +f(0) =T + 0 =T; (1) =T +f(1) =T + 1 =T + 1; (2) =T +f(2) =T + 2 =T; (3) =T +f(3) =T + 3 =T + 1; ( ) =fx2Z 12 j (x) =Tg =f 0; 2; 4; 6; 8; 10g S =f 0; 2; 4; 6; 8; 10g ( ) =S G - H=T G=S @ @ @ R G=S =H=T =Z 4 =< 2> f :Z 9 ! Z 3 ;f(1) = 1 (f) Z 9 = (f) Z 9 = (f) =Z 3 :Z 18 ! Z 18 ; (1) = 10 ( ) Z 18 = ( ) Z 18 = ( )! (Z 18 ) :C! D (C) =fd2Dj9c2C;d = (c)g D C= ( ) =D d 1 ;d 2 2 (C) d 1 d 2 1 2 (C) d 1 2 (C) =)9c 1 2C (c 1 ) =d 1 d 2 2 (C) =)9c 2 2C (c 2 ) =d 2 d 2 1 = [ (c 2 )] 1 = (c 2 1 ) d 1 d 2 1 = (c 1 ) (c 2 1 ) = (c 1 c 2 1 ) (c =c 1 c 2 1 ) = (c) =)d 1 d 2 1 2 (C) =) (C)6D C= ( ) =DM6=; P (M) M M M P (M) jMj = n P (M) = S n f;g2 S n f g2 S n f2 S n f 1 f 1 2S n I2S n a i 2M I(a i ) =a i I = a 1 a 2 ::: a n a 1 a 2 ::: a n ! : I S;n S n S S n n jS n j =n! f2S n f = a 1 a 2 ::: a n f(a 1 ) f(a 2 ) ::: f(a n ) ! : G G P (G) n S n a2G h a :G! G h a (x) =ax h a a h a G h a (x) =h a (y) =)ax =ay =)x =y =)h a y2G =)x =a 1 y h a (x) =a(a 1 y) =y =)h a h a (h a ) 1 =h a 1 (h a h a 1)(x) =h a (h a 1(x)) =h a (a 1 x) =aa 1 x =x; (h a 1h a )(x) =h a 1(h a (x)) =h a 1(ax) =a 1 ax =x: f : G ! P (G) a 2 G f(a) = h a f f a;b2G f(a) =f(b) =)h a =h b =)8x2G;h a (x) =h b (x) =)8x2G;ax =bx =)a =b: f a;b2G x2G [f(ab)](x) =h ab (x) = (ab)x =a(bx) =h a (bx) =h a (h b (x)) = (h a h b )(x) = [f(a)f(b)](x) f(ab) = f(a)f(b) f : G! P (G) f : G! f(G) f(G)6P (G) G =f(G) G P (G) G =f 1; 1;i; ig h 1 = 1 1 i i 1 1 i i ! ;h ( 1) = 1 1 i i 1 1 i i ! h i = 1 1 i i i i 1 1 ! ;h ( i) = 1 1 i i i i 1 1 ! P (G) 4! = 24 P (G) h 1 h ( 1) h i h ( i) h 1 h 1 h ( 1) h i h ( i) h ( 1) h ( 1) h 1 h ( i) h i h i h i h ( i) h ( 1) h 1 h ( i) h ( i) h i h 1 h ( 1) h G f :G! f(G);f(a) =h a U;P (M) x;y2 M f(x) = y f2 U x y U x U ! y U;P (M) ;6=T M U T x;y2T x U ! y x2T x U ! z z2MnT x U ! y ’ & $ % ’ & $ % T M z M =fa 1 ;a 2 ;:::;a n g fa 1 ;a 2 ;:::;a n g =fa i1 ;:::;a ir g[fa i;r+1 ;:::;a is g[::: f2 S n S n U =hfi m f m (a 1 ) = a 1 a 1 ;f(a 1 );f 2 (a 1 );:::;f m 1 (a 1 ) U M M =f 1; 2;:::;ng =fa 1 ;a 2 ;:::;a r g[fb 1 ;b 2 ;:::;b s g[::: b 1 ;f(b 1 );f 2 (b 1 );:::;f s 1 (b 1 ) i = 1; 2;:::;r 1 a i+1 =f(a i ) f(a r ) =a 1 i = 1; 2;:::;s 1 b i+1 =f(b i ) f(b s ) =b 1 : f a 1 ;a 2 ;:::;a r a2M a a = n b2Mj k2Z f k (a) =b o = a;f(a);f 2 (a);:::;f m 1 (a) a a a;f(a);f 2 (a);:::f m 1 (a) a i2f 1; 2;:::;ng (i) S 3 f 4 = a b c b c a ! f 4 a;b c f 4 f 4 3 = (1) a f 4 a;f 4 (a);f 4 2 (a) =fa;b;cg b f 4 b;f 4 (b);f 4 2 (b) =fb;c;ag c f 4 c;f 4 (c);f 4 2 (c) =fc;a;bg fa;b;cg f 4 f 4 = (abc) f 3 = a b c b a c ! a = b =fa;bg c =fcg f 3 f 3 = (ab)(c) M =fa;b;c;d;eg f = a b c d e b c a e d ! 2 S 5 a = b = c =fa;b;cg d = e =fd;eg f f = (abc)(de) f = a 1 a 2 a 3 ::: a n a 2 a 1 a 3 ::: a n ! a 1 a 2 f = (a 1 a 2 ) = a b c d e f b a c d f e ! = (ab)(ef) = a b c d e f b a c e f d ! = (ab)(def) f2S n f(a);f 2 (a);:::;f m (a) a f a f f2S n f f(a);f 2 (a);:::;f m (a) f(a);f 2 (a) f(a);f 3 (a) ::: f(a);f m (a) f (1; 2)(1; 3) = (1; 2; 3): (1; 2)(1; 3) = (1; 3; 2) f g x f g (fg)(i) =g(f(i)) x f f(x) xf x(fg) = (xf)g f 4 2S 3 f 4 = (abc) = (ab)(ac) = (ca)(cb) (ab)(ac) = a b c b a c ! a b c c b a ! = a b c b c a ! = (abc): S n A n A n n A 1 =A 2 f (1)g;A 3 =f (1); (1; 2; 3); (1; 3; 2)g A 4 =f (1); (123); (132); (124); (142); (134); (143); (234); (243); (12)(34); (13)(24); (14)(23)gA n S n n> 2 jA n j = n! 2 A 1 A 2 f 1; 1g :S n !f 1; 1g ( ) = ( 1; 1; ( ) = A n S n =A n =f 1; 1g jA n j = n!=2 f = (a 1 ;a 2 ;:::;a r ) =)f 1 = (a r ;a r 1 ;:::;a 2 ;a 1 ) fa 1 ;a 2 ;:::;a r g fb 1 ;b 2 ;:::;b s g (a 1 ;a 2 ;:::;a r )(b 1 ;b 2 ;:::;b s ) = (b 1 ;b 2 ;:::;b s )(a 1 ;a 2 ;:::;a r ) n n m n (m;n) 2S n 1 i j i = j 1 i j i ( 1 ) =i( ) = (i ) =j = (1; 5; 6)(2; 3) = (1; 4; 3) 1 = (4; 5; 6)(2; 1) ; 2S n 1 = 2 S n 1 i i 1 = S 7 = (1; 3; 4)(2; 6)(5; 7) = (1; 6)(2; 5; 7)(3; 4) = (1; 3; 4)(2; 6)(5; 7) # # # = (2; 5; 7)(3; 4)(1; 6) = (1; 2; 3; 5)(4; 7; 6) 1 = = (5; 7; 2)(1; 6)(4; 3) n> 5 A n n> 5 A n H A n H6=f (1)g H =A n n> 5 A n (a 1 ;a 2 ) e;a 1 ;a 2 (a 1 ;a 2 ) = (e;a 1 )(e;a 2 )(e;a 1 ) A n (a 1 ;a 2 )(b 1 ;b 2 ) n> 5 16 e6 n e (a 1 ;a 2 )(b 1 ;b 2 ) = (e;a 1 )(e;a 2 )(e;a 1 )(e;b 1 )(e;b 2 )(e;b 1 ) = (e;a 1 ;a 2 )(e;a 1 ;b 1 )(e;b 2 ;b 1 ) H A n H H =A n (a;b;c)2H x;y;z2f 1; 2;:::;ng x;y;z 2S n a x b y c z 1 (a;b;c) = (x;y;z) H A n (x;y;z)2H n> 5 a;b;c e;f = ( e;f) 1 (a;b;c) = 1 (e;f)(a;b;c)(e;f) = 1 (a;b;c) = (x;y;z) (x;y;z)2H x;y;z H S n H =A n H A n H H =A n = (a 1 ;a 2 )(a 3 ;a 4 )2H n> 5 e2f 1; 2;:::;ng = (a 1 ;a 2 ;e) 2A n 1 = (a 2 ;e)(a 3 ;a 4 )2H 1 1 = (a 1 ;a 2 )(a 3 ;a 4 )(a 2 ;e)(a 3 ;a 4 ) = (a 1 ;a 2 )(a 2 ;e) = (a 1 ;e;a 2 )2H H H =A n n> 5 A n H A n H 6= f (1)g (1) 6= 2 H 1 ; 2 ::: k = 1 2 k 1 ; 2 ::: k i = 1; 2;:::;k 1 i > i+1 1 = (a 1 ;a 2 ;:::;a m ) m> 3 = (a 1 ;a 2 ;a 3 ) 2A n 2 ; 3 ;:::; k 1 2A n H A n = 1 1 = k 1 ::: 1 1 (a 2 ;a 3 ;a 1 ;a 4 ;:::;a m ) 2 ::: k = 1 1 (a 2 ;a 3 ;a 1 ;a 4 ;:::;a m ) = (a m ;a m 1 ;:::;a 2 ;a 1 )(a 2 ;a 3 ;a 1 ;a 4 ;:::;a m ) = (a 1 ;a 2 ;a 4 ) 2Hm = 3 2 1 = (a 1 ;a 2 ;a 3 ) 2 = (a 4 ;a 5 ;a 6 ) = (a 2 ;a 3 ;a 4 )2A n 1 1 = 1 1 2 1 ::: k 1 (a 1 ;a 3 ;a 4 )(a 2 ;a 5 ;a 6 ) 3 4 ::: k = 1 1 2 1 (a 1 ;a 3 ;a 4 )(a 2 ;a 5 ;a 6 ) = (a 1 ;a 3 ;a 2 )(a 4 ;a 6 ;a 5 )(a 1 ;a 3 ;a 4 )(a 2 ;a 5 ;a 6 ) = (a 1 ;a 4 ;a 2 ;a 3 ;a 5 )2H m = 3 2 ;:::; k 1 = (a 1 ;a 2 ;a 3 ) 2 = (a 1 ;a 2 ;a 3 ) 2 = (a 1 ;a 3 ;a 2 )2H i 1 = (a 1 ;a 2 ); 2 = (a 3 ;a 4 ) = (a 2 ;a 3 ;a 4 ) ( 1 ) 1 = (a 1 ;a 3 )(a 4 ;a 2 ) 3 ::: k k 1 ::: 1 1 = (a 1 ;a 3 )(a 4 ;a 2 )(a 1 ;a 2 )(a 3 ;a 4 ) = (a 1 ;a 4 )(a 2 ;a 3 )2H R f(x) =x + 1 g(x) =x 2 h(x) =x 3 = 1 2 3 4 5 6 7 8 3 7 5 6 1 2 8 4 ! S 3 A =h(1; 2)i B =h(1; 3)i AB S 3 b S n =ff2S n jf(n) =ng S n S n 1 f = 1 2 3 4 5 6 2 1 4 5 6 3 ! 2 S 6 S =f 1; 2; 3; 4; 5; 6g f f n> 3 S n H6S n H S 5 = (1; 2; 3)(4; 5) A 3 A n =S n n = 1 f h R R R g = (1; 3; 5)(2; 7; 8; 4; 6) = (1; 3)(1; 5)(2; 7)(2; 8)(2; 4)(2; 6) A =f (1); (1; 2)g;B =f (1); (1; 3)g AB =f (1); (1; 2); (1; 3); (1; 2; 3)g (1; 3)(1; 2) = (1; 3; 2) = 2AB AB S =f 1; 2;:::;n 1;ng b S n f;g2 b S n g 1 2 b S n fg 1 (n) =g 1 (f(n)) =g 1 (n) =n fg 1 2 b S n b S n 6S n b S n =S n 1 :S n 1 ! b S n g2 S n 1 (g) = g (n) g (n) (f) = (g) =)f (n) =g (n) =)f =g (fg) =fg (n) =f (n) g (n) = (f) (g) g 0 2 b S n g 0 =g (n) g2S n 1 (g) =g 0 1 = 2 =f 1; 2g; 3 = 4 = 5 = 6 =f 3; 4; 5; 6g f (1; 2); (2; 1); (3; 4; 5; 6); (4; 5; 6; 3); (5; 6; 3; 4); (6; 3; 4; 5) f = (1; 2)(3; 4; 5; 6) = (1; 2)(3; 4)(3; 5)(3; 6) n> 3 = (1; 2)2 S n = (1; 3)2 S n = (1; 2; 3) = (1; 3; 2) S n H6S n H 2H H =H H H H C = 2S 5 : 1 = C =f (1); (1; 2; 3); (1; 3; 2); (1; 2; 3)(4; 5); (1; 3; 2)(4; 5); (4; 5)g: (1) (123) (132) (1) (1) (123) (132) (123) (123) (132) (1) (132) (132) (1) (123) n > 1 S n = (1; 2) A n 6= S n A 1 = S 1 A n =S n =)n = 1R6=; R \ (R; +; ) (R; +) a;b;c2R a (b c) = (a b) c (R; ) a;b;c2R a (b +c) = (a b) + (a c) (a +b) c = (a c) + (b c) a;b2R a b =b a R a2R a e =e a =a e2R e R a b ab R 1 R 1 (Z; +; ); (Q; +; ); (R; +; ); (C; +; ) S = x +y p 2jx;y2Z T = u +v 3 p 2 +w 3 p 4ju;v;w2Q (Z n ; +; ) c a +b n a +b =c d a b n a b =d T =f 0;eg e e e e e e e (T; +; ) (R; +; ) 0 R 0 a + a a + ( b) a b L = Z Z Z =f (a;b;c)ja;b;c2Zg (a;b;c) + (d;e;f) = (a +d;b +e;c +f); (a;b;c)(d;e;f) = (ad;bd +ce;cf): (L; +; ) (L; +) (0; 0; 0) (a;b;c) (a;b;c) = ( a; b; c) (a;b;c) (d;e;f)(g;h;i) = (a;b;c) (dg;eg +fh;fi) = (adg;bdg +ceg +cfh;cfi) = (ad;bd +ce;cf)(g;h;i) = (a;b;c)(d;e;f) (g;h;i) (a;b;c) (d;e;f) + (g;h;i) = (a;b;c) (d +g;e +h;f +i) = (ad +ag;bd +bg +ce +ch;cf +ci) = (a;b;c)(d;e;f) + (a;b;c)(g;h;i) (L; +; ) (a;b;c)(x;y;z) = (a;b;c) =) (ax;bx +cy;cz) = (a;b;c) =)ax =a;bx +cy =b;cz =c =)x = 1;z = 1;y = 0 (1; 0; 1)(a;b;c) = (a;b;c (L; +; ) (1; 0; 0)(0; 1; 0) = (0; 0; 0) (0; 1; 0)(1; 0; 0) = (0; 1; 0) (L; +; ) (a +b) c =c (a +b) =c a +c b =a c +b c: 02 R 0 0 2 R R 0 a2 R 0 +a = a a = 0 0 0 + 0 0 = 0 0 0 0 a2R a + 0 0 =a a = 0 0 + 0 0 = 0 0 = 0 0 R a;b;c2R a +c =b +c a =b c +a =c +b a =b a +x =b x =b a R a (R; +; ) (Rnf 0g; ) (R; +; ) (R; +; ); (C; +; ); (Q; +; ) (Z; +; ) R a;b2R a0 = 0a = 0 ( a)b =a( b) = (ab) ( a)( b) =ab a0 =a(0+0) =a0+a0 0+a0 =a0+a0 0 =a0 0a = 0 ( a)b +ab = ( a +a)b = 0b = 0 (ab) = ( a)b a( b) = (ab) ( a)b =a( b) b ( b) ( a)( b) =a( ( b)) =ab O R = 1 R R =f 0g O R = 1 R a2R 0 = 0 a = 1 a =a R =f 0g R ;6=S R S;R S R Z Q Q R 2Z Z R ;6=S R S () S; + a2S a2S (=)) S S a2S a2S (( =) S + a2S a2S R S R + S + S + 02 S a2S a2S + a + ( a) = 02S S (R; +; ) (S; ; ) : R! S a;b2R (a +b) = (a) (b) a;b2R (a b) = (a) (b) :Z 20 ! Z 30 ; (x) = 6x x;y2Z 20 x +y =r xy =s Z x +y = 20q +r xy = 20t +s q;t2Z 06r;s< 20 Z 6x + 6y = 6(x +y) = 120q + 6r 6(xy) = 120t + 6s Z 30 (x +y) = 6(x +y) = 6r = 6x + 6y = (x) + (y); (xy) = 6(xy) = 6s = 6 6s = 6 6(xy) = (6x)(6y) = (x) (y) R = (Z; +; );S = (2Z; +; ) :R! S; (x) = 2x (x +y) = 2(x +y) = 2x + 2y = (x) + (y) =) (xy) = 2xy6= 2x2y = (x) (y) : Z 5 ! Z 10 ; (x) = 5x (3 3) = (4) = 0; (3) (3) = 5 5 = 5: :R! S R S :R! R H = ( a b b a ! :a;b2R ) : C! H; (a +bi) = a b b a ! C H a;b;c;d2R ((a +bi) + (c +di)) = ((a +c) + (b +d)i) = a +c b +d b d a +c ! = a b b a ! + c d d c ! = (a +bi) + (c +di); ((a +bi)(c +di)) = ((ac bd) + (ad +bc)i) = ac bd ad +bc ad bc ac bd ! = a b b a ! c d d c ! = (a +bi) (c +di): (a +bi) = (c +di) =) a b b a ! = c d d c ! =)a =c;b =d =)a +bi =c +di a b b a ! 2H (a +bi) = a b b a ! C H R u2R u U(R) (U(R); ) U(Z) =f 1; 1g;U(Q) =Qnf 0g U(C) =Cnf 0g;U(Z 6 ) =f 1; 5g R a2R a n = 0 n2Z + a H x2H (1 x) (1 +x) x2H x n = 0 (1 x)(1 +x + +x n 1 ) = 1 +x + +x n 1 x x 2 x n = 1 x n = 1 (1 x) 1 x (1 +x)(1 x +x 2 :::x n 1 ) = ( 1 +x n ; n 1 x n ; n 1 x n = 1 1 +x (Z; +; ) 1 +1 (Q; +; ) Z 7 1 1 = 1; 2 1 = 4; 3 1 = 5; 4 1 = 2; 5 1 = 3; 6 1 = 6 R a;b2 R a6= 0;b6= 0 a b = 0 a b R a b R (Z 6 ; +; ) 2 3 = 0Z n n m2Z n m n fm;ng = (m;n) =d6= 1 m n d = n d m 0 (mod n) m6= 0 n d 6= 0 m (Z 12 ; +; ) 2 6 = 0 =) 4 6 = 0 =) 8 3 = 0 =) 9 4 = 0 =) 10 6 = 0 =) f 2; 3; 4; 6; 8; 9; 10g p Z p Z p (Q; +; ); (Z; +; ) (R; +; ) F a;b2F;a6= 0;ab = 0 b = 0 F a 1 2F ab = 0 =)a 1 (ab) =a 1 0 =) (a 1 a)b = 0 =)eb = 0 =)b = 0 F =f 0; 1;a 1 ;a 2 ;:::;a n g F =Fnf 0g x2F a 1 a 1 F =fa 1 ;a 1 a 1 ;a 1 a 2 ;:::;a 1 a n g F 9i a 1 =a 1 a i a i = 1 9i;j i6= j a 1 a i = a 1 a j a 1 a i a 1 a j = 0 a 1 (a i a j ) = 0 a i a j = 0 a i =a j a 1 F =F 12F 9j a 1 a j = 1 (a 1 ) 1 =a j Z 5 =f 0; 1; 2; 3; 4g p Z p R a2R na = 0 n R (R) n (Q) = (Z) = 0 (Z 8 ) = 8 2Z) = 0 R e n () n ne = 0 (=)) a2R na = 0 e2R ne = 0 (( =) ne = 0 n a2R na = 0 na =a +a + +a | {z } n =ae +ae + +ae | {z } n =a (e +e + +e) | {z } n =a(ne) =a 0 = 0 D kar(D) = n n 1 < n 1 ;n 2 < n n =n 1 n 2 e2D ne = 0 =) (n 1 n 2 )e = 0 =) (n 1 e)(n 2 e) = 0 =)D n 1 e = 0 n 2 e = 0 n 1 e = 0 n 1 1 e x ; x< 1 g(x) = ( cosx; x> 1 0; x< 1 f6=O;g6=O fg =O a;b;c2Z a b =a +b 12Z: (a b) c = (a +b 1) c =a +b +c 2 =a (b +c 1) =a (b c) a b =a +b 1 =b +a 1 =b a a 1 = 1 a =a + 1 1 =a a (2 a) =a + 2 a 1 = 1 a 2 a a b =a +b ab2Z(a b) c = (a +b ab) c =a +b ab +c ac bc +abc; a (b c) =a (b +c bc) =a +b +c bc ab ac +abc a b =a +b ab =b +a ba =b a a (b c) =a (b +c 1) =a +b +c 1 ab ac +a = 2a +b +c ab ac 1; (a b) (a c) = (a +b ab) (a +c ac) = 2a +b +c ab ac 1 a 0 = 0 a =a + 0 a 0 =a 0 (Z; ; ) a b = 1 a6= 1 b = 1 a b = 1 =)a +b ab = 1 =)a +b ab 1 = 0 =) (a 1)(1 b) = 0 a6= 1 b = 1 (Z; ; ) (Z; ; ) x x 3 = 0 =)x + 3 3x = 0 =)x = 3 2 62Z x;y2I a ax =ay = 0 a(x +y) =ax +ay = 0 + 0 = 0 x +y2I a a(xy) = (ax)y = 0y = 0 xy2I a a( x) = (ax) = 0 = 0 x2I a I a axa;aya2Haxa +aya =a(xa +ya) =a(x +y)a2H x +y2R (axa)(aya) =axa 2 ya =ax1ya =a(xy)a2H xy2R (axa) = (a(xa)) =a( (xa)) =a( x)a2H x2R H f (a +b p 2) + (c +d p 2) =f (a +c) + (b +d) p 2 = (a +c) (b +d) p 2 = (a b p 2) + (c d p 2) =f(a +b p 2) +f(c +d p 2): f (a +b p 2)(c +d p 2) =f (ac + 2bd) + (ad +bc) p 2 = (ac + 2bd) (ad +bc) p 2 = (a b p 2)(c d p 2) =f(a +b p 2)f(c +d p 2): f f(a +b p 2) =f(c +d p 2) =)a b p 2 =c d p 2 =)a =c b = d a;b;c;d2Z =)a =c b =d =)a +b p 2 =c +d p 2 =)f f x+y p 22Z( p 2) a =x b = y f(a+b p 2) = f(x y p 2) =x +y p 2 f f x = 3 p 22A y = 3 p 22A xy = 3 p 462A A (x +x) 2 = x 2 +x 2 +x 2 +x 2 = x +x +x +x = x +x x +x = 0 x = x a;b2H (a +b) 2 =a +b a 2 +ab +ba +b 2 =a +b a 2 =a b 2 =b ab +ba = 0 ab = ba ba =ba ab =baA;B2R =)A +B =A4B = (A[B)n (A\B)2R (A +B) +C = (A4B)4C =A4(B4C) =A + (B +C) A4B =B4A A +B =B +A A4; =A A4A =; A =A A;B2R =)A B =A\B2R (A B) C = (A\B)\C =A\ (B\C) =A (B C) A B =A\B =B\A =B A A (B +C) = A\ (B4C) = (A\B)4(A\C) = A B +A C A2R A U =A U R U R ;6=A2R A X =A\X =U X2R A D x2D x 2 =x =)x 2 x = 0 =)x(x 1) = 0 =)D x = 0 x 1 = 0 =)x = 0 x = 1 (a +a) 2 =a 2 +a 2 +a 2 +a 2 a +a =a +a +a +a 2a = 0 a;b;c;d;e;f2Z (a +bi) + (c +di) = (a +c) + (b +d)i2G a +c;b +d2Z [(a +bi) + (c +di)] + (e +fi) = (a +bi) + [(c +di)] + (e +fi)] (a +bi) + (c +di) = (a +c) + (b +d)i = (c +a) + (b +d)i = (c +di) + (a +bi) (a +bi) + (0 + 0i) = (0 + 0i) + (a +bi) =a +bi 0 G = 0 + 0i2G 02Z (a +bi) = ( a) + ( b)i2G a; b2Z (a +bi)(c +di) = (ac bd) + (ad +bc)i2G ac bd;ad +bc2Z[(a +bi)(c +di)](e +fi) = [(ac bd) + (ad +bc)i](e +fi) = (ace bde adf bcf) + (acf bdf +ade +bce)i; (a +bi)[(c +di)(e +fi)] = (a +bi)[(ce df) + (cf +de)i] = (ace adf bcf bde) + (acf +ade +bce bdf)i (a +bi)(c +di) = (ac bd) + (ad +bc)i = (ca db) + (cb +da)i = (c +di)(a +bi) (a +bi)[(c +di) + (e +fi)] = (a +bi)[(c +e) + (d +f)i] = (ac +ae bd bf) + (ad +af +bc +be)i; (a +bi)(c +di) + (a +bi)(e +fi) = (ac bd) + (ad +bc)i + (ae bf) + (af +be)i = (ac bd +ae bf) + (ad +bc +af +be)i (a +bi)(1 + 0i) = (1 + 0i)(a +bi) =a +bi 1 = 1 + 0i2G 1; 02Z G (a+bi)(c+di) = 0 a+bi6= 0 a6= 0 b6= 0 c+di = 0 c =d = 0 ac bd = 0 ad +bc = 0 a6= 0 c d a(c 2 +d 2 ) = 0 a6= 0 c 2 +d 2 = 0 =) c = 0;d = 0 b6= 0 d c b(c 2 +d 2 ) = 0 b6= 0 c 2 +d 2 = 0 =) c = 0;d = 0G G 1 +i 1 2 1 2 i62G 1 2 62Z x = 4 p 52A y = 4 p 52A xy = 4 p 2562A A a;b;c;d;e;f2Q (a +b 3 p 2 +c 3 p 4) + (d +e 3 p 2 +f 3 p 4) = (a +d) + (b +e) 3 p 2 + (c +f) 3 p 42T a +d;b +e;c +f2Q (a +b 3 p 2 +c 3 p 4)(d +e 3 p 2 +f 3 p 4) = ad +ae 3 p 2 +af 3 p 4 +bd 3 p 2 +be 3 p 4 +bf 3 p 8 +cd 3 p 4 +ce 3 p 8 +cf 3 p 16 = (ad + 2bf + 2ce) + (ae +bd + 2cf) 3 p 2 + (af +be +cd) 3 p 42T (a +b 3 p 2 +c 3 p 4) = ( a) + ( b) 3 p 2 + ( c) 3 p 4)2T T x;y2R f(x +y) = 0 H = 0 H + 0 H =f(x) +f(y); f(x y) = 0 H = 0 H 0 H =f(x) f(y) f x;y2 (I) =)9a;b2I (a) =x; (b) =y =) (a +b |{z} 2I ) = (a) + (b) =x +y =)x +y2 (I) x;y2 (I) =)9a;b2I (a) =x; (b) =y =) (a b |{z} 2I ) = (a) (b) =x y =)x y2 (I) x2 (I) =)9a2I (a) =x =) ( a |{z} 2I ) = (a) = x ) =) x2 (I) (I);H a;b2 ( ) =) (a) = 0 H ; (b) = 0 H =) (a +b) = (a) + (b) = 0 H + 0 H = 0 H =)a +b2 ( ) a;b2 ( ) =) (a) = 0 H ; (b) = 0 H =) (a b) = (a) (b) = 0 H 0 H = 0 H =)a b2 ( ) a2 ( ) =) (a) = 0 H =) ( a) = (a) = 0 H = 0 H ) =) a2 ( ) ( );R h2H 9r2R (r) =h h2H h = (r) = (1 R r) = (1 R ) (r) = (1 R )h; h = (r) = (r1 R ) = (r) (1 R ) =h (1 R ) H (1 R ) = 1 H x 2 = 1 =) x 2 1 = 0 =) (x 1)(x + 1) = 0 =) D x 1 = 0 x + 1 = 0 =)x = 1 x = 1: D U =f 1; 1;u 1 ;u 2 ;:::;u n g x2U x 1 =x () x 2 = 1 1 1 u = ( 1)u 1 u 2 :::u n u i 1 Z p Z p U(Z p ) =f 1; 2;:::;p 1g 1 2 (p 1) = 1 (p 1)! 1 (mod p)R I R r2R rI I I Ir I I 8r2R;8x2I rx2I () I 8r2R;8x2I xr2I () I I I (Z; +; ) n2Z I =nZ r2Z x2I 9k2 Z x = nk rx = r(nk) = n(rk)2 I xr = (nk)r = n(kr)2I I (Z; +; ) (Q; +; ) x = 22 Z r = 1 5 2 Q xr = 2 5 62Z W = ( a b c d ! :a;b;c;d2Z ) I = ( a b 0 0 ! :a;b2Z ) W A = a b c d ! 2W X = x y 0 0 ! 2I AX = ax ay cx cy ! XA = xa +yc xb +yd 0 0 ! AX62I XA2I IA B R A +B =fa +b :a2A;b2Bg A B A +B R A +B r2 R a +b2 A +B A B ra;ar2 A rb;br2B r(a +b) =ra +rb2A +B (a +b)r =ar +br2A +B A +B R f 0g R I6=R I R U R U 02U u;v2U u v2U u2U r2R ur2U ru2U (=)) U U 02 U (U; +) (R; +) u;v 2 U =) u v 2 U U (( =) u;v2U 0 v = v2U u +v = u ( v)2 U uv2 U u2 R v2 U uv2 U u2 U u2 U 0 u2 U U U R N N N =R u2N u 1 2R N uu 1 = 1 R 2N N r2R r 1 R 2N r2R r2N N =R F f 0g F F 0 (H; +; );R R I = f2H :f( 1 2 ) = 0 H I h2H f2I (h f)(1=2) =h(1=2)f(1=2) =h(1=2) 0 = 0 (f h)(1=2) =f(1=2)h(1=2) = 0 h(1=2) = 0 h f;f h2I I R R f 0g R R 8 06=a2R ba = 1 b2R 06=a2R Ra =fxa :x2Rg Ra 0:a = 0 02Ra x = 0 xa;ya2Ra =)xa ya = (x y) | {z } 2R a2Ra xa2Ra;r2R r(xa) = (rx)a2Ra rx2R R (xa)r =x(ar) =x(ra) = (xr)a2Ra Ra Ra =f 0g Ra =R Ra =f 0g 1:a = a2 Ra(a6= 0) Ra = R 12 R 9b2 R ba = 1 a 1 =b :H! K ( );H A;H (A) (H) B;K 1 (B) H ( ) x2 ( );h2H (xh) = (x) (h) = 0 K (h) = 0 K (hx) = (h) (x) = (h)0 K = 0 K xh;hx2 ( ) ( ) A (A) x2 (A) k2 (H) 9a2 A (a) = x 9h2 H (h) = k A ah;ha2A (ah) = (a) (h) =xk; (ha) = (h) (a) =kx =)xk;kx2 (A) (A) B K 1 (B) =fx2Hj (x)2Bg (0 H ) = 0 K 0 K 2B 0 H 2 1 (B) x;y2 1 (B) =) (x); (y)2B B (x) (y)2B (x y)2B x y2 1 (B) x2 1 (B) h2H (x)2B (h)2K B (xh) = (x) (h)2B (hx) = (h) (x)2B xh;hx2 1 (B) 1 (B) F R : F! R ( ) F F f 0g F ( ) =f 0g ( ) =F R N =fx2R :9n2Z + ;x n = 0g 02 N 0 1 = 0 x2 N;r2 R x n = 0 n2Z + R (rx) n = (xr) n =x n r n = 0r n = 0 xr;rx2 N x;y2 N x m = 0 y n = 0 m;n2Z + R (x y) n+m = n+m X i=0 n +m i ! x n+m i ( 1) i y i : n +m i>m i>n (x y) n+m = 0 x y2N N R I6=R a;b2R ab2I =)a2I b2I I R f 0 R g ab = 0 =)a = 0 b = 0 Z 1 1 ab2P =nZ P a2nZ b2nZ nja njb 1 m y > n N N N (x y) m+n 2N x y2 p N r2R x2 p N x n 2N R N (rx) n = (xr) n = x n |{z} 2N r n |{z} 2R 2N rx;xr2 p N p N p p N = p N n;m2Z + nm2Z + 12Z + x2 q p N =)x n 2 p N =) (x n ) m 2N =)x nm 2N =)x2 p N; x2 p N =)x 1 2 p N =)x2 q p N; p p N = p N x2K 0 x = 0 02Ann(K) a;b 2 Ann(K) x 2 K ax = bx = 0 (a b)x = ax bx = 0 a b2Ann(K) r2R;a2Ann(K) x2K ax = 0 R x2K (ra)x =r(ax) =r 0 = 0 (ar)x = (ra)x =r(ax) = 0 ra;ar2Ann(K) Ann(K)Ann(f 3g) =fa2Z 12 : 3a = 0g =f 0; 4; 8g Ann(f 4g) =fa2Z 12 : 4a = 0g =f 0; 3; 6; 9g Ann(f 3; 4g) =fa2Z 12 : 3a = 0; 4a = 0g =f 0g b2B 0 b = 02A 02A :B x;y2A :B b2B xb;yb2A (x y)b =xb yb2A x y2A :B x2A :B r2R b2B xb2A R A (xr)b = (rx)b =r( xb |{z} 2A )2A rx;xr2A :B A :B a2A;b2B =)a2A;b2R;A ab2A; a2A;b2B =)a2R;b2B;B ab2B: ab2A\B =f 0g ab = 0