*Lecture-2

*Euclidean

*Euclidean a, b, , l 0( a, b, , l )GCD ( a, b, , l)a, b, , l 0[a, b, , l ]LCM ( a, b, , l)Euclideanba>b a=qb+r 0

*Euclidean :[595, 493]

: (595, 493)Euclidean

595 493102 4931024+85102=85+17 85=175(595, 493)=17; [595, 493]=(595493)/17=17255

(595, 493)=17 =102-85 =102-(493-4102) =5102-493 =5(595-493)-493 =5595(-6)493

*abmabm pp. 24mm

25 4(mod 7); 12 5(mod 7); 2512=300 6(mod 7) 45(mod 7) m=3

*ABfaAbBfABbafafABbBaAb=f(a)fABAfBa1=a2f(a1)=f(a2)fABffAAfAAAA

*f(A, )(B,*) f(a1 a2) =f(a1) *f(a2) a1 ,a2 A, f(a1) ,f(a2) B fABAB fABfA A

*(R,)(R+,),f: RR+f(x)=10x, fRR+

Proofy R+,x=lgyf(x)=y,fRR+x, y R10x =10y x=yfRR+fRR+f(xy)=10x+y= 10x 10y =f(x)f(y)fRR+

See Slide-5

*(Z, +)(A, ), A={1, -1}, fZAxZ,

fZAProof: x, y Zx, yf(x)=1, f(y)=1 f(x+y)=1=11=f(x)f(y)(2) x, yf(x)=-1, f(y)=-1 f(x+y)=1=(-1)(-1)=f(x)f(y)(3) x yf(x)=-1, f(y)=1 f(x+y)=-1=(-1)1=f(x)f(y) x y f(x+y)=f(x)f(y) fZAQED

*QQ*: (Q, +)(Q*, )Proof (QQ*f,f(0) = xQ*.f(x) = x0

f(0+x)=f(0)f(x)=xx f(x)=xx x= xx x=1 f(0) =1

f(a)=-1 f(a+a)=(-1) (-1)=1 f(0) =1a+a=0a=0f(0)=-1f QQ* QQ*QED

*GG

G01) 2) 3) Ge,4) a

*000m mm0mm=4 m=30

*G={1, -1, i, -i}, (G, )GGGG1(-1) -1=-1, i-1=-i, (-i) -1=i

(S={1, 2, 3, 4, 6, 12}, GCD)a, b, cS, GCD(a, b) S, SGCD GCD[GCD(a, b), c] = GCD[a, GCD(b, c)] GCD(12, a)=a12GCD(1, a)=1121(S, GCD)

*Proof, pp.27

a, b G,(a b)-1=b-1 a-1Proof, pp.27

Gab a x = by a = bGx = a-1 b , y = b a-1

Ga x = a y x = y

*(Order of a Group)(Finite Group)(Infinite Group)(Abelian Group)n(Semigroup)()(Monoid)()Monoid(Symmetric Group)nAn!

*(Permutation Group)A={1, 2, 3}

*R1) R2) 3) m

* abab=0abZ623=6=0(mod 6); 23=03(mod 6) but 20 Z5n

*R={a+b51/2|a, bZ}p, q R, p=a1+b151/2 , q=a2+b251/2 , a1, b1 , a2,b2 Z. p+q = (a1+b151/2 )+(a2+b251/2) = (a1+a2 ) +(b1+b2 ) 51/2 R pq = (a1+b151/2 ) (a2+b251/2) = (a1a2+5b1b2 ) +(a1b2 + a2b1 ) 51/2 R (R, +) (R, ) (R, +, )

*FF1) F02) F 13) GF(q)Fqq

* p p 18111025

18320531

* 18020805

18290406

* p p p ( p ) p GF(p)

Proof: (01 p p a p Euclidean (a, p)=1=Aa + Bp p 1 Aa (mod p) Q.E.D.

*Q(21/2)={a+b21/2|a, bQ}(Q(21/2), +, )Proof (Q(21/2), +, ) 1a+b21/2 Q(21/2) ab a+b21/2

(Q(21/2), +, )QED

*GHGHGGHG pp.331aH, bH, abH2aH, aa-1Ha, bH, a-1bH

*HGHGg1, g2,Gh1, h2HH

*HGgGg()HgH (Hg)H()gAbelhg=ghmpp.33-34GHH2.4.3, pp.34H jGN=jnNnGH, pp.34 m=99 H: ; 1+H: ; 2+H:

*C12={e, g, , g11}H={e, g4, g8}He=Hg C12 g! H Hg={g, g5, g9}HHg C12HHgC12g2Hg2={g2, g6, g10}HHg Hg2 C12HHg Hg2 g3Hg3={g3, g7, g11}

HHg Hg2 Hg3= C12

C12HHHgHg2Hg3

*HGaGaHHaHGAbelHGaGa-1ha H, h H (pp.35)HGHMmm=3,3

M3 pp.36HGHGHG/HmZ/M

*HGHHGProofaHbH(aH)(bH)(ab)H(aH)(bH)cH (aH)(bH)= cH ab=(ae)(be) (aH)(bH)ab cHcH=(ab)H HGh H a G a-1 ha h (a-1H)(aH)= (a-1a)H=Ha-1 ha H HGQED