Lecture 8. A little bit of fun math Read: Chapter 7 (and 8) Finite Algebraic Structures

Transcription

1 Lecture 8 A lttle bt of fu ath Read: Chapter 7 (ad 8) Fte Algebrac Structures Groups Abela Cyclc Geerator Group order Rgs Felds Subgroups Euclda Algorth CRT (Chese Reader Theore) 2

2 GROUPs DEFINITION: A oepty set G ad (G,@) s a group f: CLOSURE: for all x,y G: y) s also G ASSOCIATIVITY: for all x,y,z G: z z) IDENTITY: there exsts detty eleet I G, such that, for all x G: x x ad I x INVERSE: for all x G, there exst verse eleet x - G, such that: x x I x - DEFINITION: A group (G,@) s ABELIAN f: COUTATIVITY: for all x,y G: y x 3 Groups (cotd) DEFINITION: A eleet g G s a group geerator of group (G,@) f: for all x G, there exsts >0, such that: x g @ g ( tes) Ths eas every eleet of the group ca be geerated by g I other words, G<g> DEFINITION: A group (G,@) s cyclc f a group geerator exsts! DEFINITION: Group order of a group (G,@) s the sze of set G,.e., G or #{G} or ord(g) DEFINITION: Group (G,@) s fte f ord(g) s fte. 4 2

3 Rgs ad Felds DEFINITION: A structure (R,+,*) s a rg f (R,+) s a Abela group (usually wth detty eleet deoted by 0) ad the followg propertes hold: *CLOSURE: for all x,y R, (x*y) R *ASSOCIATIVITY: for all x,y,z R, (x*y)*z x*(y*z) *IDENTITY: there exsts /0 R, s.t., for all x R, *x x *DISTRIBUTION: for all x,y,z R, (x+y)*z x*z + y*z I other words (R,+) s a Abela group wth detty eleet 0 ad (R,*) s a ood wth detty eleet /0. The rg s coutatve rg f *COUTATIVITY: for all x,y R, x*yy*x 5 Rgs ad Felds DEFINITION: A structure (F,+,*) s a feld f (F, +,*) s a coutatve rg ad: *INVERSE: all o-zero x R, have ultplcatve verse..e. there exsts a verse eleet x - R, such that: x * x

4 Exaple: Itegers uder addto G Z tegers { -3, -2, -, 0,, 2 } the group operator s +, ordary addto q the tegers are closed uder addto q the detty s 0 q the verse of x s -x q the tegers are assocatve q the tegers are coutatve (so the group s Abela) 7 No-zero ratoals uder ultplcato G Q - {0} {a/b} where a, b Z * the group operator s *, ordary ultplcato If a/b, c/d Q-{0}, the: a/b * c/d (ac/bd) Q-{0} the detty s the verse of a/b s b/a the ratoals are assocatve the ratoals are coutatve (so the group s Abela) 8 4

5 No-zero reals uder ultplcato G R -{0} the group operator s *, ordary ultplcato If a, b R-{0}, the a*b R-{0} the detty s the verse of a s /a the reals are assocatve the reals are coutatve (so the group s Abela) 9 Itegers od N uder addto G Z + N tegers od N {0 N-} the group operator s +, odular addto the tegers odulo N are closed uder addto the detty s 0 the verse of x s -x (N-x) addto s assocatve addto s coutatve (so the group s Abela) 0 5

6 Itegers od p (pre) uder ultplcato G Z * p o-zero tegers od p { p-} the group operator s *, odular ultplcato tegers od p are closed uder *: because f GCD(x, p) ad GCD(y,p) the GCD(xy,p) (Note that x s Z * P ff GCD(x,p)) the detty s the verse of x s u s.t. ux (od p) u ca be foud ether by exteded Euclda algorth ux + vp GCD(x,p) Or usg Ferat s lttle theore x p- (od p), u x - x p-2 * s assocatve * s coutatve (so the group s Abela) Postve Itegers uder Expoetato? G {0,, 2, 3 } the group operator s ^, expoetato closed uder expoetato the (oe-sded?) detty s, x^x the (rght-sde oly) verse of x s always 0, x^0 the tegers are NOT coutatve, x^y<>y^x (o-abela) the tegers are NOT assocatve, (x^y)^z <> x^(y^z) 2 6

7 Z * N : postve tegers od N relatvely pre to N G Z * N o-zero tegers od N {,x, -} such that gcd(x,n) Group operator s *, odular ultplcato Group order ord(z * N ) uber of tegers relatvely pre to N deoted by ph(n) tegers od N are closed uder ultplcato: f GCD(x, N) ad GCD(y,N), GCD(x*y,N) detty s verse of x s fro Eucld s algorth: ux + vn (od N) GCD(x,N) so, x - u ( x Ph(N)- ) ultplcato s assocatve ultplcato s coutatve (so the group s Abela) 3 No-Abela Groups: 2x2 o-sgular real atrces uder atrx ult- a b GL(2) {[ c d ], ad-bc 0} f A ad B are o-sgular, so s AB 0 the detty s I [ ] a b [ ] c d - 0 [ ] -c a d -b / (ad-bc) atrx ultplcato s assocatve atrx ultplcato s ot coutatve 4 7

8 No-Abela Groups (cotd) 2 5 [ ] [- 0.2 ] [ 0 30 ][ 2 ] [ 60 0 ] [ 2 ][ 0 30 ] [ ] 5 Subgroups DEFINITION: (H,@) s a subgroup of (G,@) f: H s a subset of G (H,@) s a group 6 8

9 Subgroup exaple Let (G,*), G Z* 7 {,2,3,4,5,6} Let H {,2,4} (od 7) Note:. H s closed uder ultplcato od 7 2. s stll the detty 3. s s verse, 2 ad 4 are verses of each other 4. assocatvty holds 5. coutatvty holds (H s Abela) 7 Subgroup exaple Let (G,*), G R-{0} o-zero reals Let (H,*), Q-{0} o-zero ratoals H s a subset of G ad both G ad H are groups ther ow rght 8 9

10 Order of a eleet Let x be a eleet of a (ultplcatve) fte teger group G. The order of x s the sallest postve uber k such that x k Notato: ord(x) 9 Order of a eleet Exaple: Z* 7 : ultplcatve group od 7 Note that: Z * 7 Z 7 ord() because ord(2) 3 because ord(3) 6 because ord(4) 3 because ord(5) 6 because ord(6) 2 because

11 Theore (Lagrage) Φ( ) - order of largest order of * G ay eleet! order of g :sallest teger such that g od Theore (Lagrage): Let G be a ultplcatve group of order. For ay g G, ord(g) dvdes ord(g). COROLLARY : b Φ( ) because : Φ() ord(z ord( b) ord(z thus : od b Z b Φ( ) * b ) / k Φ() / k Φ() / k * * ) / k 2 COROLLARY 2 : f p s pre the b Z ) ad 2) b p * p b od p a Z ord( a) a prtve eleet p p Exaple: Z * 3 prtve eleets are: {2,6,7,} 22

12 b Euclda Algorth Purpose: copute GCD (x,y) Recall that: b* b ultplcatve verse of b, b Ζ od b gcd( b, ) Euclda (, b) b 23 Euclda Algorth (cotd) t : r x r y 0 q " $ r / r #% r r od r q " $ r / r #% r r od r q " $ r / r #% r r od r ( r 0) 2 OUTPUT r Exaple: 24, Exaple: 23,

13 Exteded Euclda Algorth Purpose: copute GCD(x,y) ad verse of y (f t exsts) t : r x r y t 0 t 0 0 q " $ r / r #% r r od r t t t q od r q " $ r / r #% r r od r t t q t od r q " $ r / r #% r r 2od r t t 2 q t od r 0 f ( r ) OUTPUT t else OUTPUT "o verse" 25 Exteded Euclda Algorth (cotd) Theore: r t r ( ) > t r! r r " q / r + r od r t t 2 q t od r0 Exaple: x87 y I R T Q

14 Exteded Euclda Algorth (cotd)! r r " Exaple: x93 y87 q / r + r od r t t 2 q t od r0 I R T Q Chese Reader Theore (CRT) The followg syste of odular equatos (cogrueces) x a... x a od od (all -s relatvely pre). ' x a % & where : y ' % & Has a uque soluto: $ " y # *...* $ " # od od 28 4

15 5 29 CRT Exaple! " # $ % & od x od x 47 od 77 x od y od 7 od y od y y x + + 3*7*8) (5** / / 77 ] ) / 3( ) / [5(