28/02/ LECTURE Thursday MATH Linear and Abstract Algebra

Transcription

1 MATH 2022 Liner nd Abstrct Algebr 28/02/ LECTURE Thursdy

2 ntroducing groups Recll tht field F is rithmetic with t n t lest two elements hving ( in fct to ) stisfying six xioms : C ) ( H b eff = t Cbtc ) Cb )c=(bc ) c ssocitivity ) (2) Cfbff ) tb = bt b = b C commuttivity) ) ( H b c tf ) Ct 's ) c = b tc C distributivity )

3 (4) ( 0 C f) ( fff ) to = o t = == ( existence of dditive oh multiplictive identity elements ) (5) ft f) ( b EF ) t b = bt = ( existence of negtives 0 (6) ( F E f) { o } ) ( b Ef ) b = b = ( existence of multiplictive inverses b = ) of nonzero elements b = )

4 Subtrctiouinf : Define for bff b = t C T b ) n prticulrs = t G ) = 0 Division in f : Define for ff bff ){ } o :b = b = b n prticulr b :b = = bb 1=1

5 Focussing ttention on just Ze opertion leds to the notion of group A is n rithmetic with to groupe respect binry opertion A stisfying ) ( H b e EG ) C * b) * c = * ( btc ) C ssocitivity ) C) ( etg) ( Ve ) te = e # = u#icn!:::::::: C existence of n e identity )

6 A G is n rithmetic with to groupe respect binry opertion * stisfying C) ( H b e EG ) C * b) * c = * ( bite ) C ssocitivity ) C) ( e EG ) ( Ve ) te = e # = u#icn!:::::::e C existence of n e identity ) it * is :i"i= then is clled belin C fter Abel ( ))

7 from conditions it Cy) (5) listed erlier t "" "df"~ ( f t ) is n belin with group t o nd inverses group idejtityege! " " "" (F) { 's o ) is n belin group rereciprcc identity element nd inverses group for field f with

8 n the cse of modulr rithmetic where NEXT identity (Znt)isnbeliugroupwit t nd e inverses cn rech everything by dding 1 to itself s!7 **± ins t t ' t t 1+1=0 nd then the cycle n times repets

9 n the cse of modulr rithmetic where NEXT CZntlisnbeliugroupmil " nd cn rech! gte :3?! everything by dding to itself t : inverses We sy Cn t ) is cyclicgroup with genertor

10 The invert ibility condition of group G provides flexibility nd movement : f b E G then we cn " get from to b " *("*b)=@*")*b=e*b=t i ssocitive :!!y

11 This fcilittes solving equtions : # se = b ' * ( # n ) = ' * ' ) * * = ' * b e * n = * b n = ' * b : :i

12 This fcilittes solving equtions : # se = b ' * ( * n ) = ' * ' ) * * = ' * b e * n = * b n = * b s ndeed put = { invertible nxn mtrices over f } where next nd F is field nd let * be mtrix multipliction

13 ndeed put = { invertible nxn mtrices over f } where next nd F is field nd let A be mtrix multipliction f A B E G then LAB) ( B ' A " = A C BB ) A " = A At = A A ' = 80 tht AB EG with inverse CAB) = B ' AT

14 ndeed put = { invertible nxn mtrices over f } where next nd F is field nd let A be mtrix multipliction A f G he tt@yincisyitnn so tht with inverse CAB) Hence * is conditions defining group t = B ' AT binry opertion on 9 nd the re stisfied : i ) * is ssocitive eh is the identity element (3) A C A ' EG nd AA ' = A ' A

15 However for n 32 it is not commuttive eis c :: ]l ::] C ::] t ( not equl) [ : in ::3 C ::3 so this G is not belin = ( n fct most groups re not belin )

16 Exmple : Consider the group = { symmetries of the rectngle } l l 2 do 4 i : 3 Opertion on G is A which is composition of symmetries ( following one fter the other )

17 Exmple : = { symmetries of the rectngle }! Z 3 4 i G = { A B C } where = identity symmetry ( " do nothing " ) A = 1800 rottion B = reflection in verticl xis c = reflection in horizontl xis

18 bout digonl ) A B C A B C A A 1 C B B B C A C C B A The group is belin ( tble symmetric