Lecture 5 - Group Theory Notes PDF

Lecture 5 01110 Theorem Let G be a group and x.ge6 If then x y y x Theorem Let G be a group If e f es have properties that eN x e x...

Lecture 5 01110 Theorem Let G be a group and x.ge6 If then x y y x Theorem Let G be a group If e f es have properties that eN x e x 0 f e Claim Let G be a group It EG then for x of 1 Proof consider x x e since by definition x is the inverse of x Bythe above theoremthis means is theinverseof and socx.is x Theorem Let o be a group it x.ge 6 thenexy Proof Bytheprevious theorem sinceinverses are unique it is enough toshow a ix.is e cxyicy Note equation as tonous from associativity and aginitions ofinversesand identity elements gly x ycy.si Eu Theorem cancellation Law set 6 be a group andxig.ee6 1 it xy xa then y a estcancellation Eating 2 it yx ax then y z right cancellation Identity inverse Proof For 1 since a is a group there exists x tea andsoby multiplying theequation x xz onthe LHS by x i we get ways xexes Byassociativity thismeans x y ex a since n isthe inverse of x xx e meaningthe priorbecomes ey ez and since e is the identity y ey.ee For 2 some prooffollows justswap the letters There is also a case mg eanatyihiae n the e e c.ec iniiy hasone element c needs to beclosed consider the set cz ex and Cs eix.gg Findall binary operations on C2 Cs suchthat they are group Is G e with where else e a group G is clearsince cease we exe e exe eocene 62 e is the identity 63 e e Assume 62 is a group This means that there is an identity element without loss of generality woo we may assume e is the identity element e because e is the identity and so e e exe e we e x x x x e multiplication table or Cayley table All thatremains is to determine since is abinary operation 62 and so x e o note that since ex e and need an inner element Because of Woo we can switch around e x F Tinary operations for ca that make it a grip berion foreither operation sinceitis commutative x2 e For Cs similarly we may assume e is the identity element woo since we can assume is is a group e x y thisgroup is abelion catter checking forassociativity line y sggjggggta.gg e e x y Since e is identity eo e e e e e y e e y yo e y y Claim Proof By cancelation law if then e y y y butthiscan't happen some argument for y x y or e Assume e be x y I need LHS RHS cancellation If we have e we fail the property of a unique inverse sincethen we also would ass be yoy e so Is there a relationship between binary operations and groups If elements insee n n grid check this Lecture 6 02110 pdefinition ofancient statement 1There exists e e 6 such that for all g e G e g gone g uniform statement 2 For all g to there exists e e 6 such that goe 9 9 effiction is the Iffy Sets 6 CH Gx H 1g i hit 192 h2 g ga h he union intersection complement products Exit powersets subsets subgroups Desation Let 6 be a group with operation A subgroup of 0 is a subset c 6 suchthat is Hy sameoperatio a group as on 6 Lemma If HE 6 is a subgroup then eat eo proof Pick he it consider h en h since en is the identityelement on H Multiplyby n n h en n oh eo eat eowen Lemma let it 6 be a subset It is a subgroup it andonly if 1 It is closed under needto demonstrate associativity but it followsfrom this 2 If he H then n e a thismeans indirectly e e Example CR ECC I QI 03 t in E CQ CRY I group Y I 1 I 1 1 Functions Let x and Y be sets a function f x 4 is asetof ordered pairs xy such that foran ex yet Lecture 7 03110 inputoutputpairs Formal aginition function A function X Y is a subset gets Y with properties that of a soran x iii iii ego apr 9 is unique 2 it exigs get and cx.ae gets then y Example y I 3 thing tospecifyfor a function 1 is the domain R 03 2 Y the targets Y IR function 119 3 get xx function gets x I ex IR 0 IR Claim Y IR 0 x R ix 4 I x.ge R 1221 Cory 1 YER Suppose glf Y geton EX 2 gen 4 2 y intuitive idea cx.es EEKcfan e2 I To show geton is a function we must check 1 Let EX we need to show 3zez suchthat exine geton Since gets is a function a of ex there exists yet with exigseget Since gens is a function and yet there exists at a sain that sy.se gens Thus 2 getons because ex x.pe got g2 gins moresteps Definition A function f Y isonto if andonly if foran yey there exists Ex suchthat cx.gs gets surjective standardnotation x y ten y Examples 1 Taxes X x for anyset issurjective 2 taxi 3 R R Ta a is surjective 3 WONTWORK gex x2 IR R Ta a only for aso F IR i Definition A function f Y is oneto on it and only if yeY there exists at most one Ex suin that text y Caxis getit Y Examples 1 Idx x x x foranyset is insane 2 0 IR fro I Definition A function from f Y is bijective if andonly if it is surjective andinjective Definition Two functions f y and n Y are equal if and only it gets gens xxy Desention A function y has a rightinverse if thereexists a function g v sun that cogs x is equal to Iax got Y Y isequal to Iay Y Y oeaiisntinman noustinwoe t.fi g.i red function is injective rightinverse ex 2 A function h Y with a rest inverse but no rightinverse É redfunction is surgertive of inverse 9 it with both left and right inversen redfunction is bijective up right inverses ᵗ Theorem Let Y be a function 1 has a rightinverse it and only it is injective 2 has a restinverse if and only it is surjective 3 f has a rightistinverse if and only if it is bijective Homset xY functions 4 ii i i Homset 2 11 functions has a binary operation name romposition thatin associative where Iax is the identity element not commutative btw tag got 2 E Homset xx is not a groupuncers Age.se Homsee x x pqm Ex X Homset I Ex 1.2 Home 1.2 1.2 It 2 Autre 1123 1 aberion