Question

Abstract Algebra

1. Decide if each operation not, give a condition that is not met on the given set is a group. If so, verify it. If

Answer 1

given sel is and the operation (a,b) Cc,d) sde ined as, (adtbe, bd) Glosure (a,b), (c,d) e d( )) e R ] 9703 Let 80, byo and (adsbe, bd) ettoa) eR?/ dfo? Novo (a,b) d) Rc nce, btodto bd o g0, 1t holds closune Asoctakienty Let 6.), ced) e, t) t) eRo. (a,) ea) e,) - La,) (ea ctt de, db) then, RR adtt bet bde, bd) adabe)bd .e , bd-) Ladt be, bd) (e, 1) -la.b).ed))e) It holds assoctativit 80, Identity extsh Let, to,) e ) eR 0. Let, a,b) e ) eRl (a,b) (o,) and, (o,1) (a,b) (0.b+ 2.a, J. b) 2o, (o,) a,b) (a.1+ b.0, b1) 9,b) dentity elemen s the

existance o iese tet, ta.b),(C.d) e tea)e R| t. 2uppore (c,d) inverse o (a,b) be ladt be , bd) t0,1) ad+be o and bd 1 a nd a. be o be= b Sioee, bto 0 and 80 Now a b t L0,1) s the invense b a,b). The given set undem So peraton la, b) (e,d) (adtbe, bd) 17

Let G ) R to3. u nder the operakion closume Let la,b)d) e G hen, a, b)Le4) (ac, be+d).e G (9,b) Le,d)(ac, be+d) as, at o,cto a0fo) So, lt holds closume Ptadlaed Let la, b)cerd), le,3) ea then Ca,b) (Led) (e) (a,b) (ee, det3 Ataacsatisty (ace, bee t de + 9) (ac.e,(betd)e+) Cac, berd)(e,t) (Ca,b) rd))+ cet) Se, 1t bolds assocrahine propent Identty exist tet, u,) tG (a,b) *C1,) =(a,b) ard, Ci,o) La, b) - (a, b) erall Cab) rdantiy then, A S0, C1,0) s the be the en resse o (9,b) ,4en (erd) (a,b) Le,d) = 1, (ac, bet d) So, ac=1and be + do Let 1,0) + b (1,0) h So and, C a,)(1,0) is the inmense o a,b)ea 8o C 3a groeup under givan operafo oo

Novw, Let R2 unden the openatoon La, b) d) - ladt be, bd) elosume Let (a, b),ce, d) e R2 then, la,b)e,d) = (adt be, bd) e R2 14 holds closure RAative9 pant we shon thaf- It 's associative tdeutily et Co,) e R then, la,b)Lo, ) (a.1bO , b1) la,b) and, (o,1)(a, b) La,b) dentity element Yorall aib)tR the Inverse: tet + b = l0,1) a, b) - (o,1) and, ( INT imede o a,b) s the Hence, adbe, bd) Carb) (crd)