Question

e) CX/ZX = 0% f) CX/R* = Si

You have to establish the following isomoorphism by showing that the mapping/function its a homomorphism and then use the first isomoorphism theorem.

Note that for part e) Z* is {1, -1}, or u2, as G* is set of elements with a multiplicative inverse in G under multiplication so Z* is {1, -1} and for C* it is all complex numbers in C except zero under multiplication, and for R* it is everything in R under multiplication except zero.

S1 in part f is the unit circle in the complex plane under multiplication.

Hope this clears up any confusion!

Answer 1

e Define 9: Ct c* given by a Creio) reize reio € C² where & ERT &>0 & oso = 250 Clearly, a(x) = {1xl for for xER* for nFeist E- i 250 x=e iO2 = rg, og e ì 20, i202 =r, e for Also, for 2,2 E CX, let 2, 8, eio, &z=re Then, 9/2, z) = Glo, eio, & eiO2) = 9600 pilo, to,)) i 260,702) re E o Grcio) pre ) = 9(2.) (22) Also, 9(z) = 1.3 for a=reio re:20 = 1 & e: 20 = 60 250k O=0, 25 ice. kerce = { 1, -1} = 2%

اک2 Also, of is suspective as for any, we c* if w=aeir, then 9(40) = xey=w. Q is suojective homomorphism with kera = zx = c*/ zx 7 cx f Let I: C* s reio leie²= : 20 For 2, 2 E C² st, 2 =8, e io, & 2 = 8eid t(2, 2) = (8,8 2110, +0])= Zilo, t0) = (2,)t(z) Also, Iloeist) = t (rei250) = 1 T(-8)==(8) = 1 V rE R* &>0 Also, if a(z) = 1 store the iO) = 20 = 1 0 = t 25 ZER* i.kert = IR* Also, for wes' then thee id=w :. I is surjective homomorphism with beeonel R. in C/RX ES let w = eia