group theory?

Question

is C*xC* isomorphic to C* if yes prove it

ok so i started with
say just using & as a variable say phi or alpha
so &: C*xC*----->C8
then &: (a+bi,c+di)------->((a+c)+(b+d)i)
and i know i have to prove that it is one-one and onto and preserves operation

to prove one-one have to prove
that if &( a+bi,c+di)= &(e+fi,g+hi)
then (a+bi,c+di)=(e+fi,g+hi)

am i on the right track or totally wrong?

the C8 should be C*

Answer 1

Frankly, I don't think it is, for pretty much the same reasons as a two-dimensional vector space can't be isomorphic to a one-dimensional one. I.e., it would be hard for the mapping from C -->CxC to be onto.

But at the moment no crisp proof has yet come to mind.