Homomorphisms and Isomorphisms · Challenge problem

Both are uncountable, so cardinality arguments about their size do not distinguish them.

$\mathbb C^\times$ is not abelian.

$x^4=1$ has 4 solutions in $\mathbb C^\times$ but only 2 in $\mathbb R^\times$.

$\mathbb R^\times$ has no element of order 2.