Show that R is NOT isomorphic to R*. (R is real numbers and R* is the group of nonzero reals under multiplication)
Can someone show this. I can't even get started...
Thanks.
Suppose that $\displaystyle f:\mathbb{R}\to \mathbb{R}^*$ is an isomorphism. Then, $\displaystyle f(0)=1$ and $\displaystyle f(a)=-1$ for some $\displaystyle a\in\mathbb{R}$ so that $\displaystyle f(2a)=f(a)^2=1$. Since $\displaystyle f$ is injective, $\displaystyle 2a=0$. Thus, $\displaystyle a=0$ and as a consequence $\displaystyle 1=f(0)=-1$ (contradiction).