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.

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- Oct 22nd 2012, 09:31 PMjzelltNot isomorphic
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. - Oct 22nd 2012, 11:06 PMFernandoRevillaRe: Not isomorphic
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).