# Not isomorphic

• Oct 22nd 2012, 10:31 PM
jzellt
Not 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 23rd 2012, 12:06 AM
FernandoRevilla
Re: Not isomorphic
Quote:

Originally Posted by jzellt
Show that R is NOT isomorphic to R*. (R is real numbers and R* is the group of nonzero reals under multiplication)

Suppose that $f:\mathbb{R}\to \mathbb{R}^*$ is an isomorphism. Then, $f(0)=1$ and $f(a)=-1$ for some $a\in\mathbb{R}$ so that $f(2a)=f(a)^2=1$. Since $f$ is injective, $2a=0$. Thus, $a=0$ and as a consequence $1=f(0)=-1$ (contradiction).