# Math Help - Why is this not an isomorphism?

1. ## Why is this not an isomorphism?

$\gamma : G \rightarrow G$ where $G = (\mathbb{R},\times)$
$\gamma(x) = 3x$

It is trivially a bijection and $\gamma(x_1x_2) = \gamma(x_1)\gamma(x_2)$ seems to hold
as $\gamma(x_1 + x_2) = 3(x_1 + x_2) = 3(x_1) + 3(x_2) = \gamma(x_1) +\gamma(x_2)$

2. $3x_{1}x_{2}=\gamma(x_{1}x_{2})\not=\gamma(x_{1})\g amma(x_{2})=(3x_{1})(3x_{2})=9x_{1}x_{2}.$

3. sorry i meant the group was $(\mathbb{R}, +)$

4. In that case, I'd say the last line of the OP proves that it is an isomorphism.