Show that $\theta(x)=x^2$ from $\mathbb{R}^*$, the nonzero real number under multiplication, is a homomorphism.

Question

Show that $\theta(x)=x^2$ from $\mathbb{R}^*$, the nonzero real number under multiplication, is a group homomorphism.

How to deal with this one?

Answer 1

$\textbf{Hint}$: You must show that following for $\mathbb{R^*}=\mathbb{R}-\{0\}$:

• $\theta(1_{\mathbb{R^*}})=1_{\mathbb{R^*}}$

• $\theta(ab)=\theta{(a)}\theta{(b)}$ for all $a,b\in \mathbb{R^*}$

First one is obvious.

Second one just use commutativity of $\mathbb{R^*}$.

$\theta(ab)=(ab)^2=(ab)(ab)=a(ba)b=a(ab)b=(aa)(bb)=a^2b^2=\theta{(a)}\theta{(b)}$