# Using first isomorphism theorem to determine existence of onto homomorphisms

I'm trying to determine the existence of an onto group homomorphism $\alpha$ : $\Bbb{R^*}$ $\rightarrow$ $C_2$. By the first isomorphism theorem I know I have to find the $ker(\alpha)$ and create a factor group which is isomorphic to $C_2$ It seems to me that the real numbers will be pretty hard to break down into two elements but I'm having trouble expressing that. Also, if it was $\mathbb{Z}$ instead of $\mathbb{R}$ how would things be different? Any help is appreciated.

• Wouldn't $\alpha = \text{sgn}$ work, where $\text{sgn}$ is the sign function? – eepperly16 Apr 25 '16 at 19:31
• Yes I think that would work actually – ybce Apr 25 '16 at 19:34
• For $\mathbb{Z}$, I assume you mean the group of additive integers. The canonical homomorphism from $\mathbb{Z}$ to $\mathbb{Z}/2\mathbb{Z}$ should do. – eepperly16 Apr 25 '16 at 19:36