If $g:\mathbb{Z_{10}}\rightarrow U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$.

Question

Why is that if $g:\mathbb{Z_{10}}$$\rightarrow$$U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$?

Also, $g$ is a function, $\mathbb{Z_{10}}$ is the group of integers modulo $10$, and $U_{20}$ is the group of units modulo $20$.

Answer 1

Hint: $\operatorname{Im}(g)$ is a subgroup of $U_{20}$ and $1$ (as an element of $\Bbb Z_{10}$) generates $\Bbb Z_{10}$.

Answer 2

In fact, $U(20)=\mathbb Z_4\oplus \mathbb Z_2$. So $\operatorname{Im}g=\langle g(1)\rangle $ has order dividing $10$ and $4$.